<r I. A^ REPORT OF TUE FORTY-THIRD MEETING 0¥ THE BRITISH ASSOCIATION FOE TUB ADVANCEMUNT OF SCIENCE; HELD AT BRADFORD IN SEPTEMBER 1873. LONDON: JOHN MURRAY, ALBEMARLE STREET. 1874. {Office of the Association : 22 AtDraivRtT; BrREET, London, W.] PniSTED BY TAYI.OB AND FliAKCIS, EKD LION COUIIT, ri.r.KT STKF.KT. ALERE T FLAMMAM. CONTENTS. *>./> '^y^L/^/^ rvyx/N/xy Page Objects and Rules of the Association xvii Places of Meeting and Officers from commencement xxiv Presidents and Secretaries of the Sections of the Association from commencement xxx Evening Lectures xl Lectures to the Operative Classes xlii Treasiu'er's Account xliii Table showing the Attendance and Eeceipts at previous Meetings . . xliv Officers of Sectional Committees xlvi Officers and Council, 1873-74 xlvii Report of the Council to the General Committee xlviii Recommendations of the General Committee for Additional Reports and Researches in Science liii Synopsis of Money Grants Ix Place of Meeting in 1875 ; .' Ixi General Statement of Sums paid on account of Grants for Scientific Purposes Ixii Arrangement of the General Meetings Ixix Address by the President, Prof. A. W. Williamson, Ph.D., F.R.S. . . Ixx REPORTS OF RESEARCHES IN SCIENCE. Report of the Committee, consisting of Professor Caylet, F.R.S. , Pro- fessor Stokes, F.R.S., Professor Sir W. Teojison, F.R.S., Professor H. J. S. Smith, F.R.S., and J. W. L. Glaishee, B.A., F.R.A.S. (Reporter), on Mathematical Tables 1 a 2 JV CONTENTS. Page Observations on the Application of Machinery to the Catting of Coal in Mines. By William Firth, of Birley Wood, Leeds 175 Concluding Eeport on the M^tese Fossil Elephants. By A. Leixh Adams, M.B., F.E.S., F.G.S 185 Eeport of the Committee, consisting of Professor Ramsay, Professor Geikie, Professor J. Young, Professor Nicol, Dr. Bktce, Dr. Aexhur Mitchell, Professor Hull, Sir E. Griffith, Bart., Dr. Kino, Pro- fessor Hakkness, Mr. Peestwich, Mr. Hughes, Eev. H. W. Crosskey, Mr. W. Jolly, Mr. D. Milne-Home, and Mr, Peng ell y, appointed for the purpose of ascertaining the existence in different parts of the United Kingdom of any Erratic Blocks or Boulders, of indicatingon Maps their position and height above the sea, as also of ascertaining the nature of the rocks composing these blocks, their size, shape, and other particulars of interest, and of endeavouring to prevent the destruction of such blocks as in the opinion of the Committee are ■worthy of being preserved. Drawn up by the Eev. H. W. Crosskey, Secretary 188 Fourth Eeport on Earthquakes in, Scotland, drawn up by Dr. Bryce, F.G.S. The Committee consists of Dr. Bbyce, F.G.S. , Sir W. Thom- son, F.E.S., Geo, Forbes, F.E.S.E., and Mr. J. Brough 194 Ninth Eeport of the Committee for Exploring Kent's Cavern, Devon- shire, the Committee consisting of Sir Charles Lyell, Bart., F.E.S,, Professor Phillips, F.E.S., Sir John Lubbock, Bart., F.E.S., John Evans, F.E.S., Edward Vivian, M.xV., George Busk, F.E.S., William Boyd Dawkins, F.E.S., William Ayshford Sanford, F.G.S., and William Pengelly, F.E.S. (Eeporter) 198 The Flint and Chert Implements found in Kent's Cavern, Torquaj', Devonshire. By W. Pengelly, F.E.S., F.G.S 209 Eeport of the Committee, consisting of Dr. Gladstone, Dr. C. E. A. Wright, and W. Chandler Egberts, appointed for the purpose of investigating the Chemical Constitution and Optical Properties of Essential Oils. Drawn up by Dr. Wright 214 Eeport of the Committee, consisting of W. Chandler Eoberts, Dr, Mills, Dr. Boycott, and A. W. Gadesden, appointed for the purpose • of inquiring into the Method of making Gold-assays, and of stating the Eesults thereof. Drawn up by W. Chandler Egberts, Secretary 219 First Eeport of the Committee for the Selection and Nomenclature of Dynamical and Electrical Units, the Committee consisting of Sir W. Thomson, Professor G. C. Foster, Professor J. C. Maxwell, Mr. G. J. Stone Yj Professor Fleeming Jenkin, Dr. Siemens, Mr. F. J, Bram- WELL, and Professor Everett (Eeporter) 222 Eeport of the Committee, consisting of Professor Phillips, LL.D., F.E.S., Pi-ofessor Harkness, F.E.S., Henry Woodward, F.E.S., James Thom- son, John Brigg, and L. C. Miall, on the Labyrinthodonts of the Coal-measures. Drawn up by L. C. Miall, Secretary to the Com- mittee 225 Eeport of the Committee appointed to construct and print Catalogues . of Spectral Eays arranged upon a scale of Wave-numbers, the Cora- CONTENTS. V niittee coDsistiug of Dr. HxiGGiKe, J. N. Lcckyee, Professor Reynolds, Professor Swan, and G. Johnstone STONE^(Eeporter) 249 llcport of tte Committee, consisting of Sir John Lubbock, Bart., Pro- fessor Phillips, Professor Hughes, and "VV. Botb Dawkins, Secretary, appointed for the purpose of exploring the Settle Caves. Drawn up by Mr. Botd Dawkins 250 . Sixth Eeport of the Committee, consisting of Prof. Eveeett, Sir "W. Thomson, F.R.S., Sir Chaeles Ltell, Bart, F.R.S., Prof. J. Cleek Maxwell, F.E.S., Prof. Phillips, F.E.S., G. J. Symons, F.M.S., Prof. Ramsay, F.R.S., Prof. A. Geikie, F.R.S., James Glaishee, F.R.S., Rev. Dr. Geaham, Geoege Maw, F.G.S., W. Pengelly. F.R.S., S. J. Mackie, F.G.S., Prof. Hull, F.R.S., Prof. Ansted, F.R.S., and J. Peestwich, F.R.S., appointed for the purpose of investigating the Rate of Increase of Underground Temperature downwards in various Localities of Dry Land and under Water. Drawn up by Prof. Eveeett, D.C.L., Secretary 252 Report on the Rainfall of the British Isles for the years 1872-73, by a Committee, consisting of C. Beooke, F.H.S. (Chairman), J. Glaishee, F.R.S., Prof. J. Phillips, F.R.S., J. F. Bateman, C.E., F.R.S., R. W. Mylne, C.E., F.R.S., T. Hawksley, C.E., Prof. J. C. Adams, F.R.S., Prof. J. J. Sylvestee, F.R.S., C. Tomlinson, F.R.S., R. Field, C.E., Dr. Pole, C.E., F.R.S., Prof. D. T. Ansted, F.R.S., A. Buchan, F.R.S.E., G. J. Symons, Secretary. Drawn up by G. J. Symons .... 257 Seventh Report of the Committee appointed for the purpose of continuing Researches in Fossil Crustacea, consisting of Professor P. Maetin Duncan (M.B. Lond.), F.R.S., Heney Woodwaed, F.R.S., and Robeet Etheeidge, F.R.S. Drawn up by Heney "Woodavaed, F.R.S 304 Report on Recent Progress in Elliptic and Hyperelliptic Functions. By W. H. L. Russell, F.R.S 307 Report of the Committee, consisting of the Rev. H. F. Baenes, H. E. Dressee (Secretary), T. Haeland, J. E. Hahting, T. J. Monk, Pro- fessor Newton, and the Rev. Canon Teisteam, appointed for the purpose of continuing the investigation on the desirability of establishing a " Close Time " for the preservation of indigenous animals 346 Report of the Committee, consisting of James Glaishee, F.R.S., of the Royal Observatory, Greenwich, Robeet P. Geeg, F.G.S., and Alex- andee S. Heeschel, F.R.A.S., on Observations of Luminous Meteors, 1872-73; drawn up by Alexandee S. Heeschel, F.R.A.S 349 On the Visibility of the dark side of Venus. By Professor A. Schafaeik, of Prague 404 Report of the Committee, consisting of Dr. Rolleston, Dr. Sclatee, Dr. Anton Dohen, Professor Huxley, Professor Wyville Thomson, and E. Ray Lankestee, for the foundation of Zoological Stations in dif- ferent parts of the Globe. Drawn up by Anton Dohen, Secretary 408 Second Report of the Committee, consisting of Professor Haekness, William Jolly, and Dr. James Beyce, appointed for the purpose of collecting Fossils from localities of difficult access in North-western Scotland. Drawn up by William Jolly, Secretary 412 Vl CONTENTS. Page rifth Report of the Committee on the Treatment and Utilization of Sewage, consisting of Eichakd B. GRANtnAM, C.E., F.G.S. (Chair- man), F. J. Bramwell, C.E., F.B.S., Professor W. H. Corfielb, M.A., M.D. (Oxon.), J. Bailey De.\ton, C.E., F.G.S., J. H. Gilbert, Ph.D., F.R.S., F.C.S., W. Hope, V.C, Professor A. W. Williamson, Ph.D., F.R.S., F.C.S., and Professor J. T. Wat 413 Report of the Committee for superintending the Monthly Reports of the Progress of Chemistry, consisting of Professor A. W. Williamsox^, F.R.S., Professor Fbankland, F.R.S., and Professor Roscoe, F.R.S. 451 On the Bradford Waterworks. By Charles Gott, M.Inst.C.E 451 Report of the Committee appointed to consider the possibility of Improving the Methods of Instruction in Elementary Geometry, the Committee consisting of Professor Syltesteb, Professor Cayley, Pro- fessor Hirst, Rev. Professor Bartholomew Price, Professor H. J. S. Smith, Dr. Spottiswoode, Mr. R. B. Hayavard, Dr. Salmon, Rev. R. TowNSENH, Professor Fuller, Professor Kelland, Mr. J. M. Wilson, and Professor Clifford (Secretary) 459 Interim Report of the Committee appointed for the purpose of making Experiments on Instruments for Measuring the Speed of Ships, &c. 460 Report of the Committee, consisting of Dr. Crum Brown, Mr. J. Deavar, Dr. Gladstone, Prof. A. W. Williamson, Sir AV. Thomson, and Prof. Tait, appointed for the purpose of Determinating High Temperatures by means of the Refraugibihty of the Light evolved by Fluid or Solid Substances. Drawn up by James Dewar, Reporter 461 On a Periodicity of Cyclones and Rainfall in connexion with the Sun- spot Periodicity. By Charles Meldrum 460 Fifth Report of the Committee appointed to investigate tlie Structure of Carboniferous-Limestone Corals. Drawn up by James Thomson, Secretary. The Committee consists of Professor Harkness, F.R.S., James Thomson, F.G.S., Dr. Duncan, F.R.S., and Thomas Davidson, F.R.S 479 Report of the Committee, consisting of Colonel Lane Fox, Dr. Beddoe, Mr. Franks, Mr. Francis Gaiton, Mr. E. W. Brabrook, Sir J. Lub- bock, Bart., Sir Walter Elliot, Mr. Clements R. Markham, and Mr. E. B. Tylor, appointed for the purpose of preparing and publishing brief forms of Instructions for Travellers, Ethnologists, and other Anthropological Observers. Drawn up by Colonel A. H. Lane Fox . 482 Preliminary Note from the Committee, consisting of Professor Balfour, Convener, Dr. Cleghorn, Mr, Robert Hutchison, Mr. Alexander BucHAN, and Mr. John Sadler, on the Influence of Forests on the Rainfall 488 Report of Sub-Wealden Exploration Committee, appointed at the Brighton Meeting, 1872, consisting of Henry Willett, R. A. Godwin- AusiEN, F.R.S., W. TopLEY, F.G.S., T. Davidson, F.R.S.. J. Pbest- wicH, F.R.S., W. Boyd Dawkins, F.R.S., and 'Henry Woodward, F.R.S, Drawn up by Henry Willett and W. Topley 490 C'UNTKNTS. Vii llcport of the Committoe, consisting of Mr. Er.vncis Galton, Mr. W. Feotjde, Mr. C. W. Merrtfield, and Professor EAJfKiNE, appointed to consider and Eeport on Machinery for obtaining a Record of the llonghness of the Sea and Measurement of Waves near shore 495 Report of the Committee on Science-Lectures and Organization, — the Committee consisting of Prof. Roscoe, F.R.S. (Secretary), Prof. W. G. Adams, P.R.S., Prof. Ajjdrews, F.R.S., Prof. Balfour, F.R.S., F. J. Rkamwell, F.R.S., Prof. A. Crum Broww, F.R.S.E., Prof. T. Dter, Sir \Yalxer Elliot, F.L.S., Prof. Flower, F.R.S., Prof. G. C. Foster, F.R.S., Prof. Geikie, F.R.S., Rev. R. Harley, F.R.S., Prof. Huxlet, F.R.S., Prof. Fleehing Jenein, F.R.S., Dr. Joule. F.R.S., Col. Lane Fo^, F.G.S., Dr. Lankester, F.R.S., J. K. Lockyer, F.R.S., Dr. O'Callaghan, LL.D., D.C.L., Prof. Ramsay, F.R.S., Prof. Balfour Stewart, F.R.S., H. T. Stainton, F.R.S., Prof. Tait, F.R.S.E., J. A. TiNNE, F.R.G.S., Dr. Allen Thomson, F.R.S., Sir William Thomson, F.R.S., Prof. Wyville Thomson, F.R.S., Prof. Turner, F.R.S.E., Prof. A. W. Williamson, F.R.S., and Dr. Young 495 Second Report of the Committee on Science-Lectnres and Organization, — the Committee consisting of Prof. Roscoe, F.R.S. (Secretary), Prof! W. G. Adams, F.R.S., Prof. Andreavs, F.R.S., Prof. Balfour, F.R.S., J. Baxendell, F.R.A.S., F. J. Bram-^vell, F.R.S., Prof. A. Crum Brown, F.R.S.E., Mr. T. Buchan, Dr. Carpenter. F.R.S., Prof. Core, Warren De La Rue, F.R.S., Prof. T. Dyer, Sir Walter Elliot, F.L.S., Prof. M. Foster, F.R.S., Prof. Flower, F.R.S., Prof. G. C. Foster, F.R.S., Prof. Geikie, F.R.S., Dr. J. H. Gladstone, F.R.S., Mr. Griffith, Rev. R. Harley, F.R.S., Dr. Hirst, F.R.S., Dr. Hooker, F.R.S., Dr. Huggins, F.R.S., Prof. Huxley, F.R.S., Prof. Fleeming Jenkin, F.R.S., Dr. Joule, F.R.S., Col. A. Lane Fox, F.G.S., Dr. Lankester, F.R.S., J. K. Lockyer, F.R.S., Prof. Clerk Maxwell, F.R.S., D. Milne-Home, F.R.S.E., Dr. O'Callaghan, LL.D., D.C.L., Dr. Odling, F.R.S., Prof. Ramsay, F.R.S., W. Spottiswoode, F.R.S., Prof. Balfour Stewart, F.R.S., H. T. Siainton, F.R.S., Prof. Tait, F.R;S.E., J. A. Tinn£, F.R.G.S., Dr. Allen Thomson, F.R.S., Sir William Thomson, F.R.S., Prof. Wyville Thomson, F.R.S., Prof. Turner, F.R.S.E., Col. Strange, F.R.S., Prof. A. W. Williamson, r.R.S., G. V. Vernon, F.R.A.S., and Dr. Young 507' Vlll " CONTKNTS. NOTICES AND ABSTEACTS OP MISCELLANEOUS COMMUNICATIONS TO THE SECTIONS. MATHEMATICS AND PHYSICS. Pago Address by Professor H. J. S. Smith, M.A., F.R.S., President of the Section 1 Mathehatics. Professor Cayley on tlie Mercator's Projection of a Surface of Eevolution . . 9 Professor W. K. Clitfobd on some Curves of the Fifth Class 9 '■ on a Surface of Zero Curvature and Finite Extent 9 Mr. J. W. L. Glaisiier on certain Propositions in the Theory of Numbers deduced from Elliptic- transcendent Identities 10 on the Negative Minima of the Gamma fimction . , 13 on the Introduction of the Decimal Point into Arithmetic 13 Mr. G. O. Hanlon on the Formation of an extended Table of Logarithms . . 17 The Rev. Robert Haeley on the Theory of Differential Resolvents 17 on Professor Evans's Method of solving Cubic and other Trinomial Equations 22 M. Ch. HEEMiTESur ITrrationalite de la Base desLogarithmesHyperbolit^ues 22 Professor Henby J. Stephen Smith on Modular Equations 24 Mr. W. Spottisw'oode on Triple Tangent Planes 24 The Rev. Henhy Wage on the Calculation of Logarithms 24 Mechanics akd Physics. Dr. Robert Stawell Ball on a Geometrical Solution of the following problem: — A quiescent rigid body possessing three degrees of freedom receives an impulse ; determine the instantaneous screw about which the body commences to twist 26 on the Theory of Screws 27 Professor J. D. Everett on the Kinematics of a Rigid Body 28 CONTENTS, IX Page Piufesisor G. Fobbes on certaiu couuexions between the Molecular Properties of Metals 29 Professor J. Clerk Maxwell on the Final State of a System of Molecules in Motion subject to Forces of any kind '. •. 29 Mr. John NE^^LLE on the Axis of least Moments in a Rectangular Beam . . 32 Professor Osborne Reynolds on certain Phenomena of Impact 32 Professor Balfour Stewart on iEthereal Friction 32 . . ASTHOKOMT. Mr. W. R. BiRT on the Importance and Necessity of continued Systematic Observations on the Moon s Surface 34 Dr. William Huggins on the Proper Motions of Nebulse 34 M. Janssen on the Application of Photography to show the Passage of Venus across the Sun's Disk So Mr. J. Norman Lockyer on the Results of some recent Solar Investigations 35 Professor A. Schafarik on the Visibility of the Dark side of the Planet 'S'enus So Light. Mr. Philip Braham on Light with circularly ruled plates of Glass 36 Mr. W. S. Davis on some Abnormal Effects of Binocular Vision 30 Professor J. D. Eatsbett on a Refraction-Spectrum without a Prism 37 Professor G. Forbes on Irradiation 38 Dr. Gladstone on Photographs of Fluorescent Substances 38 Mr. J. Norman Lockyer on the Dresser-Rutherford Diffraction-grating £8 Professor Clerk Maxwell on the Relation of Geometrical Optics to other Branches of Mathematics and Physics 38 Lord Rayleigh on a Natural Limit to the Shai-pness of the Spectral Lines. . 39 Mr. Arthur Schuster on the Influence of Temperature and Pressure on the Widening of the Lines in the Spectra of Gases 39 . on a curious Phenomenon observed on the top of Snowdon ■ 40 Heat. Professor G. Forbes on Thermal Conductivity 40 Professor A. S. Hehschel on the Thermal Conductivities of certain Rocks. . 40 Professor Zengeb on the Correlation between Specific Weight and Specific Heat of Chemical Elements 40 Electeicitx xnn Magketism. Mr. W. F. Barrett on the Molecular Changes that accompany the Magneti- zation of Iron, Nickel, and Cobalt 40 on the Relationship of the Magnetic Metals, Iron, Nickel, and Cobalt 40 Professor Ch. V. Zenger on Symmetric Conductors, and the constmction of Lightning-conductors 41 CONTENTS. MliTEOBOLOGr &C. Page Dr. William B. Carpenteh on the Undercurrents of the Bosphorus and Dardanelles 41 IMr. W. S. Davis on the Refraction of Liquid Waves 43 Mr. J. Park Harimson on Lunar Influence on Clouds and Rain 43 M. AsTUUo DE Marcoartu on the Application of Telegraphy to Navigation and Meteorology 43 Mr. C. Meldrum on a Periodicity of Cyclones and Rainfall in connexion with the Sun-spot Periodicity 43 Mr. S. B. J. Skertchly on Experiments on Evaporation and Temperaturo made at Wisbeach 44 3Ir. G. M. Whipple on the Passage of Squalls across the British Isles 44 Iksteuments. Dr. Robert Stawell Ball on Dynamometers in Absolute Measure 44 Captain J. E. Davis on an Improvement in the Sextant 44 Mr. A. E. DoNKiN on an Instrument for the Composition of two Harmonic Curves 4,5 Mr. Rogers Field on an Improved Form of Aneroid for determining Heights, with a means of adjusting the Altitude-scale for various Temperatures 40 Mr. G. W. Hope on Eckhold's Omnimeter, a new Surveying-Instrument 47 Mr. G. J. Symons on Negretti and Zambra's Test-gauge Solar-Radiation Thermometer 47 Mr. S. C. TisLEY on a Compound-Pendulum Apparatus 48 Professor A. S. IIerschel on a new form of Pendulum for exhibiting Super- posed Vibrations 43 Mr. F. _H. Wenham on the Influence of Temperature on the Elastic Force of certain forms of Springs 49 Mr. G. M. Whipple on a New Form of Rutherford's Minimum Thermometer, devised and constructed by Mr. James Hicks : . . . 50 on a New Electrical Anemograph 50 Mr. C. J. WooBWARD on an improved form of Oxyhydrogeu Lantern for the use of Lectui-ers 50 CHEMISTRY. Address lay W. J. Russell, Ph.D., F.R.S., President of the Section 52 Mr. Alfred H. Allen on the Detection of Adulteration of Tea 62 Dr. Henry E. Armstrong on Alpha- and Beta-Naphthylic Sulphide 62 ] on the Action of Sulphuric acid on Ethylaniline and Dimethylaniline q.~> on Cresol Derivatives (33 Professor Dr. Crum Brown on the Action of Sulphide of Methyl on Brom- acetic Acid po Dr. J. II. Gladstone on Black Deposits of Metals go Mr. A. Vernon Harcourt and F. W. Fison on a Continuous Process for Purifying Coal-gas and obtaining Sulphur and Ammonium Sulphate G4 CONTENTS. XI Page Mr. Charlks Horner on the Spectra of certain Boric aud Phosphoric Acid Blowpipe Beads 64 Mr. J. Norman Lockyer on the Elements in the Sun 65 Mr. W. T. M'^GowEN on the Sewage of Manufacturing Towns 65 Dr. Paul and Mr. A. D. Cownley on the Valuation of Commercial Crude Anthracene 05 Mr. W. n. Pike on several Homologues of Oxaluric Acid 65 Mr. W. Chandler Roberts on Horn Silver 66 Professor Schafarik on the Constitution of some Silicates 66 Mr. John Spiller on Artificial Magnetite 66 Mr. C. J. Woodward on a Form of Gas-generator 66 Mr. C. R. A. Wright on new Derivatives from Codeine and Morphine .... 67 GEOLOGY. Address by John Phillips, M.A., D.C.L. Oxon., LL.D. Cambridge and Dubliu, r.R.S., F.G.S 70 The Rev. J. F, Blake on additional Remains of Pleistocene Mammals in Yorkshire 75 Mr. W. T. Blanford on some Evidence of Glacial Action in Tropical India in Palseozoic (or the oldest Mesozoic) times 76 Mr. Henry B. Brady on Arclicediscus Karreri, a New Type of Carboniferous Foraminifera ' 76 Mr. John Brigg on such of the Industries of Bradford as relate to its Geolo- gical Position 76 Mr. A. Champernowne on the Discovery of a Species of Starlish in Devonian ; Beds of South Devon ; with a Note by Henry Woodward 77 Mr. J. R. Dakyns on the Geology of part of Craven 78 Mr. W. Boyd Dawkins on the Rate at which Stalagmite is being accumulated in the Ingleborough Cave 80 Mr. J. W. Ellis on the Stump-Cross Caverns at Greenhow near Patety Bridge 80 Mr. W. GoMERSALL on the Round Boulder Hills of Craven 80 The Rev. J. Gunn on the Probability of finding Coal in the Eastern Counties 81 Professor Harkness on the Occurrence of Faults in the Permian Rocks of the lower portion of the Vale of the Eden, Cumberland 81 Mr. Henry Hicks on the Arenig and Llandeilo Rocks of St. David's 82 Mr. John Hopkinson on some Graptolites from the Upper Arenig Rocks of Ramsey Island, St. David's 82 on the Occurrence of numerous Species of Graptolites in the Ludlow Rocks of Shropshire 83 Mr. _W. HoRNE on the Occurrence in the Yoredale Rocks of Wensleydale of Fish and Amphibian Remains 84 Mr. J. Logan Lobley on the British Palasozoic Ai'cadas 84 Dr. T. Moffat on a Horn and Bones found in a Cutting in a Street in Maiden- head, Berks 84 ' on Geological Systems and Endemic Diseases 84 Dr. John Phillips on the Ammonitic Spiral in reference to the power of Flotation attributed to the Animal 85 XU CONTKNTS, Page Dr. John Phillips on the Ammonitic Septa in relation to Geological Time . 86 Barou von RiCHTHorEN on the Loess of Northern China, and its Eelation to the Salt-basins of Central Asia 86 Mr. li. Russell on the Geology of the Country round Bradford, Yorkshire. . 88 Mr. J. E. Taylor on the Occurrence of Elephant-remains in tlio Basement Beds of the Red Crag 91 Mr. W. ToPLEY on the Correspondence between some Areas of apparent Up- heaval and the Thickening of subjacent Beds 91 and Ml-. G. A. Leboub on the Whin Sill of Northumberland 92 Mr. W. AVhitaker on the Occurrence of Thanet Sand and of Crag in the S.W. part of Sufiblk (Sudbury) 92 Mr. Henry Wood-ward and Mr. Robert Etheridge, jun., on some Speci- mens oi Dithi/7-ocar is from the Carboniferous Limestone Series, East Kilbride, and from the Old Red Sandstone (?) of Lanarkshire; with Notes on their Geological Position &c 92 ■ on new Facts bearing on the Inquiry concerning Forms intermediate between Birds and Reptiles 93 BIOLOGY. Address by George J. Allman, M.D., LL.D., F.R.S., F.R.S.E., M.R.I.A., F.L.S., President of the Section 94 Botany. Mr. W. Archer on Parasitic Algfe 10-4 Mr. T. Baines on a Tree-Aloe from South-East Africa 104 Professor Thiselton Dyer on the Plants collected in Bermuda by Mr. II. N. Moseley 104 Professor Gulliver on the Crystals in the Testa and Pericaqi of certain Plants 104 Mr. Charles P. Hobkirk on the Mosses of the West Riding of Yorkshire. . 104 Dr. J. D. PIooKKR on the Subalpine Vegetation of Kilimanjaro, E. Africa. . . . 105 Professor Lawson on Plants collected by the Voyager Dampier 105 on a Course of Practical Instruction in Botany 105 Mr. II. N. Moseley on the Vegetation of Bermuda 105 Mr. John Shaw on some of the Changes going on in the South-African Vege- tation through the Introduction of the JNIerino Sheep 105 Professor W. C. Williamson on Fern-stems and Petioles of the Coal-measures 106 Dr. Willis on the Flora of the Environs of Bradford 106 Zoology. Professor Allman on some Recent Results with the Towing-net on the South Coast of Ireland 106 Mr. W. T. Blanford on the Distribution of the Antelopes in Southern and Western Asia 110 on the Fauna of Persia 110 Mr. J. Gwyn Jeffreys on the MoUusca of the Mediterranean Ill CONTENTS. ^11^ Pnge Mr. E. Ray Lankesteu on a Peach-coloured Bacterium 110 on the Genealogy of the MoUusca HG Mr. T. Lister on Birds observed in the West Riding of Yorkshire in former and recent years Mr. R. MacLachlan on a new Insect belonging to the Family Ephemend3> , _ with Notes on the Natural History of that Family - H'' Anatomy and Physiology. Professor Ruthehfoiid's Address to the Department of Anatomy and Phy- siology 1^"^ Mr. Alfred W. Bennett on the Movements of the Glands of Drosera .... 123 Dr. BiNZ on the Action of Alcohol on Warm-blooded Animals 124 Dr. Lauder Brunton on the Nature of Cholera 124 Mr. A. S. Davis on some Abnormal Effects of Binocular Vision 126 Dr. Dewar and Dr. MacKendrick on the Action of Light on the Retina and other Tissues 126 Professor P. Martin Duncan on the Motion of Protoplasm in the Fiisaceous Algfe 126 Dr. David Ferrier on the Localization of Function in the Brain 126 Dr. J. MiLNER FoTHERGiLL ou the Heart and Brain 127 Dr. Thomas R. Eraser on the Physiological Action of Crystalline Aconitia and pseudo- Aconitia , 128 Sir G. Duncan Gibb on the Vocal Organs in Living Centenarians 128 Dr. J. Goodman on White Corpuscles, their Nature and Origin in the Animal Organism '... 129 Dr. George Harley on the Mode of Formation of Renal Calculi 130 Mr. E. Ray Lankesteb on the Structure of the Egg, and the early Develop- ment of the Cephalopod Lolixjo 131 Dr. John Ross on Microzymes as partial Bionta 131 Dr. BuRDON Sanderson on Huizinga's Experiments on Abiogenesis 131 on the Electrical Phenomena which accompany the Contractions of tlie Leaf of Dionma muscipula 13.3 Professor 0. A. Struthers on the Diverticulum of the Small Intestine in Man, considered as a Rudimentary Structure 134 Mr. C. S. Tomes on the Development of the Armadillo's Teeth 134 Dr. ^loRRisoN Watson on the Anatomy and Phvsiology of the Indian Ele- phant ' 134 Anthropology. Dr. John Beddoe's Address to the Department of Anthropology 134 Note on the Iberians „ . . 140 Mr. A. W. Buckland on the Serpent in connexion with Primitive Metallurgy 140 Mr. 0. H. E. Carmichael on Professor Gennarelli's Paper " On the Exist- ence of a Race of Red Men in Northern Africa and Southern Europe in Prehistoric Times " , , , 141 XIV CONTENTS. Page Mr. Hyde Clarke on Prehistoric Names of Weapons 141 ■ on the Comparative Chronology of the Migrations of Man in America in relation to Comparative Philology 141 on the Ashautee and Fantee Languages 142 on the Report concerning Bushman researches of Dr. W". H. Bleek, Ph.D 142 Mr. W. Boyd Dawkins on the Northern llange of the Iberians in Europe . . 142 Mr, Robert Dunn on Ethnic Psychology 143 The Rev. W. Wyatt Gill on Ooral-Ca-\'es with Human Bones in Stalagmite . on Mangaia, South Pacific 144 Mr. J. Park Harrison on the Passage of Eastern Civilization across the Pacific 146 Dr. J. Sinclair Holden on a hitherto undescribed Neolithic Implement . . 14G Mr, J, Kaines on a true Cerebral Theoiy necessary to Anthropology 146 Mr. JcfSiN S. Phen£ on an Age of Colossi 147 Mr. F. W. RuDLER on Stone Implements from British Guiana 148 Mr. Edward B. Tylor on the Relation of Morality to Religion in the Early Stages of Civilization , , , . 148 GEOGEAPHY. Address by Sir Rutherford Alcock, K.O.B,, President of the Section .... l-OO Dr. Charles T. Beke on the true Position and Physical Characters of Mount Sinai * 161 Mr. W, T. Blanford on the Physical Geographj- of the Deserts of Persia and Central Asia 162 Dr. William B. Carpenter on the Physical Geography of the Mediterranean, considered in relation to that of the Black Sea and the Caspian 163 : on the Physical Geography of the Caspian Sea, in its relations to Geology 165 Sig-nor Guido Cora on the Equatorial Lakes of Africa 167 Mr. G. H, Darwin on a Portable Globe, and on some Maps of the World . . 167 Captain J. E. Davis on the Scientific Voyage of the ' Challenger' 167 Mr. Ney Elias on Trade-routes through Mongolia and Zungaria 169 The Rev, W, Wyatt Gill on Three Visits to New Guinea 169 Colonel Sir Frederic Goldsmid on recent Travel in Persia 171 Major Beresford Lovett on a Visit to Koh-Khodja 172 Dr, J. M'CosH on Assam, and an Overland Communication with China .... 1 72 Mr. Clements R. Markham on Recent Arctic Explorations 172 Captain J. Moresby on Discoveries at the Eastern End of New Guinea .... 172 Mr. E, Delmar Morgan on Russian Accounts of Khiva and Turcomania , , 172 ]Mr. E. L, Oxenham on a Journey from Peking to Han-kow 172 Baron von Riohthofen on the Distribution of Coal in China 17-3 CONTENTS. Xr Page faptain Rokeby ou the Survej^ for a Telegrapli-line between Berber and Soualdm ^'"^ Major St. John on Trade-routes in Persia 173 Major Evan Smith on tlie Livingstone East-Coast Aid Expedition 173 on the Trade of the East-Afiican Coast 173 Mr. J. Thomson on the Gorges and Rapids of the Upper Yangtsze 173 ECONOMIC SCIENCE and STATISTICS. Address by the Right Hon. W. E. Forsteh, M.P., President of the Section. . 174 Major-General Sir James Alexandeb on tlie Use and Abuse of Peat 183 Dr. C. E. Appleton on some of the Economical Aspects of Endowments of Education and Original Research 183 Mr. S. C. T. Babtley on the Poor-Law and its Effect on Thrift 185 Mr. J. Arthur Binns on Benefit Building Societies 185 Mr. "William Botly on Dwellings for the Industrial Classes 186 Mr. Hyde Clarke on the Influence of Large Centres of Population on Intel- lectual Manifestation 186 Mr. F. Hahn Danchell on Peat .^ 186 Mr. Frank P. Fellows's Statistics and Observations on the National Debt and our Disbursements from the Revolution in 1688 to the present time, showing the advisability of ascertaining our Annual Governmental Capital and Current Expenditm'e 186 Mr. J. G. Fitch on the Savings-Bauk in the School 187 Mr. Thomas IIaig on the East Morley and Bradford Savings-Bank 188 Mr. T. G. P. Hallett on the Income-Tax Question 188 Mr. James Hanson on Educational Statistics of Bradford 189 Mr. "W, ILiSTiNGS on Postal Reform 191 Mr. B, Haughton on Railways Amalgamated in Competing Groxips 191 Mr. W. D, Henderson on Commercial Panics 193 Mr. Samuel Jubb on the Shoddy Trade 194 Mrs. E. M. King on Confederated Homes and Cooperative Housekeeping. . . . 195 Professor Leone Levi on the Effect of the Increase of Prices of certain Neces- saries of Life on the Cost of Livings and its Relation to the Rates of Wages and Salaries 196 Mr. J. M. D. Meiklejohn on the Economic Use of Endowments 196 Mr. W. Morris on Capital and Labour 196 Mr. Archibald Neill on the Bradford Building Trades 196 Mr. R. H. Inglis Palgrave on the Relation of the Banking Reserve of the Bank of England to the Cm-rent Rate of Interest 199 Major-General Millington Synge on Puritv and Impurity in the Use and Abuse of Water .' ' 200 XVI CONTENTS. MECHANICAL SCIENCE. Page Address by W. H. Barlow, Escj., C.E., F.R.S., President of tlie Section .... 200 Mr. W. H. Barlow, Jun., on the Lisbon Steam Tramways, 1873 210 Mr. DANrEL Bateman on the Manufacture of Cards for Spinning Purposes . . 210 Mr. 0. Bergeron on the Saint-Gotthard Tunnel 210 The Rev. E. L. Bkrthon on the Hydrostatic Log 210 Mr. F. J. Bramwell on Huggett's System of Manufacturing Horse-nails .... 210 Dr. W. J. Clapp on the Nant-y-glo Coal-cutting Machine 213 Mr. Hyde Clarke on the Progress of the Through Railway to India 213 Mr. Samuel Davis ouBrain's System of Mining by means of Boring-machinery, Dynamite, and Electric Blasting 213 Mr. R. Eaton on further Results on the Working of Locomotives with Heated Air and Steam 213 Mr. C. Le Neve Foster on the " Duty " of Arrastres in reducing Gold Ore in Italy 214 Mr. P. Le Neve Foster, Jun., on the Irrigation of the Casale District .... 214 Mr. S. C. Lister on the Mechanical Treatment of Fibrous Substances 214 Mr. James R. Napier on Napier's Pressure Log 214 Mr. Archibald Neill on Stone-dressing in Bradford 214 Mr. W. E. Newton on the Sand-Blast Process for Cutting and Ornamenting Stone, Glass, and other Hard Substances 215 Mr. John Plant on the Burleigh Rock-driU 21(5 Prof. Osborne Reynolds on the Resistance of the Screw Propeller as aftected by Immersion 210 on the Friction of Shot as affected by diflerent kinds of Rifling 210 Mr. Robert Sutcliff'e on the Economical Generation of Steam 210 ^ on the Economical Utilization of Steam 217 Mr. W. Cave Thomas on the Centre-rail Railway 210 Mr. John "Waugh on the Prevention of Incrustation in Steam-Boilers 210 Mr. Thomas Webster on the Advancement of Science by Industrial Inven- tion 219 on the Assimilation of the Patent Systems of Great Britain and of the United States '. 219 Mr. John White on a Form of Channel Steamer 219 Mr. Joseph Willcock on the History, Progress, and Description of the Bowling Ironworks , 219 APPENDIX. Prof. A. S. Herschel and G. A. Lbbour on the Conducting-powers for Heat of certain Rocks, with Remarks on the Geological Aspects of the Investigation 223 I EEEATA IN REPOET FOE 1872. Omitted from Index I. Gaussian constants for the year 1829, report on the, or a theory of terrestrial magnetism founded on all available observations, 1. Mascarene Islands, second supplementary report on the extinct birds of the, by A. Newton, 23. Progress of chemistry, report of the Committee for superintending the monthly reports of the, 24. EEEATA IN THE PEESENT VOLUME. Ik the Repoets. Page 369, line 22 from bottom, for Duncan read Dunkin. 382, „ 8, after 11-09 insert per cent. 383, „ 4, for Biichner read Buchner. 384, „ 16, for Arnaud read Amand. 390, „ 23 from bottom, for Persii read Persei. 396, „ 13 from bottom, after Professor Baden Powell insert a nofa, thus t. 399, „ 20, for intrastellar read interstellar. In the footnote of the Table of " Numbers of Meteors seen&c. in August 1872" (facing p. 395), observation of an aurora at Eothbury, for August 10th read August 9th. In the Tbansactions of the Sections. Page 43, fourth line from bottom, for Asturo read Arturo. _ 64, tenth line from bottom, for uranium oxide 1 1;^, If, &c. read uranium oxide 1^, if, &o. 70, line 11, /or which it accom- read which it has accom- 173, lines 5 and 7, for Major Evan Smith read Major Euan Smith. LIST OF PLATES. PLATES I. II., III. Illustrative of the Report of the Committee on the Labyrinthodonts of the Coal-measures. OBJECTS AND RULES OF THE ASSOCIATION. OBJECTS. The Association contemplates no interference with the ground occupied by- other institutions. Its objects are : — To give a stronger impulse and a more systematic direction to scientific inquiry, ^to promote the intercourse of those who cultivate Science in different parts of the British Empire, with one another and with foreign philosophers, — to obtain a more general attention to the objects of Science, and a removal of any disadvantages of a public kind which impede its progress. RULES. Admission of Members and Associates. All persons who have attended the first Meeting shall be entitled to be- come Members of the Association, upon subscribing an obligation to con- form to its Rules. The Fellows and Members of Chartered Literary and Philosophical So- cieties publishing Transactions, in the British Empii-e, shall be entitled, in like manner, to become Members of the Association. The Officers and Members of the Councils, or Managing Committees, ot Philosophical Institutions shall be entitled, in like manner, to become Mem- bers of the Association. AU Members of a Philosophical Institution recommended by its Council or Managing Committee shall be entitled, in like manner, to become Mem- bers of the Association. Persons not belonging to such Institutions shall be elected by the General Committee or Council, to become Life Members of the Association, Annual Subscribers, or Associates for the year, subject to the approval of a General Meeting. Compositions, Subscriptions, and Privileges. Life Members shall pay, on admission, the sum of Ten Pounds. They shall receive gratuitously the Reports of the Association which may be pub- 1873. 6 XVlll RULES OF THE ASSOCIATION. lished after the date of such payment. They are eligible to all the offices of the Association. Annual Subsceibeks shall pay, on admission, the sum of Two Pounds, and in each following year the sum of One Pound. They shall receive (jratuitoushj the Eeports of the Association for the year of their admission and for the years in which they continue to pay without intermission their Annual Subscription. By omitting to pay this Subscription in any particu- lar year, Members of this class (Annual Subscribers) lose for that and all future years the privilege of receiving the volumes of the Association yratis : but they may resume their Membership and other privileges at any sub- sequent Meeting of the Association, ])aying on each such occasion the sum of One Pound. They are eligible to all the Offices of the Association. Associates for the year shall pay on admission the sum of One Pound. They shall not receive gratuitously the Reports of the Association, nor be eligible to serve on Committees, or to hold any office. The Association consists of the following classes : — 1. Life Members admitted from 1831 to 1845 inclusive, who have paid on admission Five Pomids as a composition. 2. Life Members who in 1846, or in subsequent years, have paid on ad- mission Ten Pounds as a composition. 3. Aimual Members admitted from 1831 to 1839 inclusive, subject to the payment of One Pound annually. [May resume their Membership after in- termission of Annual Payment.] 4. Annual Members admitted in any year since 1839, subject to the pay- ment of Two Pounds for the first year, and One Pound in each following year. [May resume their Membership after intermission of Annual Payment.] 5. Associates for the year, subject to the payment of One Pound. 6. Corresponding Members nominated Ijy the Council. And the Members and Associates wiU be entitled to receive the annual vohime of Eeports, yratis, or to purchase it at reduced (or Members') price, according to the following specification, viz. : — 1. Gratis. — Old Life Members who have paid Five Pounds as a compo- sition for Annual Payments, and previous to 1845 a further sum of Tm^o Pounds as a Book Subscription, or, since 1845, a further sum of Five Pounds. IN'ew Life Members who 'have paid Ten Pounds as a com.position. Annual Members who have not intermitted their Annual Sub- scription. 2. At reduced or Memlers' Prices, viz. two thirds of the Publication Price. — Old Life Members who have paid Five Pounds as a composition for Annual Payments, but no further sum as a Book Subscription. Annual Members who have intermitted their Annual Subscription. Associates for the year. [Privilege confined to the volume for that year only.] 3. Members may purchase (for the purpose of completing their sets) any of the first seventeen volumes of Transactions of the Associa- tion, and of which more than 100 copies remain, atone third of the Publication Price. Application to be made at the Office of the Association, 22 Albemarle Street, London, W. KULES OF THE ASSOCIATION. XIX Volumes not claimed within two years of the date of publication can only be issued by direction of the Council. Subscriptions shall be received by the Treasurer or Secretaries. Meetings. The Association shall meet annually, for one week, or longer. The place of each Meeting shall be appointed by the General Committee two years in advance ; and the Arrangements for it shall be entrusted to the Officers of the Association. General Committee. The General Committee shall sit during the week of the Meeting, or longer, to transact the business of the Association. It shall consist of the following persons : — ' CtAss A. Permanent Members. 1. Members of the Council, Presidents of the Association, and Presidents of Sections for the present and preceding years, with Authors of lleports in the Transactions of the Association. 2. Members who by the publication of Works or Papers have furthered the advancement of those subjects which are taken into consideration at the Sectional Meetings of the Association. With a vieiv of submittivg neiu claims under this Rule to the decision of the Council, they must he sent to the Assistant General Secretary at least one month before the Meeting of the Association. The decision of the Council on the claims of any Member of the Association to he placed on the list of the General Committee to hejinal. Class B. Temporary Members. 1. The Presidentfor the time beiAg of any Scientific Society publishing Trans- actions or, in his absence, a delegate representing him. Claims under this Rule to he sent to the Assistant General Secretary before the opening of the Meeting. 2. Office-bearers for the time being, or delegates, altogether not exceeding three, from Scientific Institutions established in the place of Meeting. Claims under this Ride to he approved by the Local Secretaries before the opening of the Electing. 3. Foreigners and other individuals whose assistance is desired, and who are specially nominated in writing, for the Meeting of the year, by the Pre- sident and General Secretaries. 4. Yice-Presidents and Secretaries of Sections. Organizing Sectional Committees*. The Presidents, Vice-Presidents, and Secretaries of the several Sections are nominated by the Council, and have power to act until their names are submitted to the General Committee for election. Prom the time of their nomination they constitute Organizing Committees for the purpose of obtaining information upon the Memoirs and Eeports likely to be submitted to the Sections f, and of preparing Reports thereon, * Passed by the General Committee, Edinburgh, 1871. + Notice to Conirifnitors of Mcnwirs.— Au\hcrs are reminded that, under an arrange- ment dating from 1871, the acceptance of MeiKoirs, and the days on which they are to be 62 XX RULES or THE ASSOCIATION. and on the order iu which it is desirable that they should be read, to be pre- sented to the Committees of the Sections at their first Meeting. An Organizing Committee may also hold such preliminary Meetings as the President of the Committee thinks expedient, but shall, under any circum- stances, meet on the first Wednesday of the Annual Meeting, at 11 a.m., to settle the terms of their Report, after which their functions as an Organizing Committee shaE cease. • Constitution of the Sectional Committees^. Oh the first day of the Annual Meeting, the President, Vice-Presidents, and Secretaries of each Section having been appointed by the General Com- mittee, these Officers, and those previous Presidents and Vice-Presidents of the Section who may desire to attend, are to meet, at 2 p.m., in their Com- mittee Rooms, and enlarge the Sectional Committees by selecting individuals from among the Members (not Associates) present at tl\c Meeting whose as- sistance they may particularly desire. The Sectional Committees thus con- stituted shall have power to add to their number from day to day. The List thus formed is to be entered daily iu the Sectional Minute-Book, and a copy forwarded without delay to the Printer, who is charged with publishing the same before 8 a.m. on the next day, in the Journal of the Sectional Proceedings. Business of the Sectional Committees. Committee Meetings are to be held on the Wednesday at 2 p.m., on the following Thursday, Friday, Saturday, Monday, and Tuesday, from 10 to 11 A.M., punctually, for the objects stated in the Rules of the Association, and specified below. The business is to be conducted in the following manner : — At the first meeting, one of the Secretaries will read the Minutes of last year's proceedings, as recorded in the Miniite-Book, and the Synopsis of Recommendations adopted at the last Meeting of the Association and printed in the last volume of the Transactions. He wiU next proceed to read the Report of the Organizing Committee t- The List of Communications to be read on Thursday shall be then arranged, and the general distribution of business throughout the week shall be provisionally appointed. At the close of the Committee Meeting the Secretaries shall forward to the Printer a List of the Papers appointed to be read. The Printer is charged with publishing the same before 8 a.m. on Thursday in the Journal. On the second day of the Annual Meeting, and the following days, the read, are now as far as possible determined by Organizing Committees for the several Sections before the beginning of the. Meeting. It has therefore become necessary, in order to give an opportunity to the Committees of doing justice to the several Communications, that each Author should prepare an Abstract of his Memoir, of a length suitable for in- sertion in the publislied Transactions of the Association, and that he should send it, toge- ther with the original Memoir, by book-post, on or before , addressed thus — "General Secretaries, British Association, 22 Albemarle Street, London, W. For Section " If it should be inconvenient to the Author that his Paper should be read on any particular days, he is requested to send information thereof to the Secretaries in a separate note. ■ * Passed by the General Committee. Edinburgh, 1871. t This and the following sentence were added by the General Committee, 1871. RULES OF THE ASSOCIATION. XXI Secretaries are to correct, ou a copy of the Journal, the list of papers which have been read on that day, to add to it a list of those appointed to be read on the next day, and to send this copy of the Journal as early in the day as possible to the Printers, who are charged with printing the same before 8 a.m. next morning in the Journal. It is necessary that one of the Secretaries of each Section should call at the Printing Office and revise the proof each evening. Minutes of the proceedings of every Committee are to be entered daily in the Minute-Book, which should be confirmed at the next meeting of the Committee. Lists of the Eeports and Memoirs read in the Sections are to be entered in the Minute-Book daily, which, with all Memoirs and Copies or Abstracts of Memoirs furnislied by Authors, are to be forwarded, at the close of the Sec- tional Meetings, to the Assistant General Secretary. The Vice-Presidents and Secretaries of Sections become ex officio temporary Members of the General Committee {vide p. xix), and will receive, on ap- pHcation to the Treasurer in the Reception Eoom, Tickets entitling them to attend its Meetings. The Committees will take into consideration any suggestions which may be offered by their Members for the advancement of Science. They are specially requested to review the recommendations adopted at preceding Meetings, as published in the volumes of the Association and the communi- cations made to the Sections at this Meeting, for the purposes of selecting definite points of research to which individual or combined exertion may be usefully directed, and branches of knowledge on the state and progress of which Reports are wanted ; to name individuals or Committees for the exe- cution of such Reports or researches ; and to state whether, and to what de- gree, these objects may be usefully advanced by the appropriation of the funds of the Association, by application to Government, Philosophical Insti- tutions, or Local Authorities. In case of appointment of Committees for special objects of Science, it is expedient that all Members of the Committee shoidd be named, and one of them appointed to act as Secretary, for insuring attention to business. Committees have power to add to their number persons whose assistance they may require. The recommendations adopted by the Committees of Sections are to be registered in the Forms furnished to their Secretaries, and one Copy of each is to be forwarded, without delay, to the Assistant General Secretary for pre- sentation to the Committee of Recommendations. Unless this be done, the Recommendations cannot receive the sanction of the Association. N.B. — Recommendations which may originate in any one of the Sections must first be sanctioned by the Committee of that Section before they can be referred to the Committee of Recommendations or confirmed by the General Committee. Notices Regarding Grants of Money. Committees and individuals, to whom grants of money have been entrusted by the Association for the prosecution of particular researches in Science, are required to present to each following Meeting of the Association a Report of the progress which has been made ; and the Individual or the Member first named of a Committee to whom a money grant has been made must (pre- viously to the nCT;t meeting of the Association) forward to the General Xxii RULES OF THE ASSOCIATION. Secretaries or Treasurer a statement of the sums which have been expended, and the balance which remains disposable on each grant. Grants of money sanctioned at any one meeting of the Association expire a week before the opening of the ensuing Meeting; nor is the Treasurer authorized, after that date, to allow any claims on account of such grants, unless they be renewed in the original or a modified form by the General Committee. No Committee shall raise money in the name or under the auspices of the British Association without special permission from the General Committee to do so ; and no money so raised shall be expended except in accordance with the rules of the Association. In each Committee, the Member first named is the only person entitled to call on the Treasurer, W. Spottiswoode, Esq., 50 Grosvenor Place, Loudon, S.W., for such portion of the sums granted as may from time to time be required. In grants of money to Committees, the Association does not contemplate the payment of personal expenses to the members. In aU cases where additional grants of money are made for the continua- tion of Eesearches at the cost of the Association, the sum named is deemed to include, as a part of the amount, whatever balance may remain unpaid on the former grant for the same object. All Instruments, Papers, Drawings, and other property of the Association are to be deposited at the Office of the Association, 22 Albemarle Street, Piccadilly, London, W., when not employed in carrying on scientific inquiries for the Association. Business of the Sections. The Meeting Room of each Section is opened for conversation from 10 to 11 daily. The Section Rooms and approaches thereto can he used for no notices, exhibitions, or other pwposes than those of the Association. At 11 precisely the Chair wiU be taken, and the reading of communica- tions, in the order pre\iously made public, be commenced. At 3 p.m. the Sections will close. Sections may, by the desii'e of the Committees, divide themselves into Departments, as often as the number and nature of the communications de- livered in may render such divisions desirable. A Eeport presented to the Association, and read to the Section which originally called for it, may be read in another Section, at the request of the Officers of that Section, with the consent of the Author. Duties of the Doorkeepers. 1. — To remain constantly at the Doors of the Rooms to which they are ap- pointed during the whole time for which they are engaged. 2. — To require of every person desirous of entering the Rooms the exhibi- tion of a Member's, Associate's or Lady's Ticket, or Reporter's Ticket, signed by the Treasurer, or a Special Ticket signed by the Assistant General Secretary. 3. — Persons unprovided with any of these Tickets can only be admitted to any particular Room by order of the Secretary in that Room. No person is exempt from these Rules, except those Officers of the Asso- ciation whose names are printed in the Programme, p. 1. RULES OF THE ASSOCIATION. XXIU Duties of the Messengers, To remain constantly at the Eooms to which they are appointed, during the whole time for which they are engaged, except when employed on mes- sages by one of the Officers directing these E,ooms. Committee of Recommendations. The General Committee shall appoint at each Meeting a Committee, which shall receive and consider the Eecommendations of the Sectional Committees, and report to the General Committee the measures which they would advise to he adopted for the advancement of Science. AH Recommendations of Grants of Money, Requests for Special Researches, and Reports on Scientific Subjects shall be submitted to the Committee of Recommendations, and not taken into consideration by the General Committee unless previously recommended by the Committee of Recommendations. Local Committees. Local Committees shall be formed by the Officers of the Association to assist in making arrangements for the Meetings. Local Committees shall have the power of adding to their numbers those Members of the Association whose assistance they may desire. Officers. A President, two or more Vice-Presidents, one or more Secretaries, and a Treasurer shall be annually appointed by the General Committee. Council. In the intervals of the Meetings, the affairs of the Association shall be ma- naged by a Council appointed by the General Committee. The Council may also assemble for the despatch of business dming the week of the Meeting. Papers and Communications. The Author of any paper or communication shall be at liberty to reserve his right of property therein. Accounts. The Accounts of the Association shall be audited annually, by Auditors appointed by the General Committee. a P CU TO 5J "5 l=! O g «! 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Cd O O) K ■ " ^CS » «feS 4; "H u b,' - .-03 •^« y en i, M « o c J; O "-^ A . h .3 H '-sP«03 c o ^ KPSS o o ^ si ^■ :fa o o^ 4, 4) E-^03 03 BJ fa -03 . « - « £ g J S art £"> ■ • *- i; • •■^^ 4J 4> fc< (^ 03 d Pa oo Wo. ^ 4> & J K 2 E- J; C on 03 w fa o « 2: o 03 s o s < 3 03 . oi :/3 WOO or- en < q ^" o 03 <: » < u ■J < ®fa 03" cfi r W!» Pk «] Pi c . d - • « " 60 o 5 >< S .« ■-> PA O 03 cn W fa o xxx REPORT 1873. Presidents and Secretaries of the Sections of the Association. Date and Place. Presidents. Secretaries. MATHEMATICAL AIN'D PHYSICAL SCIENCES. COMMITTEE OF SCIENCES, I. MATHEMATICS AND GENERAL PHYSICS. 1832. Oxford 18.33. Cambi-idge 1834. Edinburgh Davies Gilbert, D.C.L., F.E.S.... SirD. Brewster, F.E.S Rev. W. Whewell, RR.S Rev. H. Coddington. Prof. Forbes. Prof. Forbes, Prof. Lloyd. 1835. Dublin 1836. Bristol 1837. Liverpool .. 18.38. Newcastle... 1839. Birmingham 1840. Glasgow .. 1841. Plymouth.. 1842. Manchester 1843. Cork 1844. York 1845. Cambridge.. 1846. Southampton 1847. Oxford... SECTION A. MATHEMATICS AND PHYSICS Rev. Dr. Robinson Rev. William Whewell, F.R.S.... SirD. Brewster, F.R.S 1848. Swansea . 1849. Birmingham 1850. Edinburgh. 1851. Ipswich 18.'>2. Belfast 1853. Hull 1854. Liverpool.. 1855. Glasgow ... 1856. Cheltenham 1857. Dublin Prof. Sir W. R. Hamilton, Prof. Wheatstone. Prof. Forbes, W. S. Harris, F. W. Jerrard. W. S. Harris, Rev. Prof. PoweU, Prof. Stevelly. Rev. Prof. Chevallier, Major Sabine, Prof. Stevelly. J. D. Chance, "W. Snow Harris, Prof. Stevelly. Rev. Dr. Forbes, Prof. Stevelly, Arch. Smith. Prof. Stevelly. Prof. M'Cullocb, Prof. Stevelly, Rev. W. Score.sby. J. Nott, Prof. Stevelly. Rev. Wm. Hey, Prof. Stevellv. Rev. H. Goodwin, Prof. Stevellv, G. G. Stokes. John Drew, Dr. Stevelly, G. G. Stokes. Rev. H. Price, Prof. Stevelly, G. G. Stokes. Dr. Stevelly, G. G. Stokes. Prof. Stevelly, G. G. Stokes, W. Ridout WiUs. W. J. Macquorn Rankine, Prof. Smyth, Prof. Stevelly, Prof. G. G. Stokes. S. Jackson, W. J. Macquorn Rankine, Prof. Stevelly, Prof. G. G. Stokes. Prof. Dixon, W. J. Macquorn Ran- kine, Prof. Stevelly, J. Tyndall. B. Blaydes Haworth, J. 1). Sollilt, Prof. Stevellv. J. Welsh. J. Hartnup, H. G. Puckle, Prof. Stevelly. J. Tyndall, J. Welsh. Rev. Prof. Kelland, M.A., F.R.S.]Rev. Dr. Forbes, Prof. D. Gray, Prof. L.&E. Tyndall. Rev. R. Walker, M. A., F.R.S. ...C. Brooke, Rev. T. A. Southwocd, Prof. Stevelly, Rev. J. C. TurnbuU. Rev.T. R. Robinson,D.D.,F.R.S., Prof. Curtis, Prof. Hennessy, P. A. M.R.l.A. Ninni.s, W. J. Macquorn Rankine, Prof. Stevellv. Sir J. F. W. Herschel, Bart., F.R.S. Rev. Prof. WheweU, RR.S Prof. rorbe.s, F.R.S Rev. Prof. Lloyd, F.R.S Very Rev. G. Peacock, D.D., F.R.S. Prof MCulloch, M.R.LA The Earl of Ros.se, F.R.S The Very Rev. the Dean of Ely . Sir John F. W. Herschel, Bart., Rev! Prof. PoweU, M.A., F.R.S. . Lord Wrottesley, F.R.S William Hopkins, F.R.S Prof. J. D. Forbes, F.R.S., Sec. R.S.E. Rev. W. Whewell, D.D., F.R.S., &c. Prof. W. Thomson, M.A., F.R.S. L.&E. The Dean of Ely, F.R.S Prof. G. G. Stokes, M.A., Sec. R.S. PRESIDENTS AND SECRETAllIES OF THE SECTIONS. XXXI Date and Place. Presidents. 1858. Leeds Eev. W.Whewell, D.D., V.P.E.S, 1859. Aberdeen .. 1860. Oxford 1861. Manchester 1862. Cambridge, 1863. Newcastle.. 1861. Bath 1865. Birmingham 1866. Nottingham The Earl of Kosse, M.A., K.P., Eev! B. Price, M.A., F.E.S G. B. Airy, M.A., D.C.L., F.E.S. Prof. G. G. Stokes, M.A., F.E.S Secretaries. Eev. S. Earnshaw, J. P. Hennessj, Prof Stevelly, H. J. S. Smith, Prof. Tyndall. J. P". Hennessy, Prof Maxwell, H. J. S. Smith, Prof Stevelly. Eev. G. 0. Bell, Eev. T. Eennison, Prof. Stevelly. Prof E. B. Clifton, Prof. H. J. S. Smith, Prof. Stevelly. Prof E. B. Clifton, Prof II. J. S. Smith, Prof. Stevelly. Prof W. J. Macqxiorn Eankine, Eev.N.Ferrers,Prof.Fuller,F. Jenkin, C.E., F.E.S. I Prof. Stevelly, Eev. C. T. Whitley. Prof. Cayley, M.A., F.E.S.jProf. Fuller, F. Jenkin, Eev. G. F.E.A.S. W. Spottiswoode, M.A., F.E.S., F.E.A.S. Prof. Wheatstone, D.C.L., F.E.S. 1867. Dundee Iprof. Sir W. Thomson, D.C.L., I T*' "R S 1868. Norwich ...Prof. J. Tyndall, LL.D., F.E.S... Buckle, Prof Stevelly. Eev. T. N. Hutchinson, F. Jenkin, G. S. Mathews, Prof. H. J. S. Smith, J. M. Wilson. Fleeming Jenkin, Prof. H. J. S. Smith, Eev. S. N. Swann. Eev. G. Buckle, Prof. G. C. Foster, Prof. Fuller, Prof. Swan. Prof. G. C. Foster, Eev. E. Harley, E. B. Hayward. Prof G. C. Foster, E. B. Hayward, W. K. Clifford. 1869. Exeter ;Prof. J. J. Sylvester, LL.D.,: FES 1870. Liverpool...' J. Clerk Maxwell, M.A., LL.D.,iProf W. G. Adams, W. K. Clifford, " " -• ' Prof. G. C. Foster, Eev. W. Allen Whitworth. Prof W. G. Adams, J. T. Bottomley, 1871. Edinburgh . 1872. Brighton ... 1873. Bradford ... F.E.S. Prof. P. G. Tait, F.E.S.E. W. De La Eue, D.C.L., F.E.S. Prof. H. J. S. Smith, F.E.S.... Prof. W. K. Clifford, Prof J. D. Everett, Eev. E. Harley. Prof W. K. Chfford, J. W.L. Glaisher, Prof A. S. Herschel, G. F. Eodwell. Prof. W. K. Clifford, Prof Forbes, J. W. L. Glaisher, Prof A. S. Herschel. CHEMICAL SCIENCE. COMMITTEE OF SCIENCES, II. CHEMISTKX, MINEEALOGT. 18.")2. Oxford 1833. Cambridge.. 1834. Edinburgh..., 1835. Dublin 1836. Bristol John Dalton.D.C.L., F.E.S John Dalton, D.C.L., F.E.S Dr. Hope James F. W. Johnston. Prof Miller. Mr. Johnston, Dr. Christison. SECTION B. CHEMISTRY AND MINEEALOGT. Dr. T. Thomson, F.E.S. Eev. Prof. Cumming 1837. Liverpool.. 1838. Newcastle.. 1839. Birmingham 1840. Glasgow ... 1841. Plymouth.. 1842. Manchester 1843. Cork 1844. York 1845. Cambridge. Michael Faraday, F.E.S Eev. William Whewell, F.E.S., Prof T. Graham, F.E.S Dr. Thomas Thomson, F.E.S. Dr.Daubeny, F.E.S John Dalton.D.C.L, F.E.S.... Prof Apjohn, M.E.I.A Prof. T. Graham, F.E.S Eev. Prof Cumming Dr. Apjohn, Prof. Johni^ton. Dr. Apjohn, Dr. C. Henry, W. Hera- path. Prof Johnston, Prof Miller, Dr. Eeynolds. Prof 'Miller, E. L. Pattinson, Thomas Eichardson. Golding Bird. M.D., Dr. J. B. Melson. Dr. E. D. Thomson, Dr. T. Clark, Dr. L. Playfair. J. Prideaux, Eobert Himt, W. M. Tweedy. Dr. L. Playfair, E. Hunt, J. Graham. E. Hunt, Dr. Sweeny. Dr. E. Playfair, E. Solly, T. H. Barker. E. Hunt, J. P. Joule, Prof.Milkr E. Sollv. XXXll REPORT — 1873. Date and Place. 1 846. Southampton 1847. Oxford ... 1848. Swansea 1849. Birmingham 1850. Edinburgh . 1851. Ipswich 1852. Belfast ... 1853. HuU . 1854. Liverpool... 1855. Glasgow ... 1856. Cheltenham 1857. Dublin 1858. L3eds 1859. Aberdeen... 1860. Oxford Presidents. Secretaries. Michael Faraday, D.C.L., F.E.S. Rev.W.V.Harcourt, M.A., F.R.S. Richard Phillips, F.R.S John Percy, M.D., F.R.S Dr. Cliristison, V.P.R.S.E Prof. Thomas Graham, F.R.S. .. Thomas Andrews, M.D., F.R.S. 1861. Manchester. 1862. Cambridge . 1863. Newcastle... 1864. Bath 1865. Birmingliam 1866. Nottingham 1867. Dundee .. 1868. Norwich .. 1869. Exeter 1870. Liverpool.. 1871. Edinburgh 1872. Brighton .. 1873. Bradford .. Prof. J. F. W. Johnston, M.A., F R S Prof. W. A. Miller, M.D., F.R.S Dr. Lyon Playfair, C.B., F.R.S. . Prof. B. C. Brodie, F.R.S Prof. Apjohn, M.D., F.R.S. M.R.LA. Sir J. F. W. Hersohel, Bart., D.C.L. Dr. Lyon Playfair, C.B., F.R.S. . Prof. B. C. Brodie, F.R.S Prof. W. A. Miller, M.D., F.R.S. Prof. W. A. Miller, M.D., F.R.S. Dr. Alex. W. Williamson, F.R.S. W. Odling, M.B., F.R.S., F.C.S. Prof. W. A. Miller, M.D.,V.P.R.S. H. Bence Jones, M.D., F.R.S. ... Prof.T.Anderson,M.D.,F.R.S.E. Prof.E .Frankland, F.R.S., F.C.S. Dr. H. Debus, F.R.S., F.C.S. ... Prof. H. E.Roscoe,B.A., F.R.S., F.C.S. Prof. T. Andrews, M.D., F.R.S. Dr. J. H. Gladstone, F.R.S Prof. W. J. Russell, F.R.S Dr. Miller, R. Hunt, W. Randall. B. C. Brodie, R. Hunt, Prof. Solly. T. H. Henry, R. Hunt, T. Williams. R. Hunt, G. Shaw. Dr. Anderson, R. Hunt, Dr. Wilson. T. J. Pearsall, W. S. Ward. Dr. Gladstone, Prof. Hodges, Prof. Ronalds. H. S. BlundeU, Prof. R. Hunt, T. J. Pearsall. Dr. Edwards, Dr. Gladstone, Dr. Price. Prof. Frankland, Dr. H. E. Roscoe. J. Horsley, P. J. Worsley, Prof. Voelcker. Dr. Davy, Dr. Gladstone, Prof. Sul- livan. Dr. Gladstone, W. Odling, R. Rey- nolds. J. S. Brazier, Dr. Gladstone, G. D. Liveing, Dr. Odling. A. Vernon Harcourt, G. D. Liveing, A. B. Northcote. A. Vernon Harcourt, G. D. Liveing. H. W. Elphinstone, W. Odling, Prof. Roscoe. Prof. Liveing, H. L. Pattinson, J. C. Stevenson. A. V. Harcourt, Prof. Liveing, R. Biggs. A. V. Harcourt, H. Adkins, Prof. Wanklyn, A. Winkler Wills. J. H. Atherton, Prof. Liveing, W. J. Russell, J. White. A. Crura Brown, Prof. G. D. Liveing, W. J. RusseU. Dr. A. Crum Brown, Dr. W. J. Rus- sell, F. Sutton. Prof. A. Crum Brown, M.D., Dr. W. J. Russell, Dr. Atkinson. Prof. A. Crum Brown, M.D., A. E. Fletcher, Dr. W. J. Russell. J. T. Buchanan, W. N. Hartley, T. E. Thorpe. Dr. Mills, W. Chandler Roberts, Dr. W. J. RusseU, Dr. T. Wood. Dr. Armstrong, Dr. MiUs, W. Chan- dler Roberts, Dr. Thorpe. GEOLOGICAL (and, until 1851, GEOGRAPHICAL) SCIENCE. COMMITTEE OF SCIENCES, III. GEOLOGY AND GEOGRAPHY. 1832. Oxford R. I. Murchison, F.R.S. 1833. Cambridge . 1834. Edinburgh . ]8.3.-.. Dublin, 1836. Bristol . G. B. Greenough, F.R.S. Prof. Jameson John Taylor. W. Lonsdale, John Phillips. Prof. Phillips, T. Jameson Torrie, Rev. J. Yates. 1837. Liverpool . SECTION C. R.J. Griffith Rev. Dr. Buckland, F.R.S.— Geo- grafhf. R. I.Murchi.son.F.R.S. Rev.Pro'r. Scdgwick,F.R.S.— Gco- y?-rt^)^y. G.B.Greenough,F.R.S. GEOLOGY AND GEOGRAPHY. Captain Portlock, T. J. Torrie. William Sanders, S. Stutchbury, T. J. Torrie. Captain Portlock, R. Hunter. — Geo- graphy. Captain H M. Denham.R.N. PRESIDENTS AND SECRETARIES OF THE SECTIONS. XXXIU Pate and Place. 1838. Newcastle... 1839. Birmingham 1840. Glasgow ... 1841. Plymouth.. 1842. Manchester 1843. Cork 1844. York 1845. Cambridge j. 184i5. Southampton 1847. O.xford 1848. Swansea ... 1849. Birmingham 1850. Edinburgh* Presidents. C. Lyell, P.R.S., Y.F.G.S.—Gco- graphii. Lord Prudhope. Rev. Drl Buckland, F.R.S.— (?«o- graph/. G-.B.Greenough.F.R.S Charles 'Lyell, F.R.S. — Geogra- fhj. G. B. Greenough, F.R.S. H.T.De la Beche, F.R.S R. I. Murchison, F.R.S Richard E. Griffith, F.R.S., M.R.I.A. Henry Warburton, M.P., Pres. Geol. Soc. Rev. Prof. Sedgwick, M.A., F.R.S. LeonardIIorner,F.R.S. — Gcogra- phij. G. B. Greenough, F.R.S. Very Rev. Dr. Buckland, F.R.S. Sir H. T. De la Beche, C.B., Sir Charles Lyell, F.R.S., F.G.S. Sir Roderick I. Marchi30n,F.R.S. Secretaries. W. C. Trevelyan, Capt. Portlock.— Gvogrcqjhi/. Capt. Washington. George Lloyd, M.D., H. E. Strickland, Charles Darwin. W. J. Hamilton, D. Milne, Hugh Murray, H. E. Strickland, John Scoular, M.D. W. J. Hamilton, Edward Moore,M.D., R. Hutton. E. W. Binney, R. Hutton, Dr. R. Lloyd, H. E. Strickland. Francis M. Jennings, H. E. Strick- land. Prof.Ansted, E. H. Bunbury. Rev. J. C. Gumming, A. C. Ramsay, RcT. W. Thorp. Robert A. Austen, J. H. Norten, M.D., Prof. Oldham. — Geography. Dr. C. T. Beke. Prof. Ansted, Prof. Oldham, A. C. Ramsay, J. Ruskin. Starling Benson, Prof. Oldham, Prof. Ramsay. J. Beete Jukes, Prof. Oldham, Prof. A. C. Ramsay. A. Keith Johnston, Hugh Miller, Pro- fessor Nieol. SECTION c (continued). — geology. 1851. Ipswich 1852. Belfast ., 18.5.3. Hull 1854. Liverpool . . 18.55. Gla.sgow ... 1856. Cheltenham 1857. Dublin 1858. Leeds 1859. Aberdeen ... 1860. Oxford 1S61. Manchester 18G2. Cambridge 1 863. Newcastle ... William Hopkins, M.A., F.R.S... Lieut.-Col. Portlock,E.E., F.R.S. Prof Sedgwick, F.R.S Prof. Edward Forbes, F.R.S. . . . Sir E. I. Murchison, F.R.S Prof. A. C. Ramsay, F.R.S The Lord Talbot de Malahide ... William Hopkins, M.A., LL.D., Sir Charles Lyell, LL.D., D.C.L., F.R.S. Rev. Prof. Sedgwick, LL.D., F.R.S., F.G.S. Sir R. I. Murchison, D.C.L., LL.D., F.R.S., &c. J. Beete Jukes, M.A., F.R.S Prof Warington W. Smyth, F.R.S., F.G.S. 0. J. F. Bunbury, G. W. Ormerod, Searles Wood. James Bryce, James MacAdam, Prof M'Coy, Prof Nicol. Prof. Harkness, William Lawton. .John Cunningham, Prof. Harkness, G. W. Ormerod, J. W. WoodaU. James Bryce, Prof Harkness, Prof. Nicol. Rev. P. B. Brodie, Rer. E. Hopworth, Edward Hull, J. Scougall, T.Wright. Prof Hai'kness, Gilbert Sanders, Ro- bert H. Scott. Prof. Nicol, H. C. Sorby, E. W. Shaw. Prof Harkness, Rev. J. Longmuir, H. C. Sorby. Prof Harkness, Edward Hull, Capt. Woodall. Prof Harkness, Edward Hull, T. Ru- pert Jones, G. W. Ormerod. Lucas Barrett, Prof T. Rupert Jones, H. C. Sorby. E. F. Boyd, John Daglish, H. C. Sor- by, Thomas Sopwith. * At a Meeting of the General Committee held in 1850, it was resolved "That tlio subject of Gcographv be separated from Geology and combined with Ethnology, to consti- tute a separate Section, under the title of the " Geographical and Ethnological Section." for Presidents and Secretaries of which see page xxxvi. 1873. c SXXIV REPORT — 1873. Date and Place. Presidents. Secretaries. 1864. Bath 1865. Birmingham 1866. Nottingham 1867. Dundee... 1868. Norwich 1869. Exeter ... 1870. Liverpool... 1871. Edinburgh 1872. Brighton . 1873. Bradford . Prof. J. Phillips, LL.D., F.E.S., E.G.S. SirE. I. Murchison, Bart.,K.C.B. Prof. A.C. Eamsay, LL.D., F.E.S. Archibald Geikie, E.E.S., E.G.S. E. A. C. Godwin-Austen, F.E.S., 17 G S Prof. E.Harkness, RE.S., E.G.S. Sir Phili]3 de M. Grey Egerton, Bart, M.P., RE.S. Prof. A. Geikie, F.E.S., RG.S... E. A. C. Godwin-Austen, F.E.S. Prof. J. Phillips, D.C.L., F.E.S., RG.S, W. B. Dawkins, J. Johnston, H. C. Soi-by, W. Pengelly. Eev. P. B. Brodic, J. Jones, Eev. E. Myers, H. C. Sorby, W. Pengelly. E. Etheridge, W. Pengelly, T. Wil- son, G. H. Wright. Edward Hull, W. Pengelly, Henry Woodward. Eev. O. Fisher, Eev. J. Gunn, W. Pengelly, Eev. H. H. Winwood. W. Pengelly, W. Boyd Dawkins, Eev. H. H. Winwood; W. Pengelly, Eev. H. H. Winwood, W. Boyd"Dawkins, G. H. Morton. E. Etheridge, J. Geikie, J. McKenny Hughes, L. C. Miall. L. C. Miall, George Scott, William Topley, Henry Woodward. L. C. Miall, E. H. Tiddeman, W. Topley. BIOLOGICAL SCIENCES. COMMITTEE OE SCIENCES, IV. ZOOLOGY, BOTANY, PHYSIOXOeY, ANATOMY. 1832. Oxford 18.33. Cambridge* 183-1. EcUnburgh Eev. P. B. Dimcan, RG.S Eev. W. L. P. Garnons, F.L.S., Prof. Graham Eev. Prof. J. S. Henslow. C. C. Babington, D. Don. W. Yarrell, Prof. Bm-nett. 1835. Dublin , 1836. Bristol . 1837. Liverpool.. 1838. Newcastle.., 1839. Brimingham 1840. Glasgow .. 1841. Plymouth.. 1842. Manchester SECTION D. — Dr. Allman ; Eev. Prof. Henslow . W. S. MacLeay Sir W. Jardine, Bart, Prof. Owen, F.E.S Sir W. J. Hooker, LL.D 1843. Cork. 1844. York. 1845. Cambridge 1846. Southampton 1847. Oxford.... John Eichardson, M.D.,RE.S... Hon. and Very Eev. W. Herbert LL.D., F.L.S. Wilham Thompson, F.L.S , Very Eev. The Dean of Manches- ter. Eev. Prof. Henslow, F.L.S Sir J. Eichardson, M.D., F.E.S. H. E. Strickland, M.A., F.E.S..., •ZOOIOGY AND BOTANY. J. Curtis, Dr. Litton. J. Curtis, Prof. Don, Dr. Eiley, S. Eootsey. C. C. Babington, Eev. L. Jenyns, W. Swainson. J. E. Gray, Prof. Jones, E. Owen, Dr. Eicliard.son. E. Forbes, W. Ick, E. Patterson. Prof. W. Couper, E. Forbes, E. Pat- terson. J. Couch, Dr. Lankester, E. Patterson. Dr. Lankester, E. Patterson, J. A. Turner. G. J. Allman, Dr. Lankester, E. Pat- terson. Prof. Allman, H. Goodsir, Dr. King, Dr. Lankester. Dr. Lankester, T. V. Wollaston. Dr. Lankester, T. V. Wollaston, H. Wooldridge. Dr. Lankester, Dr. Melville, T. V. ' Wollaston. SECTION D (continued). — zoology and botany, including physiology. [For the Presidents and Secretaries of the Anatomical and Pliysiological Subsections and the temporary Section E of Anatomy and Medicine, see p. sxxvi.] 1848. Swansea 1849. Birmingham 1850. Edinburgh.. L. W. Dillwyn, F.E.S. Dr. E. Wilbraham Falconer, A. Hen- frey. Dr. Lankester. Dr. Lankester, Dr. Eussell. Prof. J. H. Bennett, M.D., Dr. Lan- kester, Dr. Douglas Maclagan. * At this Meeting Physiology and Anatomy were made a separate Committee, for Presidents and Secretaries of wliich see p. xxxvi. William Spence, RE.S Prof. Goodsir, F.E.S. L. &E. ... PRESIDENTS AND SECRETARIES OF THE SECTIONS. XXXV Date and Place. Presidents. Secretai'ies. 1851. 1852. 1853. 1854. 1855. 1856. 1857. 1858. 1859. 1860. 1861. 1802. 1863. 1864. 1865. Ipswich. Belfast . Eev. Prof. Henslow, M.A., F.R.S, W. Ogilby Hull Liverpool ... Q-lasgow . . . Cheltenham, Dublin Leeds Aberdeen ... Oxford Manchester.. Cambridge... Newcastle . . . Bath Birmingham 1866. Nottingham. 1867. Dundee . 1868. Norwich 1869. Exeter ^70. Liverpool . . 1871. Edinburgh 1872. Brighton C. C. Babington, M.A., E.R.S... Prof. Balfour, M.D., F.R.S EeT. Dr. Eleeming, F.E.S.E. ... Thomas BeU, E.E.8., Pres.L.S. . . , Prof.W.H. Harvey, M.D., P.E.S, C. C. Babington, M.A., F.E.S.... Sir W. Jardine, Bart., F.E.S.E.. Eev, Prof. Henslow, F.L.S Prof. C. C. Babington, F.E.S. ... Prof. Huxley, F.E.S Prof. Balfour, M.D., F.E.S Dr. John E. Gray, F.E.S T. Thomson, M.D., F.E.S SECTION D (contimied).- Prof. Huxley, LL.D., F.E.S.— Physiological Dep. Prof. Hum- phry, M.D., V.'R.B.— Anthropo- logical Dep. Alfred E. Wallace. F.E.G.S. Prof. Sharpey, M.D., Sec. E.S.— JDep. of Zool. and Bot. George Busk, M.D., F.E.S. Rev. M. J. Berkeley, F.L.S. Dep. of 'Physiology. W, H. Flower, F.E.S. George Busk, F.E.S., F.L.S Dep. of Bot. and Zool. C. Spence Bate, F.E.S.— i>(?j). of Ethno. E. B. Tvlor. Prof. G. Eolleston, M.A., M.D. F.E.S..F.L.S.— i)f:p. A7wt. and Ph/siol. Prof. M. Foster, M.D., F.L.S.— Dfp. of Ethno. J. Evans, F.E.S. Prof Allen Thomson,M.D.,F.E.S. — Dep. of Bot. and Zool. Prof. Wyville Thomson, F.E.S.— Dep. of Anthropol. Prof. W. Turner, M.D. Sir John Lubbock, Bart., P.E.S. — Dep. of Avat. and Physiol. Dr. Burdon Sanderson, F.E.S. — Dep of Anthropol. Col. A. Prof. Allman, F. VV. Johnston, Dr. E. Lankester. Dr. Dickie, George C. Hyndman, Dr. Edwin Lankester. Eobert Harrison, Dr. E. Lankester, Isaac Byerley, Dr. E. Lankester. William Keddie, Dr. Lankester. Dr. J. Abercrombie, Prof. Buokman, Dr. Lankester. Prof. J. E.Kinahan, Dr. E. Lankester, Eobert Patterson, Dr. W. E. Steele. Henry Denny, Dr. Heaton, Dr. E. Lankester, Dr. E. Perceval Wright. Prof. Dickie, M.D., Dr. E, Lankester, Dr. Ogilvy. W. S. Churcli, Dr. E. Lankester, P. L. Sclater, Dr. E. Perceval Wright. Dr. T. Alcock, Dr. E. Lankester, Dr. P. L. Sclater, Dr. E. P. Wriglit. Alfred Newton, Dr. E. P. Wright. Dr. E. Charlton, A. Newton, Eev. H. B. Tristram, Dr. E. P. Wright. H. B. Brady, C. E. Broom, H. T. Stainton, Dr. E. P. Wright. Dr. J. Anthony, Eev. C. Clarke, Rev H. B. Tristram, Dr. E. P. Wright.' -BIOLOGY*. Dr. J. Beddard, W. Felkin, Eev. H. B. Tristram, W. Turner, E. B. Tylor, Dr. E. P. Wright. C. Spence Bate, Dr. S. Cobbold, Dr. M. Foster, H. T. Stainton, Rev. H. B, Tristram, Prof. W. Turner. Dr. T. S. Cobbold, G. W. Firth, Dr. M. Foster, Prof. Lawson, H. T. Stainton, Rev. Dr. H. B. Tristram, Dr. E. P. Wright. Dr. T. S. Cobbold, Prof. M. Foster, M.D., E. Ray Lankester, Professor Lawson, H. T. Stainton, Rev. H. B. Tristram. Dr. T. S. Cobbold, Sebastian Evans, Prof. Lawson, Thos. J. Moore, H, T. Stainton, Rev. H. B.Tristram, C. Stanilaud Wake, E. Ray Lan- kester. Dr. T. R. Eraser, Dr. Arthur Gamgee, E. Ray Lankester, Prof. Lawson, H. T. Stainton, C. Staniland Wake, Dr. W. Rutherford, Dr. Kelburne King. Prof. Thiselton-Dyer, H. T. Stainton, Prof. Lawson, F. W. Rudler, J. H. Lamprey, Dr. Gamgee, E. Ray Lan- kester, Dr. Pye Smith. Lane Fox, F.G.S. * At a Meeting of the General Committee in 1865, it was resolved: — "That the title of Section D be changed to Biology ; " and " That for the word 'Subsection,' in tho rules for conducting the business of the Sections, the word ' Deportment' be substituted, (■2 XXXV 1 REPORT — 1873. Date and Place. 1873. Bradford ... Presidents. Prof. Allman, F.n.S.—Bep. of Anat. and Phi/siol. Prof. Ru- therford, M.b.—Dcp. of An thropol. Dr. Beddoe, F.E.S. Seeretaries. Prof. Thiselton-Dyer, Prof. Lawson, B. M'Lacblan, Dr. Pye-Smith, E. Ray Lankester, F. W. Eudler, J. H. Lamprey. ANATOMICAL AND PHYSIOLOGICAL SCIENCES. COMMITTEE OP SCIENCES, T. ANATOMY AND PnTSlOLOGY. 1833. 1834. 1835. 1836. 1837. 1838. 1839. 1840. Cambridge... Edinburgh... Dr. Haviland Dr. Abercrombie Dr. Bond, Mr. Paget. Dr. Roget, Dr. William Thomson, Dublin Bristol Liverpool . . . Newcastle . . . Birmingham Glasgow . . . SECTION E. (UNTin 1847.) ANATOMY AND MEDICINE. Dr. Pritchard 'Dr. Harrison, Dr. Hart. Dr. Roget, F.R.S Prof. W. Clark, M.D T. E. Headlam, M.D John Yelloly, M.D., F.R.S. James Watson, M.D 1841. Plymouth.. 1843. 1843. 1844. Manchester. Cork York P. M. Roget, M.D., Sec.R.S. Edward Holme, M.D., P.L.S. Sir James Pitcairn, M.D J. C. Pritchard, M.D Dr. Symonds. Dr. J. Carson, jun., James Long, Dr. J. R. W. Vose. T. M. Greenhow, Dr. J. R. W. Vose. Dr. G. O. Rees, F. Ryland. Dr. J. Brown, Prof. Couper, Prof. Reid. Dr. J. Butter, J. Fuge, Dr. R. S. Sargent. Dr. Chaytor, Dr. R. S. Sargent. Dr. John Popham, Dr. R. S. Sargent. I. Erichsen, Dr. R. S. Sargent. SECTION E. PHYSIOLOGY. 184.'5. Cambridge .{Prof. .T. Haviland, M.D. 1 846. Southam pton 1847. Oxford* ... Prof. Owen, M.D., F.R.S.. Prof. Ogle, M.D., F.R.S.. . Dr. R. S. Sargent, Dr. Webster. C. P. Koele, Dr. Layeock, Dr. Sargent. Dr. Thomas K. Cliambers, W. P. Ormerod. 18.50. 18.55. 1857. 1858. 1859. 1860. 1861. 1862. 1863. 1864. 1865. Edinburgh Glasgow . . . Dublin Leeds Aberdeen ... Oxford Manchester . Cambridge . Newcastle... Bath Birminghmf. PHYSIOLOGICAL SUBSECTIONS OF SECTION D. IProf. Bennett, M.D., F.R.S.E. IProf. Allen Thomson, F.R.S. ... iProf. R. Harrison, M.D Sir Benjamin Brodie, Bart.. F.R.S. IProf. Sharpev, M.D., Sec.R.S. ... |Prof. G. Rolleston, M.D., F.L.S. Dr. John Daw, F.R.S.L. & E. ... C.E.Paget, M.D Prof. Rolleston, M.D., F.R.S. ... Dr. Edward Smith, LL.D., F.R.S. Prof. Aclaud, M.D., LL.D., F.R.S. Prof. J. H. Corbett, Dr. J. Struthers. Dr. R. D. Lvons, Prof. Redfern. C. G. Wheelhouse. Prof. Bennett, Prof. Redfern. Dr. R. M'Donnell, Dr. Edward Smith. Dr. W. Roberts, Dr. Edward Smith. G. F. Helm, Dr. Edward Smith. Dr. D. Embleton, Dr. W. Turner. J. S. Bartrum, Dr. W. Turner. Dr. A. Fleming, Dr. P. Heslop, Oliver Pembleton, Dr. W. Turner. GEOGRAPHICAL AND ETHNOLOGICAL SCIENCES. [For Presidents and Secretaries for Geography previous to 1851, .see Section C, p. rrrii.] 1846. Southampton 1847. Oxford 1S4S. Swansea ... 1849. Birmingham 1850. Edinburgh.. ETHNOLOGICAL SUBSECTIONS OF SECTION D. Dr. Pritchard IDr. King. Prof. H. H. Wilson, M.A IProf. Buckley. G. Grant Francis. IDr. R. G. Lath.am. Vicc-Admiral Sir A. Malcolm ...iDaniel Wilson. * By direction of the G eneral Committee at Oxford, Sections D and E Were incorporated under tlie name of " Section D— Zoology and Botany, including Phvsiology " (seep, ssxiv). Tn>; Section being then vacant was assigned in 1851 to Geography." T Vide note on preceding page. PRESIDENTS AND SECRETARIES OF THE SECTIONS. XXXvIi Date and Place. Presidents. Secretaries. SECTION E. GEOGEAPHT AND ETHNOLOGY. 1851. 1852, 1853. 1854. 1855. 1856. 18.57. 1858. 1859. 1860. 1861. 1862. 1863. 1864. 1865. 1866. 1867. 1868. 1869. 1870. 1871. 1872. 1873, Ipswich . . . Belfast Hull Liverpool . . . Glasgow . . . Cheltenham Dublin Leeds Aberdeen ... Oxford Manchester . Cambridge . Newcastle... Bath Birmingham Nottingham Dundee Norwich ... Exeter Liverpool . . . Edinburgh. Brighton ... Bradford ... Sir R. I. Murchison, E.E.S., Pres R.G.S. Col. Chesney, E.A., D.C.L., R. G. Latham, M.D., F.R.S. Sir E. L Murchison, D.C.L., E E S Sir J. Eichardson, M.D., F.E.S. Col. Sir H. C. Eawlinson, K.C.B. HenthawnTodd, Pres. Rev. Dr. J E.I.A. Sir E. I F.R.S. Rear-Admir.al Sir James Ross, D.C.L., F.R.S. Sir E, L Murchison, F.R.S. John Crawfurd, F.R.S. , Murchison, G.C.St.S. Clerli Francis Gulton, F.R.S. .. K.C.B. K.C.B. Sir R. I. Murchison, F.R.S. Sir R. I. Murchison, F.R.S. Major-General Sir R. Rawlinson, M.P., K.C.B., F.R.S. Sir Charles Nicholson, Bart., LL.D. Sir Samuel Baker, F.R.G.S Capt. G. H. Richards, R.N., F.E.S , E. Cull, Eev. J. W. Donaldson, Dr- Norton Shaw. E. Cull, E. MacAdam, Dr. Norton Sliaw. E. Cull, Rev. H. W. Kemp, Dr. Nor- ton Shaw. Richard Cull, Rev. H. Higgins, Dr. Ihne, Dr. Norton Shaw. Dr. W. G. Blaekie, R. Cull, Dr. Nor- ton Shaw. E. Cull, F. D. Hartland, W. H. Eum- sey, Dr. Norton Shaw. E. Cull, S. Ferguson, Dr. E. E. Mad- den, Dr. Norton Shaw. E.Cull, Francis Gallon, P.O'Callaghan, Dr. Norton Sliaw, Thomas Wright. Eichard Cull, Professor Geddes, Dr. Norton Shaw. D.C.L., Capt. Burrows, Dr. J. Hunt, Dr. C. Lempriere, Dr. Norton Shaw. Dr. J. Hunt, J. Kingsley, Dr. Norton Shaw, W. Sj)ottiswoode. J. W. Clarke, Eev. J. Glover, Dr. Hunt, Dr. Norton Shaw, T. Wright. C. Carter Blake, Hume Greenfield, C. E. Markham, E. S. Watson. H. W. Bates, C. E. Mnrkbam, Capt. E. M. Murchison, T. Wriglit. II. W. Bates, S. Evans, G. Jabet, C. R. Markliam, Tlionias 'Wright. H. W. Bates, Eev. E. T. Cirsius, E. H. Major, Clements E. Markliam, D. W. Nash, T. Wright. H. W. Bates, Cyril Graham, C. E. Markham, S. J.Mackie, E. Sturrock. T. Baincs, H. W. Bates, C. E. Mark- ham, T. Wright. SECTION E (continued) Sir Bartle Frere, K.C.B., LL. F.R.G.S. SirE. I. Murchison. Bt.,K.C. LL.D., D.C.L., F.R.S., F.G Colonel Yule, C.B., F.R.G.S. . Francis Galton, F.R.S Sir Rutherford Alcock, K.C.B. -GEOGEAPHT. D., H. W. Bates, Clements E. Markham, J. H. Thomas. .B., H. W. Bates, David BiLxton, Albert .S. J. Mott, Clements E. Markham. Clements E. Markham, A. Buchan, J. H. Thomas, A. Keith Johnston. H. W. Bates, A. Keith Johnston, Eev. J. Newton, J. H. Thomas. H. W. Bates, A. Keith Johi ,ston. Cla- ra ents E. Markham. STATISTICAL SCIENCE. COMMITTEE OF SCIENCES, TI. STATISIICS 1833. Cambridge .IProf. Babbage, F.E.S 1834. Edinburgh .{Sir Charles Lemon, Bart. ... J. E. Drin'cwater. Dr. Cleland, C. Hope Maclean. SECTION F. STATISTICS. 183.5. Dublin 'Charles Babbage, F.R.S 1836. Bristol Sir Charles Lemon, Bart., F.E.S. W. Grog, Prof. Longfiekl. Eev. J. B. Bromby, C. B. Fripp, James Hevwood. XXXVlll REPOET 1873. Date and Place. 1837. Liverpool... 1838. Newcastle... 1839. Birmingham 1810. Glasgow ... 1841. Plymouth... 1812. Manchester. 1843. Cork 1844. York 184.5. Cambridge . 1846. Southampton 1847. Oxford 1848. .Swansea . . . 1849. Birmingham 1850. Edinburgh .. 18.51. Ipswich 1852. Belfast 1853. Hidl 1854. Liverpool ... 1855. Glasgow Presidents. Secretaries. Et. Hon. Lord Sandon W. E. Greg, W. Langton, Dr. W. C. Tayler. W. Cargill, J. Heywood, W. E. Wood. F. Clarke, E. W. Eawson, Dr. W. C. Tayler. C. E. Baird, Prof. Eamsay, E. W. Eawson. Eev. Dr. Byrth, Eev. E. Luney, E. W. Eawson. Eev. E. Luney, G. W. Ormerod, Dr. W. C. Tayler. Dr. D. Biilleu, Dr. W. Cooke Tayler. J. Fletcher, J. Heywood, Dr. Laycook. J. Fletcher, W. Cooke Tayler, LL.D. J. Fletcher, F. G. P. Neison, Dr. W. C. Tayler, Eev. T. L. Shapcott. Eev. W. H. Cox, J. J. Danson, F. G. P. Neison. J. Fletcher, Capt. E. Shortrede. Dr. Finch, Prof. Hancock, F. G. P. Neison. Prof. Hancock, J. Fletcher, Dr. J. Stark. J. Fletcher, Prof. Hancock, the Archbishop of Prof. Hancock, Prof. Ingram, James MacAdam, Jun. Edward Chesliire, WiUiam Newmarch, E. Cheshire, J. T. Danson, Dr. W. H. Duncan, W. Nevnnarch. J. A. Campbell, E. Cheshire, W. New- march, Prof. E. H. Walsh. Colonel Sykes, F.E.S Henry Hallam, F.E.S Et. Hon. Lord Sandon, F.E.S.. M.P. Lieut.-Col. Sykes, F.E.S G. W. Wood, M.P., F.L.S Sir C. Lemon, Bart., M.P Lieut.-Col. Sykes, F.E.S., F.L.S. Et. Hon. The Eavl Eitzwilliam... G. E. Porter, F.E.S Travers Twiss, D.C.L., F.E.S. ... J. H. Vivian, M.P., F.E.S Et. Hon. Lord Lyttelton , Very Eev. Dr. John Lee, V.P.E.S.E. Sir John P. BoUeau, Bart. His Grace Dublin. James Heywood, M.P., F.E.S< Thomas Tooke, F.E.S E. Monckton Milnes, M.P. .. SECTION p (continued). — economic science and statistics. 1856. Cheltenham 1857. Dublin 1858. Leeds 1859. Aberdeen .. 1860. Oxford 1861. Manchester 1862. Cambridge.. 1863. Newcastle . . . 1864. Bath 1865. Birmingham 1866. Nottingham 1867. Dundee 18G8. Norwich ... Et. Hon. Lord Stanley, M.P. .. His Grace the Archbishop of Dubhn, M.E.I.A. Edward Baines Col. Sykes, M.P., F.E.S. ... Nassau W. Senior, M.A. ... WiUiam Newmarch, F.E.S. Edwin Chadwick, C.B William Tite, M.P., F.E.S. Et. Hon. Lord Stanley, LL.D., M.P. Prof. J. E. T. Eogers Eev. C. H. Bromby, E. Cheshire, Dr. W. N, Hancock Newmarch, W. M. Tartt. Prof Cairns, Dr. H. D. Hutton, W. Newmarch. T. B. Baines, Prof. Cairns, S. Brown, Capt. Fishbourne, Dr. J. Strang. ..[Prof. Cairns, Edmmid Macrory, A. M. Smith, Dr. John Strang. Edmund Macrory, W. Newmarch, Eev. Prof. J. E. T. Eogers. David Chadwick, Prof. E. C. Christie, E. Macrory, Eev. Prof. J. E. T. Eogers. H. D. Macleod, Edmund Macrory. ..:T. Doublcday, Edmund Macrory, I Frederick Purdy, James Potts. William Farr, M.D., D.C.L., E. Macrory, E. T. Pavne, F. Purdy. F.E.S. G. J. D. Goodman, G. J. Johnston, E. Macrory. E. Birldn, Jun., Prof. Leone Levi, E. Macrory. M. E. Grant Duff, M.P jProf. Leone Levi, E. Macrory, A. J. Warden. Samuel Brown, Pres. Instit. Ac-:Eev. W. C. Davie, Prof. Leone Levi, tuaries. I PRESIDENTS AND SECllETARIES OF THE SECTIONS; XXXIX Date and Place. Presidents. Secretaries. 1869. Exeter 1870. Liverpool... 1871. Edinburgh 1.^72. Brighton ... 1S73. Bradford ... Et. Hon. Sir Stafford H. North- cote, Bart., C.B., M.P. Prof. W. Stanley Jevons, M.A. . , Et. Hon. Lord Neaves Prof. Henry Fawcett, M.P. ... Rt. Hon. W. E. Forster, M.P. Edmund Macrory, Frederick Purdy, Charles T. D. Acland. Chas. E. Dudley Baxter, E. Macrory, J. Miles Moss. J. G. Fitch, James Meikle. J. G. Fitch, Barclay Phillips. J. G. Fitch, Swire Smith. MECHANICAL SCIEI^CE. SECTION G. MECHAjSTICAL SCIENCE. 1836. 18:37. 1838. 1839. Bristol ... Liverpool ... Newcastle Birmingham 1840. Glasgow 1841. 1842. 1843. 1844. 1845. 1846. 1847. 1848. 1849. 1850. 1851. 1852. 1853. 1854. 1855. 1856. 1857. 18.58. 1859. 1860. 1861. 1862, 1863. 1864. 1865. 1866. 1867. 1868. Plymouth . . . Manchester . Cork York Cambridge Southami)ton Oxford Swansea Birmingham Edinburgh .. Ipswich Belfast Hull Liverpool ... Glasgow ... Cheltenham Dublin Leeds Aberdeen ... Oxford Manchester . Cambridge .. Newcastle . . . Bath Birmingham Nottingham Dundee Norwich ... Davies Gilbert, D.C.L., F.E.S. Eev. Dr. Eobinson Charles Babbage, F.E.S. John Taylor, F.E.S Eev. Prof. Willis, F.E.S Prof. J. Macneill, M.E.I. A.... John Taylor, F.E.S George Eennie, F.E.S Eev. Prof. Willis, M.A., F.E.S. . Eev. Prof. Walker, M.A., F.E.S Eev. Prof. Walker, M.A., F.E.S. Eobert Stephenson, M.P., F.E.S, Eev. Dr. Eobinson , WiUiara Cubitt, F.E.S John Walker,C.E., LL.D., F.E.S. T. G. Bunt, G. T. Clark, W. West. Charles Vignoles, Thomas Webster. E. Hawthorn, C. Vignoles, T. Webster. Prof. Willis, F^E.S., and EobertW. Carpmael, WiUiam Hawkes, Tho- Stephenson. mas Webster. Sir John Eobinsson J. Scott Eu.sseU, J. Thomson, J. Tod, C. Vignoles. Henry Chatfield, Thomas Web.ster. J. F. Bateman, J. Scott Eussell, J. Tliomson, Charles Vignoles. James Thomson, Eobert Mallet. Charles Vignoles, Thomas Webster. Eev. W. T. Kingsley-. WiUiam Betts, Jim., Charles Manby. J. Glynn, E. A. Le Mesurier. R. A. Le Mesm-ier, W. P. Struve. Charles Manby, W. P. Marshall. Dr. Lees, David Stephenson. John Head, Charles Manby. John F. Bateman, C. B. Hancock, Charles Manby, James Thomson. James Oldham, J.Thomson, W. Sykes Ward. John Grantham, J. Oldham, J. Thom- son. L. Hill, Jun., William Eamsay, J, Thomson. C. Atherton, B. Jones, Jun., H. M. Jeffery. Prof. Downing, W. T. Doyne, A. Tate, James Thomson, Henry Wright. J. C. Dennis, J. Dixon, H. Wright. E. Abernethy, P. Le Neve Foster, H. Wright. P. Le Neve Foster, Rev. F. Harrison, Henry Wright. P. Le Neve Foster, John Eobinson, H. Wright. W. M. Fawcett, P. Le Neve Foster. P. Le Neve Foster, P. Westmacott, J. F. Sijencer. P. Le Neve Foster, Eobert Pitt. P. Le Neve Foster, Henry Lea, W. P. Marshall, Walter May. Le Neve Foster, J. F. Iselin, M. A. Tarbottom. Lo Neve Foster, John P. Smith, W. W. Urquhart. Le Neve Foster, J. F. Iselin, C. Manby, W. Smith. William Fairbaim, C.E., F.E.S.. John Scott Eussell, F.E.S W. J. Macquorn Eankine, C.E.. F.E.S. George Eennie, F.E.S. f The Eight Hon. The Earl of Eosse, F.E.S. WiUiam Fairbaim, F.E.S Eev. Prof. WiUis, M.A., F.E.S. . Prof. W. J. Macquorn Eankine, LL.D,, F.E.S. J. F. Bateman, C.E., F.E.S William Fairbau-n, LL.D., F.E.S. Eev. Prof. Willis, M.A., F.E.S. . J. Hawkshaw, F.E.S Sir W. G. Armstrong, LL.D., F.E.S. Thomas Hawksley, V.P.Inst. C.E., F.G.S. Prof. W. J. Macquorn Eankine, LL.D., F.E.S. G. P. Bidder, C.E., F.E.G.S. ... xl REPORT 1873. Date and Place. Presidents. Secretaries. 1869. Exeter 1870. Liverpool... 1871. Edinburgh 1872. Brighton ... 1873. Bradford... C. W. Siemens, F.E.S. Chas. B. Vignoles, C.E. Prof. Fleeming Jenkin, P. J. Bramwell, C.E.... rF.E.S.'! F.E.S... . P. Le Neve Foster, H. Bauerman. H. Bauerman, P. Le Neve Foster, T. King, J. N. Shoolbred. H. Bauerman, Alexander Leslie, J. P. Smith. H. M. Brunei, P. Le Neve Foster, W. H. Barlow, F.E.S. J. G. Gamble, J. N. Shoolbred. Crawford Barlow, IT. Bauerman, S. II. Carbult, J. C. Hawkshaw, J. N. Shoolbred. List of Evening Lectures. Date and Place. 1842. Manchester 1843. Cork , 1844. York, 1845. Cambridge ., 1846. Southampton 1847. Oxford 1848. Swansea . 1849. Birmingham 1850. Edinburgh. 1851. Ipswich 1852. Belfast ... 1853. Hull Lecturer. Charles Yignoles, F.E.S. . Sir M. L Brunei E. I. Murchison Prof. Owen, M.D., F.E.S. Prof. E. Forbes, F.E.S. ., Dr. Eobinson Charles Lyell, F.E.S Dr. Falconer, F.E.S G. B. Airy, F.E.S., Astron.Eoyal E. I. Murchison, F.E.S Prof. Owen, M.D., F.E.S Charles Lyoll, F.E.S W. E. Grove, F.E.S Eev. Prof. B. Powell, F.E.S. Prof. M. Faraday, F.E.S. ... Hugh E. Strickland, F.G.S. John Percy, M.D., F.E.S W. Carpenter, M.D., F.E.S. ... Dr. Faraday, F.E.S Eev. Prof. WiUis, M.A., F.E.S Prof. J. H. Bennett, M.D., F.E.S.E. Dr. Mantell, F.E.S Prof. E. Owen, M.D., F.E.S. G. B. Airy, P.E.S., Astron. Eoy. Prof. G.G. Stokes.D.C.L., F.E.S Colonel Portlock, E.E., F.E.S. Prof. J. Phillips, LL.D., F.E.S. F.G.S. Eobert Hunt, F.E.S Subject of Discourse. The Principles and Constmiction of Atmospheric Eailways. The Thames Tunnel. The Geology of llussia. The Dinornis of New Zealand. The Distribution of Animal Life in the jEgean Sea. The Earl of Eosse's Telescope. Geology of North America. The Gigantic Tortoise of the Siwalik Hills in India. Progress of Terrestrial Magnetism. Geology of Eussia. Fossil Mammalia of the British Isles. Valley and Delta of the Misi?issippi. Properties of the Explosive substance discovered by Dr. Schonbein ; also some Eesearches of his own on the Decomposition of Water by Heat. Shooting-stars. Magnetic and Diamagnetic Pheno- mena. The Dodo (Bidus inepfus). Metallurgical operations of Swansea and its neighboin'liood. Eecent Microscopical Discoveries. Mr. Gassiot's Battery. Transit of different Weights with _ varying velocities on Eailwaj's. Passage of the Blood through the minute vessels of Animals in cou- nexion with Nutrition. Extinct Birds of New Zealand. Distinction between Plants and Ani- mals, and their changes of Form. Total Solar Eclipse of July 28, 1851. Eecent discoveries in the properties of Light. Eecent discovery of Eock-.salt at Car- rickfergus, and geological and pr;ic- ticalconsiderationseonnectedwith it. Some peculiar phenomena in the Geo- logy and Physical Geography of Yorkshire. Tlie present stato of Photography. LIST OP EVENING LECTURES. xli Date and Place. 1854. Liverpool ... 1855. Glasgow 1856. Clieltenhani Lecturer. Prof. II. Owen, M.D., F.R.S. ... Col. E. Sabiue, V.P.E.S Dr. W. B. Carpenter, F.R.S. ... Licut.-Col. n. Eawlinson Col. Sir H. Eawlinson , 1857. Dublin 1858. Leeds 1859. Aberdeen .. 1860. Oxford 1861. Manchester 1862. Cambridge 1863. Newcastle- on-Tyne. 1864. Bath 1865. Birmingham 1866. Nottingham 1867. Dundee Subject of Discourse. 1868. Norwich .... 1869. Exeter 1870. Liverpool ... 1871. TSdinburgh 1872. Brighton .., 1873. Bradford .. W. E. Grove, P.E.S Prof. W. Thomson, RE.S Eev. Dr. Livingstone, D.C.L. .. Prof. J. Phillips, LL.D., F.E.S Prof. E. Owen, M.D., F.E.S. .. SirE.I.Murchison, D.C.L Eev. Dr. Eobinson, F.E.S Eev. Prof. Walker, F.E.S Captain Shcrard Osborn, E.N. . Prof. W. A. Miller, M.A., F.E.S, G. B. Airy, F.E.S., Astron. Eoy. . Prof. Tyndall, LL.D., F.E.S. ... Prof. Odling, F.E.S Prof. Williamson, F.E.S James Glaisher, F.E.S. Prof. Eoscoe, F.E.S Dr. Livingstone, F.E.S. J. Beete Jukcs, F.E.S. ... William Huggins, F.E.S Dr. J. D. Ilooker, F.E.S Archibald Geikie, F.E.S Alexander Ilcrsehcl, F.E.A.S. J. Fcrgu.sson, F.R.S Dr. W. Odling, F.E.S Prof. J. Phillips, LL.D., F.E.S. J. Norman Lockyer, F.E.S.... Prof J. Tyndall, LL.D., F.E.S Prof. W. J. Macquorn Eankine, LL.D., F.E.S. F. A. Abel, F.E.S E. B. Tylor, F.E.S Prof. P. Martin Duncan, M.D., FES Prof. W. K. Chfford Anthropomorphous Apes. Progress of researches in Terrestrial Magnetism. Characters of Species. As.syrian and Babylonian Antiquities and Ethnology. Eecent discoveries in As.syria and Babylonia, with the results of Cunei- form research up to the present time. Correlation of Physical Forces. The Atlantic Telegraph. Eecent discoveries in Africa. The Ironstones of Yorkshire. The Fossil Mammalia of Australia. Geology of the Northern Highlands. Electrical Discharges in highly rare- fied Media. Physical Constitution of the Sun, Arctic Discovery. Spectrum Analysis. The late Eclipse of;the Sun. The Forms and Action of Water. Organic Chemistry. Tlie chemistry of the Galvanic Bat- tery considered in relation to Dy- namics. The Balloon Ascents made for the British Association. The Chemical Action of Light. Eecent Travels in Africa. Probabilities as to the position and extent of the Coal-measures beneath the red rocks of the Midland Coun- ties. The residts of Spectrum Analysis applied to Heavenly Bodies. Insular Floras. The Geological origin of the present Scenery of Scotland. The ]3resent state of knowledge re- garding Meteors and Meteorites. Archivology of tlie early Buddhist Monuments. Reverse Chemical Actions. Vesuvius. The Physical Constitution of the Stars and Nebulae. The Scientific Use of the Imagination. Stream-lines and Waves, in connexion with Naval Architecture. Some recent investigations and appli- cations of Explosive Agents. The Eelation of Primitive to Modern Civilization. Insect Metamorphosis. Prof. W. C. Williamson, F.E.S. Prof Clerk Maxwell F.R.S The Aims and Instruments of Scien- tific Thought. Coal and Coal Plants. Molecules. xlii REPORT — 1873. Date and Place. Lecturer. Subject of Discourse. Lectures to the Operative Classes. 1867. DuTidee.. 1868. Norwicli 1869. Exeter .. 1870. Liverpool , 1872. Brigbton , 1873. Bradford . Prof. J. Tyndall, LL.D., F.E.S. Prof. Huxley, LL.D., F.E.S. .. Prof. MiUer, M.D., F.E.S Sir John Lubbock, Bart., M.P., F.E.S. William Spottiswoode, LL.D., P T? S C. W.' SiemenB, D.C.L., F.E.S... Matter and Force. A jiiece of Chalk. Experimental illustrations of the modes of detecting the Composi- tion of the Sun and other Heavenly Bodies by the Spectrum. Savages. Sunshine, Sea, and Sky. Fuel. 1-1 ^ o o •* 0» CO O iO rt (M CI O -^ i-< %n CO o T)< -^ "^ I— t CO in o oooooooooooooooooo oooooooooooooooooo OCOOiOO'(^0^000»'^00000 ooooc^irac^iooMO.-iw»rau3cic^w as iM xliv REPORT 1873. Table shoivinr/ the Attendance and Receipts Date of Meeting. 1831, Sept. a/ . 1832, June 19 ., 1833, June 25 ., 1834, Sept. 8 ., 1835, Aug. 10 .. 1836, Aug. 22 .. 1837, Sept. n .. 1S38, Aug. 10 .. 1839, Aug. 26 .. 1840, Sept. 17 .. 1 841, July 20 .. 1842, June 23 .. 1843, Aug. 17 .. 1844, Sept. 26 .. 1845, June 19 .. 1S46, Sept. 10 .. 1847, June 23 .. 1S48, Aug. 9 1849, Sept. 12 .. 1850, July 21 .. 1851, July 2 1852, Sept. 1 1853, Sept. 3 .. 1854, Sept. 20 .. 1855, Sept. 12 .. 1856, Aug. 6 1857, Aug. 26 .. 185S, Sept. 22 .. 1859, Sept. 14 .. i860, June 27 .. 1 861, Sept. 4 1862, Oct, 1 1863, Aug. 26 .. 1864, Sept, 13 .. J865, Sept. 6 .. 1866, Aug. 22 .. 1867, Sept. 4 .., 1868, Aug. ig .., 1869, Aug. 18 .., 1870, Sept. 14 ... 187I; Aug. 2 1872, Aug. 14 .. 1873, Sept. 17 .. 1874, Aug. 19 •• Where held. York Oxford Cambridge Edinburgh Dublin Bristol Liverpool Newcastle-on-Tyne .. Birmingliam Glasgow Plymouth Manchester ..., Cork York Cambridge Southampton Oxford Swansea Birmingham .... Edinbiu-gh .... Ipswich Belfast Hull Liverpool Glasgow Cheltenham Dublin Leeds Aberdeen Oxford Manchester .... Cambridge .... Newcastle-on-Tyne .. Bath Birmingham .... Nottingham .... Dundee Norwich Exeter Liverpool Edinburgh .... Brighton Bradford , Belfast Presidents. The Earl Pitzwilliam, D.C.L. .. The Eev. W. Buckland, P.R.S. , The Rev. A. Sedgwick, RE.S... Sir T. M. Brisbane, D.C.L The Eev. Provost Lloyd, LL.D, The Marquis of Lansdowne The Earl of Burlington, RE.S. The Duke of Northumberland.. The Eev. W. Vernon Harcourt The Marquis of Breadalbane .. The Eev. W. Whewell, RE.S... The Lord Francis Egerton The Earl of Eosse, RE.S The Eev. G. Peacock, D.D Sir John P. W. Herschel, Bart. Sir Roderick I. Murchison, Bart Sir Eobert H. Inglis, Bart The Marquis of Northampton . . , The Eev. T. E. Eobinson, D.D. . Sir David Brewster, K.H G. B. Airy, Esq., Astron. Eoyal . Lieut.-General Sabine, F.E.S. ... William Hopkins, Esq., RE.S. . The Earl of Harrowby, RE.S. .. The Duke of Argyll, RE.S Prof. C. G. B.Daubeny, M.D.... The Eev. Humphrey Lloyd, D.D. Richard Owen, M.D., D".C.L. ... H.E.H. The Prince Consort . . . The Lord Wrottesley, MA William Pairbairn, LL.D.,F.E.S. The Eev. Prof. Willis, M.A. Sir William G. Armstrong, C.B. Sir Charles Lyell, Bart., MA... Prof. J. PhilHps, M.A.,LL.D... William E. Grove, Q.C., P.E.S. The Duke of Buccleuch, K.C.B Dr. Joseph D. Hooker, P.R.S. . Prof. G. G. Stokes, D.C.L Prof. T. H. Huxley, LL.D Prof Sir W. Thomson, LL.D.... Dr. W. B. Carpenter, P.R.S ... Prof A. W. Williamson, P.R.S. Prof. J. Tyndall, LL.D, P.R.S. Old Life Members. New Life Members. 169 6? 303 169 109 28 226 150 3'3 36 241 10 314 18 149 3 227 12 235 9 172 8 164 10 141 13 238 23 194 33 182 14 236 15 222 4^ 184 27 286 21 321 113 239 15 203 36 287 40 292 44 207 31 167 25 196 18 204 21 314 39 246 28 24s 36 212 27 ATTENDANCE AND RECEIPTS AT ANNUAL MEETINGS. xlv at Annual Meetings of the Association. Attended by Amount received during the Meeting. Sums paid on Account of Grants for Scientific Purposes. Old N Annual An ew nual Asso ciates. La dies. Fore igners. Total. Members. Men ibers. £ s. d. £ s. d. •• .. • • 353 900 1298 20 167 434 14 918 14 6 1350 1840 ,, ., .. ......... 1 [OO* 2400 34 1438 1-0 1353 891 28 1315 956 12 2 1595 11 1546 16 4 1235 1° I' 1449 17 8 1565 10 2 981 12 8 830 9 9 685 16 208 5 4 275 I 8 46 3 75 3 71 1 45 I 94 65 197 54 17 76 85 90 22 60* 33t 531* 160 9t ^07 270 4-95 376 160 172 196 203 197 35 1079 36 857 53 1260 15 9^9 39 40 25 707 93 33 +47 ^37 22 1071 963 159 19 6 iz8 42 510 ^73 H 1241 10S5 ° ° 345 18 61 47 244 '41 37 7'° 620 391 9 7 63 60 510 192 9 1108 iog5 304 6 7 56 57 367 .36 6 876 903 205 121 1 21 765 524 10 l8o2 1882 33° 19 7 142 ] 01 1 094 543 26 2133 2311 480 16 4 104 48 412 346 9 ii'5 1098 734 13 9 156 ) 20 900 569 26 2022 2015 507 15 3 III 9' 710 509 13 1698 1931 618 18 2 125 1 79 I 206 821 22 2564 2782 684 II 1 177 59 636 463 47 1689 1604 1241 7 1S4 1 25 I 589 791 15 3139 3944 iiii 5 10 150 57 433 242 25 1161 1089 1293 16 6 154 : .09 I 704 I 004 25 3335 3640 1608 3 10 182 1 03 1 119 I 058 13 2802 2965 1289 IS 8 215 1 49 766 5c8 23 1997 2227 1591 7 10 218 1 05 960 771 II 2303 2469 1750 13 4 193 J 18 I 163 771 7 2444 2613 1739 4 226 17 720 682 45 t 2°°4 2042 1940 229 ] 07 678 600 17 1856 1931 1572 303 95 1 103 910 14 2878 3096 1472 2 6 311 27 976 754 21 2463 2575 1285 ° 280 80 937 912 43 2533 2649 16S5 237 99 796 601 11 1983 2102 * Ladies were not admitted by purcba.scd Tickets until 1843. t Tickets for admission to Sections only. J Including Ladies. xlvi REPORT — 1873. OPPICEES OP SECTIONAL COMMITTEES PEESENT AT THE BRADPOED MEETING. SECTION A. MATHEMATICS AND PHYSICS. Pm/(7««1— Professor Henry J. S. Smith, M.A., LL.D., F.R.S. Vice-Presidents.— Pioiessov Caylev, M.A., F.R.S. ; James Glaislier, F.R.S. ; Pro- fessor G. Carey Foster, F.R.S. ; Professor R. Harley, F.R.S. ; Professor Hem-ici ; W. Hugo-ins, F.R.S. ; Professor Clerk-Maxwell, M. A., F.R.S. ; Professor Balfom- Stewart, F.R.S. /Sec;-rf«>7"c.s.— Professor W. K. Clifford, M.A. ; Professor Forbes, B.A., F.R.S.E. ; J. W. L. Glaislier, B.A., F.R.A.S. ; Professor A. S. Herscliel, B.A., F.R.A.S. SECTION B. CHEMISTET AND MINERALOGT, INCLtTDING THEIR APPLICATIONS TO AGKICULIURE AND THE ARTS. P;-es«V7ewi!.— Professor W. J. Russell, F.R.S. Vice-Presidents.— Bv. J. H. Gilbert, F.R.S. ; Dr. Gladstone, F.R.S. ; A. Vernon Harcourt, F.R.S. ; James Young, F.R.S. j Professor G. 0. Foster, B.A., F.R.S. ; Dr. C. W. Siemens, F.R.S. Secretaries.— Br. Armstrono-, F.C.S. ; Dr. Mills, F.C.S.j W. Chandler Roberts, F.C.S.j Dr. TlioriDe, F.R.S.E. SECTION 0. GEOLOGY, PmjWen;;.— Professor Phillips, M.A., LL.D., D.C.L., F.R.S., F.G.S. Vice-Presi(Ie?ds.—Siv Phillip Egerton, Bart, F.R.S. ; Professor T. M'K. Hughes, M.A., F.G.S. ; J. Gwvn Jeffreys, F.R.S. ; W. PengeUy, F.R.S., F.G.S. ; Professor W. 0. Williamson, F.R.S. Secretaries.—!.. C. MiaU, F.G.S. ; R. H. Tiddeman, F.G.S. ; W. Topley, F.G.S. SECTION D. BIOLOGY. Presided. — Professor Allman, M.D., LL.D., F.R.S. Vice-Presidenfs.-Fi-oiessov Balfour, F.R.S. ; Dr. Beddoe, F.R.S. ; Sir Walter Elliott, K.C.S.I. ; Dr. Hooker, C.B., F.R.S. ; Professor Rutherford, M.D. j Dr. Burdon Sanderson, F.R.S. ; A. R. Wallace, F.R.G.S. Secretaries. — Professor Thiselton-Dyer, B.A., B.Sc, F.L.S. ; Professor Lawson, M.A., F.L.S. ; R. M'Lachlan, F.L^S. ; Dr. Pye-Smith ; E. Ray Lankester, M.A. : F. W. Rudler, F.G.S.; J. H. Lamprey. SECTION E. GEOGRAPHY AND ETHNOLOGY. President. — Sir Rutherford Alcock, K.C.B. Vice-Presidcnts.—Admiral Sir Edward Belcher, F.R.S. ; F. Gallon, F.R.S. ; Cap- tain M. S. Nolloth, R.N., F.R.G.S.; Admiral E. Oumianney, C.B., F.R.8., Major-General Strachey, F.R.S. Secretaries.— U. W. Bates, F.L.S., F.R.G.S.; A. Keith Johnston, F.R.G.S.: Clements R. Markham, C.B., F.R.S., F.R.G.S. SECTION E. ECONOMIC SCIENCE AND STATISTICS. Presidetd.— Right Hon. W. E. Forster, M.P. Vice-Presidents.— Lord Haughton, D.C.L., F.R.S., F.R.G.S.; Edwaid Baines, M.P. ; Sir James Alexander ; Edward Miall, M.P. ; F. S. Powell, M.l". ; Duncan McLaren, M.P. ; Samuel Brown, F.S.S. ; James Heywood, M.A., F.R.S. Secretaries. — J. G. Fitch ; Swire Smith. SECTION G. — MECHANICAL SCIENCE. President— W. H. Barlow, F.R.S. Vice-Presidents.— F. J. Bramwell, F.R.S.; Admiral Sir E. Belcher, K.C.B. ; P. le Neve Foster, M.A. ; Sir John Hawkshaw, F.R.S. ; C. W. Merritield, F.R.S. : James R. Napier, F.R.S. ; C. W. Siemens, F.R.S. ; Thomas Webster, Q.C. , F.R.S. > ^ > Secretaries.— Gr&vdordi Barlow, B.A. ; H. Bauevman, F.G.S. ; E. H. Carbutt, C.E. ; John Clarke Hawkshaw, M.A., F.G.S.: C. W. Siemens, F.G.S. ; J. N. Shool- bred, F.G.S. > > , > , OFFICERS AND COUNCIL, 1873-74. TRUSTEES (PERMANENT). General Sir Edward Sabike, K.C.B.. E.A., D.C.L., F.E.S. Sir Pnil-iP DE M. Gkev-Egeetox, Bart., M.P., F.E.S., F.G.S. Sir Joi»- LUUBOCX, Bart., M.P., F.E.S., F.L.S. PRESIDENT. PEOFESSOE A. W. WILLIAMSON, Pn.D., F.E.S., F.C.S. VICE-PRESIDENTS. The Right Hon. theEABL of Eosse, F.E.S.,F.E.A.S. | Sir JoHx HA-n-KSnAW, F.E.S^.PXJ.SJ The Ei^ht Hon. LoKD Houghton, D.C.L., F.E.S. The Eight Hon. W. E. FoESTEE, M.P. Matthkw W. Thompson, Esq., Mayor of Bradford. J. P. Gassiot, Esq., D.C.L., LL.D., F.E.S. Professor Phillips, D.C.L., LL.D., F.E.S. PRESIDENT ELECT. PROFESSOR J. TYNDALL, D.C.L., LL.D., F.E.S. VICE-PRESIDENTS ELECT. The Eight Hon. the Eael of Enniskillen, D.C.L., F.E.S. The Right Hon. the Eabl of EoSSE, F.E.S., F.E.A.8. Sir EiCHAED Wallace, Bart,, M.P. The Eev. Dr. Henby, President of Queen's College, Belfast. Dr. T. Akdeews, F.R.S., F.C.S. Eev. Dr. Eobikson, F.E.S., P.R.A.S. Professor Stokes, D.C.L., Sec.E.S. LOCAL SECRETARIES FOR THE MEETING AT BELFAST. W. QiTABTUS EwART, Esq. Dr. P. Eedfebn. T. Sinclaib, Esq., J. P. LOCAL TREASURER FOR THE MEETING AT BELFAST. William J. C. Allen, Esq. ORDINARY MEMBERS Beddoe, John, M.D., F.E.S. BeamweLL, F. J., Esq., C.E., F.E.S. Debus, Dr. H., F.E.S. De La Eue, Waeeen, Esq., D.C.L., T.E.S. Evans, .John, Esq., F.E.S. Fitch, J. G., Esq., M.A. Flo-wee, Professor W. H., F.E.S. FosTEE, Prof. G. C, r.E.S. Galton, Feancis, Esq., F.E.S. HiEST, Dr. T. Aecher, F.E.S. HUGGINS, William, Esq., D.C.L., F.E.S. Jeffbeys, J. GwYN, Esq, F.E.S. LOCKYEB, J. N., Esq., F.E.S. OF THE COUNCIL. Maxwell, Professor J. Clebk, P.H.S, Meeeifield, C. W., Esq., F.E.S. KoBTHCOTE,Rt.Hon.SirSTAFFonDH.,Bt.,M.P. Omm.^kney, Admiral E., C.B., F.E.S. Pengelly, W., Esq., F.E.S. Peestwich, J., Esq., F.E.S. EusSELL, Dr. W. J., F.E.S. ScLATEE, Dr. p. L., F.E.S. Siemens, C. W., Esq, D.C.L., F.E.S. Smith, Professor H. J. S., F.E.S. Steacuey, Mojor-General, F.E.S. SiEANGE, Lieut.-Colonel A., F.E.S. EX-OFFICIO MEMBERS OF THE COUNCIL. The President and President Elect, the Vice-Presidents and Vice-Presidents Elect, the General and Assistant General Secretaries, the General Treasurer, the Trustees, and the Presidents of former years, viz. : — Richard Owen, M.D., D.C.L. Sir W. Fairbairn, Bart., LL.D. The Eev. Professor Willis, F.E.S. Sir W. G. Armstrong, C.B., LL.D. Sir Chas. Lyell, Bart, M.A.,LL.D. Professor Phillips, M.A., D.C.L. Sir William R. Grove, F.R.S. The Duke of Devonshire. The Rev. T. R. Robinson, D.D. Sir G. B. Airv, Astronomer Royal. General SirE. Sabine, K.C.B. The Earl of Harrowby. The Duke of Argyll. The Eev. H. Lloyd, D.D. The Duke of Bucclcuch, K.B. Dr. Joseph D. Hooker, D.C.L. Professor Stokes, D.C.L. Prof. Huiley, LL.D., Seo.R.S. Prof Sir W. Thomson, D.C.L. Dr. Carpenter, F.R.S. GENERAL SECRETARIES. Capt. Douglas Galton, C.E., R.E., F.R.S., F.G.S., 12 Chester Street, Grosvenor Place, Loudon, S.W Prof. Michael Foster, M.D., F.R.S., Trinity College, Cambridge. ASSISTANT GENERAL SECRETARY. Geokge Gbiffith, Esq., M.A., F.C.S., Harrow-on-the-hill, Middlesex. GENERAL TREASURER. William Spottiswoode, Esq., M.A., LL.D., F.E.S., F.E.G.S., TO Grosvenor Place, London, S.W AUDITORS. J. Gwyn Jeffreys, Esq., F.E.S. Professor Phillirs, F.E.S. Professor Syhestsr, F.E.S. xlviii REPORT — 1873, Report of the Council for the Year 1872-73 presented to the General Committee at Bradford, on Wednesday , September 1.7th, 1873. During the past year the Council have received Reports from the General Treasurer ; and his account for the year will bo presented to the General Committee this daj''. The Council have had under their consideration the three resolutions which were referred to them by the General Committee at Brighton, They beg to report upon the action they have taken upon each case. First Resolution. — " That the Council be requested to take such steps as they deem desirable to induce the Colonial Office to afford sufficient aid to the Observatory at Mauritius to enable an investigation of the Cyclones in the Pacific Ocean to be carried on there"*. In accordance with this Resolution the following correspondence took place between Dr. Carpenter, the President of the Association, and the Right Honourable the Earl of Kimberley, Secretary of State for the Colonies : — " British Association for tbc Advancement of Science, 22 Albemarle Street, W., December 20, 1872. " My Loud, — On behalf of the British Association, I have the honour to bring under your Lordship's notice the following statement respecting the position of the Observatory at the Mauritius : — " The Mauritius Observatory is for the most part a Meteorological and Magnetical Observatory. As a Meteorological station, Mauritius is most important ; and tlie present Director of the Observatory, Mr. Charles Meldrum, has taken advantage of his position to work out several important Meteoro- logical problems as far as his means have allowed him. '•'■ He has fostered the growth, if he did uot originate, the Meteorological fjociety of Mauritius, of which he is the active Secretarj-, and his researches have \>QQa materially aided by these means. " He has collated the logs of vessels crossing the Indian Ocean, extending over a period of between twenty and thirty years, and has derived from these some very important results. In the first place, it has been almost established by these observations that the behaviour of the barometer at the ilauritius affords an indication of storms taking place between that island and the Cape of Good Hope. By a study of these logs of ships he is also able to tell in what directions such storms travel, and thus he is able to give very valuable advice to ships' masters who should happen to be at the Mauritius. Moreover, Mr. Meldrum's recent observations tend to show that the cyclones in the Indian Ocean are periodical, and occur most frequently during those years when there are most sun-spots. " In addition to this work, Mr. Meldrum's duties require him constantly to attend to the routine work of his observatory, to keep the time, &c. He is almost unprovided with assistants ; and if he happens to be unwell the current work of the observatory is liable in a measure to be stopped. On account of overwork, Mr. Meldrum has lately been unwell for two months, although uot so unwell as to put a stop to all his scientific labours. ^ ♦ The resolution -was adopted by the Council, with the following modification: — "That the Council take steps to induce the Colonial Office to afford sulficient pecuniary aid to the Observatory at Slauritius to enable an investigation of Cyclones to be carried on there." REPORT OF THE COUNCIL. xlix " The importance of maintaining the sequence of the observations in the Mauritius Observatory, of further collating the logs of ships, and of con- tinuing the inquiry into the periodicity of cyclones, has induced the British Association to urge upon your Lordship the necessity of affording additional assistance to Mr. Meldrum, to enable him to pursue these labours and perform his duties in a satisfactory manner. " It may be assumed that such assistance, to be efficient, will cost about ^300 a year beyond the present cost of the establishment ; and if it is to be of value for the purpose of the investigation into the periodicity of cyclones, this additional allowance will have to be continued for a period of about ten years. " I trust that the scientific importance of this subject will induce your Lordship to give this matter your favourable consideration, and to place Mr. Meldrum in a position to complete the inquiries he has commenced with so much success. " I have the honour to be, " My Lord, "Your most obedient Servant, (Signed) " William B. Caepenteb, President of the British Association." " The Rhjht Hon. the Earl of Kimherley, Secretary of State for Colo7iies." " Downing Street. 19th December, 1872. " Sir, — I am directed by the Earl of Kimherley to acknowledge the receipt of your letter of the 10th instant, urging, on behalf of the British Association, the necessity of affording additional assistance to Mr. Meldrum in his labours at the Mauritius Observatory. " The Colonial Government is well aware of the value of the Meteoro- logical researches now carried on at their Observatory by Mr. Meldnim ; but the state of the finances of the Colony is such that no increase can be made to any of the Government establishments except on urgent grounds. - " The Secretary of State will, however, in deference to the wish ex- pressed by the British Association, forward a copy of your letter to the Governor for his consideration and report. " I am, Sir, " Your obedient Servant, (Signed) " E. M. Meade." " Downing Street, 18th February, 1873. " Sir, — With reference to my letter of the 19th December last, I now forward to you, by the Earl of Kimberley's desire, the copy of a despatch which has been received from the Governor of Mauritius on the subject of affording assistance to Mr. Meldrum of the Mauritius Observatory. Lord Kimherley regrets that he cannot authorize any further charge for this service on the Colonial llevenue. " I am, Sir, " Your obedient Servant, (Signed) " H. T. Holland." 1873. fl 1 REPORT — 1873. Sir A. H, Gordon to tJie Earl of Kimherley. " Grovernment House, Mah^ Seychelles, 15th January, 1873. " My Lord, — I have had the honour to receive your Lordship's despatch (No. 302) of the 20th ultimo ou the subject of the assistance to be afforded to Mr. Meldrum of the Mauritius Observatory. " 2. Some slight increase was made in this year's estimates to the amount voted for this purpose, but not to the extent proposed by the British Association. " 3. The -whole stibjcct is one in respect to which I should be glad to be informed of your Lordship's views and wishes. " 4. It is admitted, and indeed the increased grant is urged by the British Association on this ground, that the benefit of Mr. Meldrum's investigations is of general application, and that it is the advancement of science, and not any special interest of Mauritius itself that is concerned. Under these cir- cumstances I confess that it seems to me hardly just that the revenue of Mauritius should bear the whole burden of these investigations, and that the Imperial Treasury, or, at aU events, the Meteorological Society, might be fairly called upon to defray a part of the expenses incurred. " I have &c., (Signed) <' Aethub Gordon." " Tlie Right Hon. tJie Earl of Kimlerley, 6fc. &fcP In consequence of this communication the Council requested the President to urge upon the Lords Commissioners of Her Majesty's Treasury the desirability of aifording such pecuniary aid to the Mauritius Observatory as would enable the Director to continue his observations on the periodicitj' of Cyclones; and an intimation has been received from Her Majesty's Govern- ment that an inquiry into the condition, size, and cost of the Establishment of the Mauritius is now being conducted by a Special Commission from England, pending which inquiry no increase of expenditure upon the Observatory can be sanctioned ; but that when the results of this inquiry shall be made known the Secretary of State for the Colonies will direct the attention of the Governor to the subject. Second Resolution. — " That, in the event of the Council having reason to believe that any changes aflPecting the acknowledged efficiency and scientific character of the botanical establishment at Kew are contemplated by the Government, the Council be requested to take such steps as in their judgment wiO. be conducive to the interests of botanical science in this country." The Council have not deemed it necessary to take any action upon this Resolution. Third Resolution. — "That the Council be requested to take such steps as they may deem desirable to urge upon the Indian Government the pre- paration of a Photoheliograph and other instruments for solar observa- tion, with the view of assisting in the observation of the Transit of Venus in 1874, and for the continuation of solar observations in India." The Council communicated with His Grace the Duke of Argyll, the Secretary of State for India, upon the subject, with the result cxi)lained in the following correspondence : — REPOKT or THE COUNCIL. " British Association for the Advancement of Science, 22 Albemarle Street, W., November 27th, 1872. " My Lord Duke, — On behalf of the British Association, I have the honour to urge upon your Grace's consideration the importance of making adequate preparation in India for the observation of the Transit of Venus in 1874, as •well as of making provision for the continuation of solar observations in India, a matter to -which the Council attach special importance. " The observations ought to comprise both eye and photograpliie records ; and the following instruments are specially recommended by the Council as those which it is desirable to procure at once. The photographic records shoiild be made in the manner determined upon by the Astronomer Royal and by M. Otto Struve for the llussian Government — namely, by means of a Photoheliograph, on the principle of the instrument which has been worked at the Ivew Observatory during ten years, but improved both in the optical and mechanical parts. " Tor eye-observations it will be desirable to have a Telescope of the greatest excellence, of 6-iuch aperture, mounted cquatorially iu the best manner, with a clockwork driver. It is also desirable to have a 4-iuch telescope, mounted cquatorially, and driven by clockwork. '* A transit instrument with clock, and one or two chronometers, and also an Altazimuth Instrument. "As the 6-inch equatorial would be available afterwards for Sun Observa- tions, it would be desirable to fit it with a Spectroscope of sufiicient dispersive power to permit of the prominences being observed efficiently. "The Council would recommend that the Heliograph should be worked continuously iu India, inasmuch as such records are calciilated to throw much light upon the causes of climatic changes, and it is impossible in any one locality to secure a coutinuous record of the sun's activity : observations of this nature are about to be proceeded with at the Eoyal Observatorj', Greenwich ; but past experience has shown that, on the average, half the days in the year are unproductive, and it is hoped that if India cooperates the gaps may be filled up. " The Council of the Association trust that the importance of the subject win induce your Grace to give the matter a favourable consideration. " I have the honour to be, «' My Lord Duke, " Your most obedient Servant, (Signed) " W. B. Caepenxeb, President of the Briiisli Association" " Bis Grace The Bule of Argyll, K.G., Secretary of State for India." " India Office, December 13th, 1872. " Sir,' — I am directed by the Secretary of State for India in Council to acknowledge the receipt of yoiu- letter of the 27th ultimo, expressing the desii'e of the Council of the British Association that provision may be made in India for observation in that country of the Transit of Ycnus in 1874, and for a continuation of solar observations in future, " In reply, I am desired by the Duke of Argyll to state that His Grace has been in correspondence with the Astronomer Eoyal and the Government of India with reference to an observation in Northern India of tlie Transit of (?2 Hi HE PORT — 1873. Venus, and that a phofcoheliograph and other instruments are now in course of preparation for this object. " With reference to the continuation of future solar observations in India, I am to add that there is a Government Astronomer in the Madras Presi- dency, and a Superintendent of the Colaba Observatory in the Bombay Presidency, besides Officers employed in the Survey Department in Bengal and the North-western Provinces, all of whom are engaged from time to time in recording observations of this nature. " I am, Sir, " Your obedient Sei-vant, (Signed) " Heksian Mekivalk." " William B. Carpenter, Esq., British Association, 22 Albemarle Street, W." " India Office, February 28th, 1873. " Sib, — "With reference to my letter of the 13th of December last, relative to an observation in India of the Transit of the planet Venus in December 1874, 1 am directed to state, for the information of the Council of the British Association for the Advancement of Science, that the Secretary of State for India in Council, having reconsidered this matter, and looking to the number of existing burdens on the revenues of India, and to the fact that the selection of any station in that country was not originally contemplated for 'eye- observations ' of the transit, has determined to sanction only the expendi- ture (=£356 7s. del.) necessary for the purchase and packing of a Photo- heliograph, and any further outlay that may be requisite for the adaptation of such instruments as may be now in India available for the pui'pose of the proposed observation. " The Duke of Argyll in Council has been led to sanction thus much of the scheme proposed by Lieut.-Colonel Tennant, in consequence of the recom- mendation submitted by the Astronomer lloyal in favour of the use of pho- tography for an observation of the transit at some place in Northern India. " I am. Sir, " Your obedient Servant, (Signed) " Herman Merivaee." " William B. Carpenter, Esq., British Association" The General Committee will recollect that a Committee was appointed at Exeter in 1869, on the Laws Regulating the Flow and Action of Water holding Solid Matter in Suspension, consisting of Mr. J. Hawksley, Professor Eankine, Mr. II. A. Grantham, Sir A. S. Waugh, and Mr. T. Login, with authority to represent to the Government the desirability of undertaking- experiments bearing on the subject. The Committee presented a Memorial to the Indian Government, who have recently intimated their intention of advancing a sum of £2000 to enable Mr. Login to carry on experiments. The Council regret to have to announce the death of their Clerk, Mr. Askham, who was always most assiduous in his attention to his duties. They have appointed Mr. H. C. Stewardson in his place. They recommend that a gratuity of .£50 be given to Mr. Askham's Widow. RECOMMENDATIONS OF THE GENERAL COMMITTEE. liii The Council have added the following list of names of gentlemen present at the last Meeting of the Association to the list of Corresponding Members ; — M. C. Bergeron. Lausanne. Paris. Liege. Professor E. CrouUebois. Professor G. Devalque. M. "VV. de Fonvielle. Paris. Professor Paul Gervais. Paris. Professor James Hall. Albany, New York. Mr. J, E. Hilgard. Coast Survey, Washington. M. Georges Lemoine. Paris. Eichter. St. Professor Victor von Petersburg. Professor Carl Semper. "Wiirtzburg. Professor A. Wurtz. Paris. The General Committee will remember that Belfast has already been selected as the place of meeting for next year. The Council have been in- formed that invitations to hold subsequent Meetings at Bristol and Glasgow win be presented to the General Committee. Kecommendatioxs adopted by the General Committee at the Bradford Meeting in September 1873. [When Committees are appointed, the Member first named is regarded as the Secretary, except there is a specific nomination.] Involving Grants of Money. That the Committee, consisting of Professor Cayley, Professor G. G. Stokes, Professor H. J. S. Smith, Professor Sir W. Thomson, and Mr. J. W. L. Glaisher (Secretary), on Mathematical Tables be reappointed, with a grant of ^100 for the completion of the tabulation of the Elliptic Functions. That the sum of £100 be granted to the Committee on Mathematical Tables towards the printing of the tables of the Elliptic Functions that have been calculated by the Committee. That Mr. Glaisher, Colonel Strange, Professor Sir W. Thomson, Mr. Brooke, ]\Ir. AYalker, M. de Fonvielle, Professor Zenger, and iCr. Mann (Secretary), be a Committee for the purpose of investigating the efficacy of Lightning- conductors, giving suggestions for their improvement, and reporting upon any case in which a building has been injured by lightning, especially where siich building was ju-ofessedly protected by a lightning-conductor, and that the sum of £50 granted last year, but not expended, be regrantcd to the Committee. That a Committee be appointed, consisting of Professor Balfour Stewart, Mr. Glaisher, and Mr. Lockyer, and that a grant of ,£100 be made to them in order to provide assistance to Mr. Meldrum in conducting meteorological researches in Mauritius. That Professor Balfour Stewart and Mr. W. F. Barrett be a Committee for the purpose of investigating the magnetization of Iron, Nickel, and Cobalt, and that the sum of £20 be placed at their disposal for the purpose. That the Committee for reporting on the Eainfall of the British Isles, con- sisting of Mr. Charles Brooke, Mr. Glaisher, Professor Phillips, Mr. G. J. Symons, Mr. J. F. Bateman, Mr. T. Hawksley, Mr. C. Tomlinson, and Mr. llogers Field, be reappointed ; that Mr. G. J. Symons be the Secretary, and that a grant of £100 be placed at their disposal for the purpose. That the Committee, consisting of Mr. James Glaisher, Mr. R. P. Greg, Mr. Charles Brooke, Professor G. Forbes, and Professor A. S. Herschel, be liv REPOET — 1873. reappointed, and the sum of £30 be placed at their disposal for the purpose of showing the radiant-points of shooting-stars on graphical cliarts. That the Committee on Thermo-Elcctricity, consisting of Professor Tait, Professor Tyndall, and Professor Balfour Stewart, be reajjpointed, and that the sum of £50 be placed at their disposal for the purpose. That Professor A. W. Williamson, Professor Sir W. Thomson, Professor Clerk Maxwell, Professor G. C. Foster, Mr. Abel, Professor F. Jenkin, Mr. Siemens, and Mr. R. Sabine be reappointed a Committee for the purpose of testing the Ifew Pyrometer of Mr. Siemens, and that the sum of ,£30 (which was granted last year and has lapsed) be regranted to the Committee. That Professor Crum Brown, Mr. Dewar, Professor Tait, Professor Sir W. Thomson, and Dr. Gladstone be a Committee for the pui'pose of conducting in- vestigations as to the determination of High Temperatures by various methods ; that Mr. Dewar be the Secretary, and that the sum of ,£70 be placed at their disposal for the purpose. That Professor "Williamson, Professor Eoscoe, and Professor Frankland be a Committee for the purpose of superintending the Monthly llecords of the Progress of Chemistry published in the Journal of the Chemical Society, and that the sum of £100 be placed at their disposal for the pui'pose. That Dr. Gladstone, Dr. C. E. A. Wright, and Mr. Chandler Eoberts be reappointed a Committee for the purpose of investigating the chemical con- stitution and optical properties of essential oils ; that Mr. Chandler Eoberts be the Secretary; that the sum of £10 be placed at their disposal for the purpose ; and that the subject of investigation be Isomeric Turpenes and their Derivatives. That Dr. H. A. Armstrong and Dr. Thorpe be a Committee for the purpose of investigating Isomeric Cresols and their Derivatives ; that Dr. Armstrong be the Secretary, and that the sum of £20 be placed at their disposal for the purpose. That Professor A. S. Herschcl and Mr. G. A. Lebour be a Committee for the purpose of conducting experiments on the conducting-power for Heat of certain rocks ; that Professor Herschel be the Secretary, and that the sum of £10 be i)laced at their disposal for the purpose. That Professor Phillips, Professor Harkness, Mr. Henry Woodward, Mr. James Thomson, Mr. John Brigg, and Mr. L. C. Miall be a Committee for the purpose of investigating and reporting upon the Labyrinthodonts of the Coal- measures ; that Mr. L. C. Miall be the Secretary, and that the sum of £10 be placed at their disposal for the purpose. That Dr. Bryce and Mr. William Jolly be a Committee for the purpose of coUectingFossilsfromlocalities of difficult access in the north-west of Scotland; that the specimens be deposited as arranged in the Eesolution of last year ; that Mr. WiUiam Jolly be the Secretary, and that the sum of £10 be placed at their disposal for the purpose. That the Eev. T. Wiltshire, Mr. J. Thomson, and Professor W. C.Williamson be a Committee for the purpose of continuing the investigation of Mountain Limestone Corals, and the preparation of plates for publication, and that the Committee be requested to direct their attention to the early publication of the results hitherto attained ; that Mr. James Thomson be the Secretary, and that the sum of £25 be placed at their disposal for the purpose. That Mr. H. Willett, Mr. E. A. C. Godwin- Austen, W. Toplcy, Mr. Da- vidson, Mr. Prestwich, Professor Boyd Dawkins, and Mr. Henry Woodward be a Committee for the purpose of- promoting the '• Sub-Wcalden Explora- tion ; " that Mr. H. Willett be the Secretary, and that the sum of £25 be placed at their disposal for the purpose. RECOMMENDATIONS OP THE GENERAL COMMITTEE. \r That Sir C. Lyell, Burt., Professor Phillips, Sir John Lubbock, Bart., Mr. J. Evans, Mr. E. Vivian, Mr. W. Peugelly, Mr. G. Busk, Mr. W. B. Dawkins, Mr. W. A. Sanford, and Mr. J. E. Lee be a Committee for the purpose of continuing the exploration of Eent's Cavern, Torquay ', that Mr. Pengelly bo the Secretary, and that the sum of <£150 be placed at theii* disposal for the purpose. That Professor Harkness, Mr. Prestwich, Professor Hughes, Eev. H. W. Crosskey, Messrs. C. J. "Woodward, W. Boyd Dawkius, George Maw, L. C. Miall, G. H. Morton, and J. E. Lee be a Committee for the purpose of re- cording the position, height above the sea, lithological characters, size, and origin of the more important of the Erratic Blocks of England and Wales, reporting other matters of interest connected with the same, and taking mea- sures for their preservation ; that the Eev. H. AV. Crosskey be the Secretary, and that the sum of =£10 be placed at their disposal for the purpose. That Mr. Henry Woodward, Professor W. C. Williamson, Mr. F. W. Budler, Mr. L. C. Miall, Mr. W. Topley, Mr. W. Whitaker, and Mr. G; A. Lebour be a Committee for the purpose of preparing a Record of Geological and Palaj- ontological Literature ; that Mr. Henry AVoodward be the Secretary, and that the sum of ^100 be placed at their disposal for the purpose. That Sir John Lubbock, Bart., Professor Phillips, Professor Hughes, Messrs. W. Boyd Dawkins, L. C. Miall, and R. H. Tiddeman be a Committee for the purpose of assisting the exploration of the Victoria Cave, Settle ; that II. H. Tiddeman be the Secretary, and that the sum of £50 be placed at their disposal for the purpose. That Mr. Stainton, Sir John Lubbock, and Professor IS'ewton be reappointed a Committee for the purpose of continuing a Record of Zoological Literature ; that Mr. Stainton be the Secretary, and that the sum of .£100 be placed at their disposal for the purpose. That Mr. Gwyn Jeffreys, Mr. G. S. Brady, Mr. Robertson, and Mr. H. Brady be a Committee for the purpose of dredging off the coasts of Durham and North Yorkshire ; that Mr. H. Brady be the Secretary, and that the sum of £30 be placed at their disposal for the purpose. That Professor Balfour, Dr. M'^Kendrick, and Mr. Dewar be a Committee for the purpose of carrying on investigations into the Physiological Action of Light ; that Dr. McKendrick be the Secretary, and that the sum of £20 be placed at their disposal for the piirpose. That Dr. Pye-Smith, Dr. Brunton, and Mr. West be a Committee for the purpose of making physiological researches on the nature of intestinal secre- tion ; that Dr. Brunton be the Secretary, and that the sum of £20 be placed at their disposal for the purpose. That Dr. M. Foster, Mr. E. Ray Lankester, Dr. Anton Dohrn, and Mr. A. G. Dew-Smith be a Committee for determining the best methods of breeding the embryos of delicate marine organisms ; that Dr. Anton Dohrn bo the Secre- tary, and that the sum of £30 be placed at their disposal for the purpose. That Colonel Lane Fox, Dr. Beddoe, Mr. Franks, Mr. Francis Galton, Mr. Edward Brabrook, Sir J. Lubbock, Bart., Sir Walter Elliot, Mr. Clements R. Markham, and Mr. E. B. Tylor be reappointed a Committee for the purpose of preparing and publishing brief forms of instruction for travellers, ethnolo- gists, and other anthropological observers ; that Colonel Lane Fox be the Se- cretary, and that the sum of £50 be placed at their disposal for the pvu'posc, £25 being the renewal of the unexpended grant of last year. That Lord Houghton, Professor Thorold Rogers, AV. Ncwmarch, Professor Fawcctt, M.P., Jacob Bchrens, F. P. FcUows, R. H. Inglis Palgrave, Archi- Ivi REPORT — 1873. bald Hamilton, and S. Bromi te a Committee for the purpose of inquiring into the economic effect of combinations of labourers or capitalists, and into the laws of Economic Science bearing on the principles on which they are foixnded ; that Professor L. Levi be the Secretary, and that the sum of .£25 be placed at their disposal for the j)urpose. That the Committee on instruments for measuring the speed of ships be reappointed ; that it consist of the following Members : — Mr. "W. Proude, Mr. JP. J. BramweU, Mr. A. E. Fletcher, Eev. E. L. Berthon, Mr. James E. Napier, Mr. C. W. Merrifield, Dr. C. W. Siemens, Mr. H. M. Brunei, Mr. W. Smith, Sir WiUiam Thomson, and Mr. J. N. Shoolbred ; that Mr. J. N. Shoolbred be the Secretary, and that the sum of .£50 be placed at their disposal for the purpose. That the sum of .£50 be granted to Mr. Askham's widow (recommended by the Council). Applications for Reports and Researches not involving Grants of Money. That Professor Sylvester, Professor Cayley, Professor Hirst, Eev. Professor Bartholomew Price, Professor H. J. S. Smith, Dr. Si^ottiswoode, Mr. E. B. Hayward, Dr. Salmon, Eev. E. Townsend, Professor Fuller, Professor Kel- land, Mr. J. M. Wilson, and Professor Clifford be reappointed a Committee (with power to add to their number) for the purpose of considering the pos- sibility of improving the methods of instruction in elementary geometry ; and that Professor Clifford be the Secretary. That the Committee, consisting of Dr. Joule, Professor Sir W. Thomson, Professor Tait, Professor Balfour Stewart, and Professor J. Clerk Maxwell, be reappointed to effect the determination of the Mechanical Equivalent of Heat. That the Committee, consisting of the following Members, with power to add to their number, — Professor Eoscoe, Professor "W. G. Adams, Professor Andrews, Professor Balfour, Mr. Baxendell, Mr. Bramwell, Professor A. Crum Brown, Mr. Buchan, Dr. Carpenter, Professor Core, Dr. DeLaEue, Professor Tbiselton Dyer, Sir Walter Elliot, Professor Flower, Professor G. C. Foster, Professor M. Foster, Colonel Lane Fox, Professor Geikie, Dr. J. H. Gladstone, Mr. Griffith, Eev. E. Harlcy, Dr. Hirst, Dr. Hooker, Dr. Huggins, Professor Huxley, Professor Fleeming Jenkin, Dr. Joule, Dr. Lankcster, Mr. J. N. Lockyer, Professor Clerk Maxwell, Mr. D. Milne-Home, Dr. O'Callaghau, Professor Odling, Professor Eamsay, Dr. Spottiswoode, Mr. Stainton, Professor Balfour Stewart, Colonel Strange, Professor Tait, Mr. J. A. Tinne, Professor Allen Thomson, Professor Sir William Thomson, Professor Wyville Tliomson, Professor Turner, Mr. G. V. Vernon, Professor A. W. Williamson, Professor Young, Professor Eoscoe being the Secretary, — be reappointed — 1", to consider and report on the best means of advancing science by Lectures, with authority to act, subject to the approval of tlie Council, in the course of the present year, if judged desirable. 2°, to consider and report whether any steps cau be taken to render scientific organization more comi^lete and effectual. That the Eclipse Committee, consisting of the President and General Officers (with power to add to their number), be reappointed. That the Committee on Tides, consisting of Professor Sir W. Thomson, Professor J. C. Adams, Mr. J. Oldham, Eear-Admiral Eichards, General Strachey, Mr. W. Parkes, Mr. Webster, and Colonel Walker, be reappointed. That the Committee on Underground Temperature, consisting of Professor RECOMMENDATIONS OF THE GENERAL COMMITTEE. Ivii Everett (Secretary), Professor Sir W. Thomson, Sir Charles Lyell, Bart., Pro- fessor J. Clerk Maxwell, Professor Phillips, Mr. G. J. Symons, Professor Eamsay, Professor Gcikie, Mr. Glaisher, Eev. Dr. Graham, Mr. George Maw, Mr. Peugelly, Mr. S. J. Maclde, Professor Edward Hull, and Professor Ansted, be reappointed, with the addition of Dr. Clement Le Neve Poster. That the Committee, consisting of Dr. Huggins, Mr. J. IST. Lockyer, Dr. Reynolds, and Mr. Stoney, on Inverse Wave-lengths, be reappointed, and that Mr. Spottiswoode, Dr. De La Eue, and Dr. W. M. "Watts be added to the Committee. That the Committee, consisting of Professor Cayley, Mr. J. W. L. Glaisher, Dr. W. Pole, Mr. Merrificld, Professor Fuller, Mr. H. M. Brunei, and Pro- fessor W. K. Clifford, be reappointed to estimate the cost of constructing Mr. Babbage's Analytical Engine, and to consider the advisability of printing tables by its means. That Mr. ^Y. 11. L. Russell be requested to continue his Report on recent progress in the Theory of Elliptic and Hyperelliptic Functions. That Professor H. J. S. Smith, Professor Clifford, Professor W. G. Adams, Professor Balfour Stewart, Mr. J. G. Fitch, Mr. George Griffith, Mr. Marshall Watts, Professor Everett, Professor G. Carey Foster, and Mr. W. F. Barrett be a Committee (with power to add to their number) to consider and report on the extent and method of teaching Physics in Schools, and that Professor G. Carey Foster be the Secretary. That Professor Sir W. Thomson, Professor Everett, Professor G. C. Foster, Professor J. Clerk Maxwell, Mr. G. J. Stoney, Professor Fleeming Jenkin, Dr. Siemens, Mr. Bramwell, Professor W. G. Adams, and Professor Balfour Stewart be a Committee for reporting on the Nomenclature of Dynamical and Electrical Units, and that Professor Everett be the Secretary. That Professor Tait be requested to prepare a Report on Quaternions. That Mr. Roberts, Dr. Mills, J. S. Sellon, Dr. Boycott, and Mr. Gadesden be a Committee for the purpose of inquiring into the method of making gold assays, and stating the results thereof ; that Mr. W. C. Roberts be the Se- cretary. That Dr. Bryee, Professor Sir W. Thomson, Mr. J. Brough, Mr. G. Forbes, Mr. D. Milne-Holme, and Mr. J. Thomson be a Committee for the purpose of continuing the Observations and Records of Earthquakes in Scotland, and that Dr. Bryce be the Secretary. That the Ruv. H. F. Barnes, Mr. Dresser, Mr. Harland, Mr. Harting, Professor Newton, and the Rev. Canon Tristram be reappointed a Committee for the purpose of inquiring into the possibility of establishing " a close time " for the protection of indigenous animals, and that Mr. Dresser be the Se- cretary. That Professor Balfour, Dr. Cleghorn, Mr. Hutchinson, Mr. Buchan, and Mr. Sadler be reappointed a Committee for the purpose of taking observations on the effect of the denudation of timber on the rainfall of North Britain ; that Mr. Hutchinson be the Secretary. That Dr. Carpenter, Professor AUman, Professor Newton, and Mr. H. B. Brady be a Committee for the purpose of inquiring into and reporting upon the possibility of increasing the scientific usefulness of the Aquaria at Brighton and Sydenham ; that Dr. Carpenter be the Secretary. That the Metric Committee be reappointed, such Committee to consist of The Right Hon. Sir Stafford H. Northcote, Bart., C.B., M.P., The Right Hon. C. B. Adderley, M.P., Sir W. Armstrong, Mr. Samuel Brown, Dr. Farr, A. Hamilton, Professor Frauklaud, Professor Heuuessy, Professor Leone Levi, Iviii EEPORT — 1873. Mr. C. "VV. Siemens, Professor A. W. Williamson, Major-Gen. Strnchcy, and Dr. Koberts ; that Professor Leone Levi bo the Secretary. That the Committee for the purpose of continuing the investigations on the Treatment and Utilization of Sewage be renewed, and that such Committee consist of Mr. E. B. Grantham, Professor Corfield, Mr. Bramwell, Dr. J. H. Gilbert, Mr. W. Hope, and Professor Williamson. That Mr. J. 11. Napier, Mr. E. J.Bramwell, Mr. C. W. Merrifield, Sir John Hawkshaw, Mr. T. Webster, Q.C., and Professor Osborne lleynolds be a Committee for the purpose of considering and reporting on British Measures in use for mechanical and other purposes, and that Mr. C. W. Merrifield be the Secretary. That Mr. Francis Galton, Mr. C. W. Merrifield, Mr, W. Fronde, and Pro- fessor Osborne Reynolds be a Committee for the purpose of obtaining a record of the varying amount of sea disturbance, and the measurement of waves near shore. That Mr. F. J. Bramwell, Mr. Hawksley, Mr. Edward Easton, Sir William Armstrong, and Mr. W. Hope be a Committee to investigate and report upon the utilization and transmission of wind and water power ; that Mr. W. Hope be the Secretary. That Mr. H.Bessemer, Mr. E. J. Bramwell, Dr. Lyon Playfair, Dr. C. W. Siemens, and Mr. T. Webster, Q.C., be a Committee for the purpose of con- sidering and reporting on the contributions to science due to inventors and invention in the industrial arts, and that Mr. T. Webster, Q.C, be the Se- cretary. That Mr. W. H. Barlow, Mr. H. Bessemer, Mr. F. J. Bramwell, Captain Douglas Galton, Sir John Hawkshaw, Mr. C. W. Siemens, Professor Abel, and Mr. E. H. Carbutt be a Committee for the purpose of considering what steps can be taken in furtherance of the objects of the Address of the President of this Section [Mechanical] as to the use of steel for structural purposes, and that Mr. E. H. Carbutt be the Secretary. Hesolittions referred to the Council for consideration and action if it seem desirable. That the Council be requested to take steps to bring the importance of the meteorological researches at Mauritius before the Government, in order that, when they become convinced of the value of these researches by the action of the Association, they may be induced to increase the assistance. That the Council be requested to take such steps as they may consider desirable for the purpose of representing to Her Majesty's Government the importance of tlie scientific results to bo obtained from Arctic Exploratiojj. That the Council be i-equested to consider the possibility and expediency of making arrangements for the constitution of an Annual Museum for the exhibition of specimens and apparatus on a similar footing to that of the Sections, and similarly provided with officers to superintend the arrange- ments. That the Council of the British Association be requested to communicate with the authorities in charge of the St, Gothard Tunnel, with the view of obtaining permission for the Committee on Underground Temperature to take observations on temperature during the progress of the works. RECOMMENDATIONS OF THE GENERAL COMMITTEE. Hx Communications ordered to he printed in exteuso in the Annual Reiiort of the Association, That Professor A. Scliafarik's paper " On the visibility of the dark side of Venus " be printed in extenso among the Eeports. That Mr. Mcldrum's paper " On a Periodicity of Cyclones and Eainfall iu connexion with the Sun-spot Periodicity " be printed in extenso among the Reports. That the Tables (extending to 3 or 4 pages) appended to Mr. Gwyn JcfTreys's paper " On Mediterranean MoUusca " be printed in the Report. That Mr. Pengelly's paper, " The Plint and Chert Implements found in Kent's Cavernj Torquay, Devonshire," read in the department of Anthro- pology, be printed in extenso in the Annual Report. That Mr. Firth's paper " On the Coal-cutting Machine " and Mr. Gott's paper (with the diagrams, on the understanding that the blocks be supplied) " Oil the Bradford AVaterworks " be printed in extenso in the Annual Volume. Besolution referred to the Parliamentanj Committee. That the Memorial from the Council of tho Leeds Philosophical and Literary Society to the General Committee of the British Association be referred to the Parliamentary Committee. [Copy.] Memorial from the Council of the Leeds Philosophical and Literary Society to the General Committee of the British Association. The Council of the Leeds Philoso^ihical and Literary Society desrie to direct the attention of the General Committee of the British Association to a question of legislation capable of affecting prejudicially a number of Societies engaged in the promotion of science. Since the British Association recognizes as one of its functions the vigilant observation through its Parliamentary Committee of current legislation affect- ing the interests of science, your memorialists have much confidence in bring- ing the subject before it. The Rating BiU introduced by Government during the last Session of Par- liament, proposed to withdraw from Scientific aiid Literary Societies the ex- emption from rating specially conferred upon tliem by an Act passed about thirty years ago. The Institution which your memorialists represent, like many others, would have suffered seriously in its capability of maintaining a large Public Museum had this Bill become law. After the discussion of the question in Parliament, your memorialists are convinced that no sufficient reason exists for thus abstracting from the funds of Scientific and Literary Societies a sum of money which is important to their efficiency, but too small to affect apprcciablj^ the question of the distri- bution of taxation. So many exemptions of religious and educational insti- tutions were admitted by the amended Bill, that it could lay no claim to uniformity in its treatment of the subject of Rating. Your memorialists respectfully invite the attention of the General Com- mittee of the British Association to this subject, Avith the view of maintain- ing the present exemption, should further legislation be undertaken. Signed, By order of the Council of the Leeds Philosophical and Literary Society, Thomas Wilson, 1 rr cv Richard Reynolds, / ^'''- ^<^C'-<'to^'^«. Sept. 9th, 1873. Ix REPORT — 1873. Synojjsis of Grants of Money apjjropriafed to Scientific Purposes by the General Committee at the Bradford Meeting in September 1873. The names of the Members who would he entitled to call on the General Treasurer for the respective Grants are prefixed. Maihematlcs and Physics. *Cayley, Professor. — Mathematical Tables 100 Cayley, Professor, — Printing Mathematical Tables 100 Glaisher, Mr. J. — Efficacy of Lightning Conductors (renewed) 50 Balfour Ste^yart, Professor. — Mauritius Observatory 100 Balfour Stewart, Professor. — Magnetization of Iron 20 *Brooke, Mr.— British Painfull 100 *Glaisher, Mr. J. — Luminous Meteors 30 *"Tait, Professor. — Thermo-Elcctricity (renewed) 50 *Williamson, Prof. A. W. — Testing Sicmens's New Pyrometer (renewed) 30 Chemistry. *Brown, Professor Crum. — High Temperature of Bodies (partly renewed) 70 *Willianison, Prof. A. W. — Records of the Progress of Chemistry (^100 renewed) 100 *Gladstone, Dr. — Chemical Constitution and Optical Properties of Essential Oils 10 Armstrong, Dr. — Isomeric Cresols and their Derivatives .... 20 Geology. Herschcl, Professor. — Thermal Conducting-power of Eocks. . 10 Phillips, Professor. — Labyrinthodonts of the Coal-measures . . 10 *Brj-ce, Dr. — Collection of Fossils in the I^orth-wcat of Scotland 10 *Wiltshire, Rev. T.— Investigation of Fossil Corals . , 25 *Willett, Mr. H.— The Sub-Wealdeu Exploration 25 *Lye]l, Sir C, Bart.— Kent's Cavern Exploration 150 *Harkness, Professor. — Mapping Positions of Erratic Blocks and Boulders 10 Woodward, Mr. H. — Record of Geological and Palseontological Literature 100 *Lubhock, Sir J. — Exploration of Victoria Cave 50 Carried forward , , £1170 * Eeappointed. SYNOPSIS OF GRANTS OF MONEY. kl Biology, Brought forward ^£1170 *Laue Fox, Col. A. — Forms of Instruction for Travellers (^25 renewed) 50 *Staintou, Mr. — Eecord of the Progress of Zoology 100 Jeffreys, Mr. Gwyn. — Dredging off the Coasts of Yorkshire . . 30 Balfour, Professor. — Physiological Action of Light 20 Pye-Smith, Dr.— The Nature of Intestinal Secretion 20 Foster, Dr. M. — Methods of Breeding the Embryos of Delicate Marine Organisms 30 'o"" Statistics a^ul Economic Science. Houghton, Lord. — Economic Effects of Trades Unions 25 Meclianics. *Froude, Mr. W. — Instruments for Measuring the Speed of Ships and Currents (renewed) 50 1495 Askham's Widow, Mr 50 Total.... ^1545 * Eeappointed. The Annual Meeting in 1874. The Meeting at Belfast will commence on "Wednesday, August 19, 1874. Place of Meeting in 1875. The Annual Meeting of the Association in 1875 will be held at Bristol, Ixii REPORT — 1873. General Statement of Sums which have been paid on Account of Grants for Scieiitific Purposes. 1834. TiJo Discussions 20 1835. TiJe Discussions ■ fi2 r.ritisli Fossil Ichthyology lOj £\iu 1836. Tide Discussions 163 Hricish Fossil Ichthyology 105 Thermometric Observations, &c. 50 Experiments on long-continued Heat 17 1 Rain-Gauges 9 13 Refraction Experiments 15 Lunar Nutation 60 Tliermometers 15 6 £434 14 1837. Tide Discussions 284 1 Chemical Constants 24 13 6 Lunar Nutation 70 Observations on Waves 100 12 Tides at Bristol... 150 Meteorology and Subterranean Temperature 89 5 Vitrification Experiments 150 Heart Experiments 8 4 6 Barometric Observations 30 Barometers 11 18 6 £918 14 6 1838. Tide Discussions 29 British Fossil Fishes 100 Meteorological Observations and Anemometer (construction) ... 100 Cast Iron (Strength of) 60 Animal and Vegetable Substances (Preservation of) ]9 1 10 Railway Constants 41 12 10 Bristol Tides 50 Growth of Plants 75 Mud in Rivers 3 6 6 Education Committee 50 Heart Experiments 5 3 Land and Sea Level 267 8 7 Subterranean Temperature 8 6 Steam-vessels 100 Meteorological Committee 319 5 Thermometers 16 4 £956 12 2 1839. Fossil Ichthyology 110 Meteorological Observations at Plymouth 63 iq T.Iechanism of Waves 144 2 Bristol Tides , 35 ijj g Meteorology and Subterranean Temperature 21 Vitrification Experiments. 9 Cast-Iron Experiments 100 Railway Constants 28 Land and Sea Level 274 Steam-vessels' Engines 100 Stars in Histoire Celeste 331 Stars in Lacaille 11 Stars in R. A. S. Catalogue Animal Secretions 10 Steam-engines in Cornwall 50 Atmospheric Air 16 Cast and Wrought Iron 40 Heat on Organic Bodies 3 Gases on Solar Spectrum 22 Hourly Meteorological Observa- tions, Inverness and Kingussie 49 Fossil Reptiles 118 Mining Statistics 50 11 4 7 7 2 1 4 18 6 16 6 10 1 7 8 2 9 £1595 11 1840. Bristol Tides 100 Subterranean Temperature 13 13 6 Heart Experiments 18 19 Lungs Experiments 8 13 Tide Discussions 50 Land and Sea Level 6 11 1 Stars (Uistoire Celeste) 242 10 Stars (Lacaille) 4 15 Stars (Catalogue) 264 Atmospheric Air 15 15 Water on Iron 10 Heat on Organic Bodies 7 Meteorological Observations 52 17 6 Foreign Scientific Memoirs 112 1 6 Working Population 100 School Statistics 50 Forms of Vessels 184 7 Chemical and Electrical Pheno- mena 40 Meteorological Observations at Plymouth 80 Magnetical Observations 185 13 1841. Observations on Waves 30 Meteorology and Subterranean Temperature 8 Actinometers '. 10 Earthquake Shocks 17 Acrid Poisons C Veins and Absorbents 3 Mud in Rivers 5 Marine Zoology 15 Skeleton Maps 20 Mountain Barometers 6 Stars (Histoire Celeste) 185 £1546 16 4 8 7 12 8 IS 6 GENERAL STATEMENT. Ixiii £ s. d. Stars (Lacaille) 79 5 Stars (Nomenclature of) 17 19 6 Stars (Catalogue of) 40 Water on Iron 50 Meteorological Observations at Inverness 20 Meteorological Observations (re- duction of) 25 Fossil Reptiles 50 Foreign Memoirs C2 Kailvvay Sections 38 1 6 Forms of Vessels 193 12 Meteorological Observations at Plymouth 55 Magnetical Observations Gl 18 8 Fishes of the Old Red Sandstone 100 Tides at Leilh 50 Anemometer at Edinburgh C9 110 Tabulating Observations 9 6 3 Races of Men 5 Radiate Animals , 2 lEr235 10 11 1842. Pynamometrlc Instruments 113 11 2 Anoplura Britannise 52 12 Tides at Bristol 59 8 Gases on Light 30 14 7 Chronometers 26 17 6 Marine Zoology 15 British Fossil Mammalia 100 Statistics of Education 20 Marine Steam-vessels' Engines... 28 Stars (Histoire Celeste) 59 Stars (Brit. Assoc. Cat. of ) 110 Railway Sections 161 10 British Belenmites , 50 Fossil Reptiles (publication of Report) 210 Forms of Vessels 180 Galvanic Experiments on Rocks 5 8 6 Meteorological Experiments at Plymouth 68 Constant Indicator and Dynamo- metric Instruments 90 Force of Wind 10 Light on Growth of Seeds 8 Vital Statistics 50 Vegetative Power of Seeds 8 1 11 Questions on Human Race 7 9 jei449 17 8 1843. Revision of the Nomenclature of Stars 2 Reduction of Stars, British Asso- ciation Catalogue 25 Anomalous Tides, Frith of Forth 120 Hourly Meteorological Observn- tionsat Kingussie andliiverness 77 12 8 Meteorological Observations at Plymouth 55 Whewell's Meteorological Ane- mometer at Plymouth 10 £ s. d. Meteorological Observations, Os- ier's Anemometer at Plymouth 20 Reduction of Meteorological Ob- servations 30 Meteorological Instruments and Gratuiti'es 39 C Construction of Anemometer at Inverness 56 12 2 Magnetic Cooperation 10 8 10 Meteorological Recorder for Kew Observatory 50 Action of Gases on Light 18 16 1 Establishment at Kew Observa- tory, Wages, Repairs, Furni- ture and Sundries 133 4 7 Experiments by Captive Balloons 81 S Oxidation oftheKailsof Railways 20 Publication of Report on Fossil Reptiles 40 Coloured Drawings of Railway Sections 147 18 3 Registration of Earthquake Shocks 30 Report on Zoological Nomencla- ture 10 Uncovering Lower Red Sand- stone near Manchester 4 4 6 Vegetative Power of Seeds 5 3 8 Marine Testacea (Habits of) ... 10 Marine Zoology 10 Marine Zoology 2 14 11 Preparation of Pieport on British Fossil Mammalia 100 Physiological Operations of Me- dicinal Agents 20 Vital Statistics 30 Additional Experiments on the Forms of Vessels 70 Additional Experiments on the Forms of Vessels 100 Reduction of Experiments on the Forms of Vessels 100 Morin's Instrument and Constant Indicator 69 Experiments on the Strength of Materials 60 i;i565 10 2 5 8 14 10 8 4 IS44. Meteorological Observations at Kingussie and Inverness 12 Completing Observations at Ply- mouth 35 Magnetic and Meteorological Co- operation 25 Publication of the British Asso- ciation Catalogue of Stars 35 Observations on Tides on the East ceust of Scotland 100 Revision of the Nomenclature of S'ars IS42 2 9 6 Maintaining the Establishment in Kew Observatory..., 117 17 3 Instruments for Kew Observatory 5G 7 3 Ixiv REPORT 1873. £ s. d. Influence of Liglit on Plants 10 SiibteiTancous Tempeiatuie in Ireland 5 Coloured Drawings of Railway Sections 15 17 G Investigation of Fossil Fislies of the Lower Tertiary Strata ... 100 Registering the Shocks of Earth- quakes 1842 23 11 10 Structure of Fossil Shells 20 Radiata and MoUusca of the jEgean and Red Seas 1842 100 Geographical Distributions of Marine Zoology 1842 10 Marine Zoology of Devon and Cornwall 10 Marine Zoology of Corfu 10 Experiments on the Vitality of Seeds 9 3 Experiments on the Vitality of Seeds 1842 8 7 3 Exotic Anoplura 15 Strength of Materials 100 Completing Experiments on the Forms of Ships 100 Inquiries into Asphyxia 10 Investigations on the Internal Constitution of Metals 50 Constant Indicator and Morin's Instrument 18 42 10 3 6 £981 12 8 1845. Publication of the British Associa- tion Catalogue of Stars 331 14 Meteorological Observations at Inverness 30 18 Magnetic and Meteorological Co- operation 16 IG Meteorological Instruments at Edinburgh 18 11 Reduction of Anemometrical Ob- servations at Plymouth 25 Electrical Experiments at Kew Observatory 43 17 Maintaining the Establishment in Kew Observatory 149 15 For Kreil's Barometrograph 25 Oases from Iron Furnaces 50 The Actinngraph 15 Microscopic Structure of Shells 20 Exotic Anoplura 1843 10 Vitality of Seeds 1843 2 Vitality of Seeds 1844 7 Marine Zoology of Cornwall ... 10 Physiological Action of Medicines 20 Statistics of Sickness and Mor- tality in York 20 Earthquake Shocks 1843 15 14 8 ~£830 9 9 184G. British Association Catalogue of Stars 1844 211 15 Fossil Fishes of the London Clay 100 I ! 7 £ s. d. Computation of the Gaussian Constants for 1829 50 Maintaining the Establishment at Kew Observatory 146 10 7 Strength of Materials CO Researches in Asphyxia 6 16 2 Examination of Fossil Shells 10 Vitality of Seeds 1844 2 15 10 Vitality of Seeds 1845 7 12 3 Marine Zoology of Cornwall 10 Marine Zoology of Britain 10 Exotic Anoplura 1844 25 Expenses attending Anemometers 11 7 6 Anemometers' Repairs 2 3 6 Atmospheric Waves 3 3 3 Captive Balloons 1844 8 19 3 Varieties of the Human Race 1844 7 6 3 Statistics of Sickness and Mor- tality in York 12 £685 16 1847. Compulation of the Gaussian Constants for 1829 50 Habits of Marine Animals 10 Physiological Action of Medicines 20 Marine Zoology of Cornwall 10 Atmospheric Waves 6 9 3 Vitality of Seeds 4 7 7 Maintaining the Establishment at Kew Observatory 107 8 6 £20 8 5 4 1848. Maintaining the Establishment at Kew Observatory 171 15 11 Atmospheric Waves 3 10 9 Vitality of Seeds 9 15 Completion of Catalogues of Stars 70 On Colouring Matters 5 On Growth of Plants 15 £275 1 8 1849. Electrical Observations at Kew Observatory 50 Maintaining Establishment at ditto 76 2 5 Vitality of Seeds 5 8 1 On Growth of Plants 5 Registration of Periodical phe- nomena 10 Bill on account of Anemometrical Observations 13 9 £159 19 6 1850. Maintaining the Establishment at Kew Observatory 255 18 Transit of Earthquake Waves ... 50 Periodical Phenomena 15 Meteorological Instruments, Azores 25 £3 IT) 18 b GENERAL STATEMENT. Ixv £ s. d. 1851. Maintaining the Establishment at Kew Observatory (includes part ofgrantin 1849) 309 2 2 TheoryofHeat 20 1 1 Periodical Phenomena of Animals and Plants 5 Vitality of Seeds 5 6 4 Influence of Solar Radiation 30 Ethnological Inquiries 12 Researches on Annelida 10 ^£391 9 7 1852. Maintaining the Establishment at Kew Observatory (including balance of grant for 1850) ...233 17 8 Experiments on the Conduction of Heat 5 2 9 Influence of Solar Radiations ... 20 Geological Map of Ireland 15 Researches on the British Anne- lida 10 Vitality of Seeds 10 6 2 Strength of Boiler Plates 10 £304 6 7 1853. Maintaining the Establishment at Kevsr Observatory 165 Experiments on the Influence of Solar Radiation 15 Reseaiches on the British Anne- lida 10 Dredging on the East Coast of Scotland 10 Ethnological Queries 5 ~£205 1854. Maintaining the Establishment at Kew Observatory (including balance of former grant) 330 15 4 Investigations on Flax 11 Effects of Temperature on Wrought Iron 10 Registration of Periodical Phe- nomena , 10 British Annelida 10 Vitality of Seeds 5 2 3 Conduction of Heat 4 2 '£380 19 7 1855. Maintaining the Establishment at Kevf Observatory 425 Earthquake Movements 10 Physical Aspect of the Moon 11 8 5 Vitality of Seeds , 10 7 II Map of the World 15 Ethnological Queries 5 Dredging near Belfast 4 lelSO 16 4 1856. Maintaining the Establishment at Kew Observatory : — 1854 £ 75 01 ,,. . . 1855 £500 oM''' " " 1873. £ ». d. Strickland's Ornithological Syno- nyms 100 Dredging and Dredging Forms... 9 13 9 Chemical Action of Light 20 Strength of Iron Plates 10 Registration of Periodical Pheno- mena 10 Propagation of Salmon 10 £734 13 9 1857. Maintaining the Establishment at Kew Observatory 350 Earthquake Wave Experiments. . 40 Dredging near Belfast 10 Dredging on the West Coast of Scotland 10 Investigations into the Mollusca ofCalifornia 10 Experiments on Flax 5 Natural History of Madagascar. . 20 Researches on British Annelida 25 Report on Natural Products im- ported into Liverpool 10 Artificial Propagation of Salmon 10 Temperature of Mines 7 8 Thermometers for Subterranean Observations 5 7 4 Life-Boats ..500 £507 15 4 1858. Maintaining the Establishment at Kew Observatory 500 Earthquake Wave Experiments.. 25 Dredging on the West Coast of Scotland 10 Dredging near Dublin 5 Vitality of Seeds 5 5 Dredging near Belfast 18 13 2 Report on the British Annelida... 25 Experiments on the production of Heat by Motion in Fluids... 20 Report on the Natural Products imported into Scotland 10 £618 18 2 1859. ^^^'^^^^ Maintaining the Establishment at Kew Observatory 500 Dredging near Dublin 15 Osteology of Birds 50 Irish Tunicata 5 Manure Experiments 20 British Medusidje 5 Dredging Committee 5 Steam-vessels' Performance 5 Marine Fauna of South and West oflreland 10 Photographic Chemistry 10 Lanarkshire Fossils 20 1 Balloon Ascents 39 1 1 £684 11 1 1860. Maintaining the Establishment of Kew Observatory 500 Dredging near Belfast 16 6 Dredging in Dublin Bay 15 e Ixvi REPORT 1873. £ s. d. Inquiry into the Performance of Steam-vessels 124 Explorations in tlie Yellow Sand- stone of Dura Den 20 Cliemico-raechanical Analysis of Rocks and Minerals 25 Researches on the Growth of Plants 10 Researches on the Solubility of Salts 30 Researches on the Constituents of Manures 25 Balance of Captive Balloon Ac- counts.. 1 13 6 ^1241 7 1861. Maintaining the Establishment of Kew Observatory 500 Earthquake Experiments 25 Dredging North and East Coasts ofScotland 23 Drtdging Committee : — ISfiO J50 Ol „ n n 1861 £22 0/ '^ " " Excavations at Dura Den 20 Solubility of Salts 20 Steam-vessel Performance 150 Fossils of Lesmahago 15 Explorations at Uriconium 20 Chemical Alloys 20 Classified Index to the Transac- tions 100 Dredging in the Mersey and Dee 5 Dip Circle 30 Photoheliographic Observations 50 Prison Diet 20 Gauging of Water 10 Alpine Ascents 6 5 1 Constituents of Manures 25 £1111 5 10 1862. Maintaining the Establishment of Kew Observatory 500 Patent Laws .'. 21 6 MoUusca of N.-W. America 10 Natural History by Mercantile Marine 5 Tidal Observations 25 Photoheliometer at Kew 40 Photographic Pictures of the Sun 150 Rocks of Donegal 25 Dredging Durham and North- umberland 25 Connexion of Storms 20 Dredging North-east Coast of Scotland 6 9 6 Ravages of Teredo 3 11 Standardsof Electrical Resistance 50 Railway Accidents 10 Balloon Committee 200 Dredging Dublin Uay 10 Dredging th"fe Mersey 5 Prison Diet 20 Gauging of Water 12 10 £ s. d. Steamships' Performance 150 Thermo-Electric Currents 5 J E1293 16 6 1863. Maintaining the Establishment of KevF Observatory 600 Balloon Committee deficiency... 70 Balloon Ascents (other expenses) 25 Entozoa 25 Coal Fossils 20 Herrings •.. 20 Granites of Donegal 5 Prison Diet 20 Vertical Atmospheric Movements 13 Dredging Shetland 50 Dredging North-east coast of Scotland 25 Dredging Northumberland and Durham 17 3 10 Dredging Committee superin- tendence 10 Steamship Performance 100 Balloon Committee 200 Carbon under pressure 10 Volcanic Temperature 100 Bromide of Ammonium 8 Electrical Standards 100 Construction and distribu- tion 40 Luminous Meteors 17 Kew Additional Buildings for Photoheliograph 100 Thermo-Electricity 15 Analysis of Rocks 8 Hydroida 10 £ 1608 3 10 1864. Maintaining the Establishment of Kew Observatory 600 Coal Fossils 20 Vertical Atmospheric Move- ments 20 Dredging Shetland 75 Dredging Northumberland 25 Balloon Committee 200 Carbon under pressure 10 Standards of Electric Resistance 100 Analysis of Rocks 10 Hydroida 10 Askham's Gift 50 Nitrite of Amyle 10 Nomenclature Committee 5 Rain-Gauges 19 15 8 Cast- Iron Investigation 20 Tidal Observations in the Humber 50 Spectral Rays 45 Luminous Meteors 20 £ 1289 15 8 1865. Maintaining the Establishment of Kew Observatory 600 Balloon Committee 100 o Hydroida 13 GENERAL STATEMENT. Ixvii £ s. d. Rain-Gauges 30 Tidal Observations in the Humber 6 8 Hexylic Compounds 20 Amyl Compounds 20 Irish Flora 25 American MoUusca 3 9 Organic Acids 20 Lingula Flags Excavation 10 Eurypterus 50 Electrical Standards 100 Malta Caves Researches 30 Oyster Breeding 25 Gibraltar Caves Researches 150 Kent's Hole Excavations 100 Moon's Surface Observations ... 35 Marine Fauna 25 Dredging Aberdeenshire 25 Dredging Channel Islands 50 Zoological Nomenclature 5 Resistance of Floating Bodies in Water 100 Bath Waters Analysis 8 10 Luminous Meteors 40 1866. Maintaining the Establishment of Ke w Observatory 600 Lunar Committee 64 Balloon Committee 50 Metrical Committee 50 British Rainfall 50 KUkenny Coal Fields 16 Alum Bay Fossil Leaf-Bed 15 Luminous Meteors 50 Lingula Flags Excavation 20 Chemical Constitution of Cast Iron 50 Amyl Compounds 25 Electrical Standards 100 Malta Caves Exploration 30 Kent's Hole Exploration 200 Marine Fauna, &c., Devon and Cornwall 25 Dredging Aberdeenshire Coast... 25 Dredging Hebrides Coast 50 Dredging the Mersey 5 Resistance of Floating Bodies in Water 50 Polycyanides of Organic Radi- cals 20 Rigor Mortis 10 Irish Annelida 15 Catalogue of Crania 50 Didine Birds of Mascarene Islands 50 Typical Crania Researches 30 Palestine Exploration Fund 100 iE1591 7 10 13 4 £1750 13 4 1867. Maintaining the EstabUshment of Kew Observatory 600 Meteorological Instruments, Pa- lestine 50 Lunar Committee 120 £ s. d. Metrical Committee 30 Kent's Hole Explorations 100 Palestine Explorations 50 Insect Fauna, Palestine ......... 30 British Rainfall 50 Kilkenny Coal Fields 25 Alum Bay Fossil Leaf-Bed 25 Luminous Meteors 50 Bournemouth, &c. Leaf-Beds ... 30 Dredging Shetland 75 Steamship Reports Condensation 100 Electrical Standards 100 Ethyle and Methyle series 25 Fossil Crustacea 25 Sound under Water 24 4 North Greenland Fauna 75 Do. Plant Beds ... 100 Iron and Steel Manufacture ... 25 Patent Laws 30 ^1739 4 1868. Maintaining the Establishment of Kew Observatory 600 Lunar Committee 120 Metrical Committee 50 Zoological Record 100 Kent's Hole Explorations 150 Steamship Performances 100 British Rainfall 50 Luminous Meteors 50 Organic Acids 60 Fossil Crustacea 25 Methyl series 25 Mercury and Bile 25 Organic remains in Limestone Rocks 25 Scottish Earthquakes 20 Fauna, Devon and Cornwall ... 30 British Fossil Corals 50 Bagshot Leaf-beds 50 Greenland Explorations 100 Fossil Flora 25 Tidal Observations 100 Underground Temperature 50 Spectroscopic investigations of Animal Substances 5 Secondary Reptiles, &c 30 British Marine Invertebrate Fauna 100 i'1940 1869. ==^ Maintaining the Establishment of Kew Observatory 600 Lunar Committee 50 Metrical Committee 25 Zoological Record 100 Committee on Gases in Deep- well Water 25 British Rainfall 50 Thermal Conductivity of Iron, &c 30 Kent's Hole Explorations 150 Steamship Performances 30 Ixviii REPORT — 1873. £, 8. d. Chemical Constitution of Cast Iron 80 Iron and Steel Manufacture ... 100 Methyl Series 30 Organic remains iu Limestone Rocks 10 Earthquakes in Scotland 10 British Fossil Corals 50 Bagshot Leaf-Beds 30 Fossil Flora 25 Tidal Observations 100 Underground Temperature 30 Spectroscopic Investigations of Animal Substances 5 Organic Acids 12 Kiltorcan Fossils 20 Chemical Constitution and Phy- siological Action Relations ... 15 Mountain Limestone Fossils 25 Utilization of Sewage 10 Products of Digestion 10 £1622 1870. Maintaining theEstablishment of Kew Observatory 600 Metrical Committee 25 Zoological Record 100 Committee on Marine Fauna ... 20 Ears in Fishes 10 Chemical nature of Cast Iron ... 80 Luminous Meteors 30 Heat in the Blood 15 British Rainfall 100 Thermal Conductivityof Iron &c. 20 British Fossil Corals 50 Kent's Hole Explorations 150 Scottish Earthquakes 4 Bagshot Leaf-Beds 15 Fossil Flora 25 Tidal Observations 100 Underground Temperature 50 Kiltorcan Quarries Fossils 20 Mountain Limestone Fossils ... 25 Utihzation of Sewage 50 Organic Chemical Compounds... 30 Onny River Sediment 3 Mechanical Equivalent of Heat 50 £1572 1871. Maintainingthe Establishment of Kew Observatoi-y 600 Monthly Reports of Progress in Chemistry 100 Metrical Committee 25 Zoological Record 100 Thermal Equivalents of the Oxides of Chloriue 10 Tidal Observations 100 FoJEil Flora 25 £ Luminous Meteors 30 British Fossil Corals 25 Heat in the Blood 7 British Rainfall 50 Kent's Hole Explorations 150 Fossil Crustacea 25 Methyl Compounds 25 Lunar Objects 20 Fossil Corals Sections, for Pho- tographing 20 Bagshot Leaf-Beds 20 Moab Explorations 100 Gaussian Constants 40 £1472 1872. Maintaining the Establishment of Kew Observatory 300 Metrical Committee 75 Zoological Record 100 Tidal Committee 200 Carboniferous Corals 25 Organic Chemical Compounds 25 Exploration of Moab 100 Terato-Embryological Inquiries 10 Kent's Cavern Exploration 100 Luminous Meteors 20 Heat in the Bio id 15 Fossil Crustacea 25 Fossil Elephants of Malta 25 Lunar Objects 20 Inverse Wave-Lengths 20 British Rainfall 100 Poisonous Substances Antago- nism 10 Essential Oils, Chemical Consti- tution, &c 40 Mathematical Tables 50 Thermal Conductivity of Meta ls 25 J 1285 1873. Zoological Record It'O Chemistry Record 200 Tidal Committee 400 Sewage Committee 100 Kent's Cavern Exploration 150 Carboniferous Corals 25 Fossil Elephants 25 Wave-Lengths 150 British Rainfall 100 Essential Oils 30 Mathematical Tables 100 Gaussian Constants 10 Sub-Wealden Explorations 25 Underground Temperature 150 Settle Cave Exploration 50 Fossil Flora, Ireland 20 Timber Denudation and Rainfall 20 Luminous Meteors 30 £1685 *. d. 2 6 2 6 GENKKAL MEETINGS. Ixix General Meelinc/s. On Wednesday Evening, September 17, at 8 p.m., in St. George's Hall, Pr. W. B. Carpenter, LL.D., F.E.S., President, resigned the office of President to Professor Alexander W. Williamson, Ph.D., F.E.S., who took the Chair, and delivered an Address, for which see page Ixx. On Thurs"day Evening, September 18, at 8 p.m., a Soiree took place in St. George's Hall. On Friday Evening, September 19, at 8.30 p.m., in St. George's Hall, Professor W. C. WiUiamson, F.R.S., delivered a Discourse on " Coal and Coal Plants." On Saturday Evening, at 8 p.m., in St. George's Hall, Dr. C. W. Siemens, F.R.S., delivered a Discourse on "Fuel" to the Operative Classes of Bradford. On Monday Evening, September 22, at 8.30 p.m., in St. George's Hall, Prof. Clerk Maxwell, F.R.S., delivered a Discourse on " Molecules." On Tuesday Evening, September 23, at 8 p.m., a Soiree took place in the Mechanic's Institute. On Wednesday, September 24, at 2.30 p.m., the concluding General Meeting took place, when the Proceedings of the General Committee, and the Grants of Money for Scientific purposes, were explained to the Members. The Meeting was then adjoiirned to Belfast*. * The Meeting is appointed to take place on Wednesday, August lO^ 1874. 1873. ADDRESS OF ALEXANDER W. WILLIAMSON, Ph.D., F.R.S., PRESIDENT. Ladies and Gentlemen, — • Instead of rising to address you on this occasion I had hoped to sit quietly amongst you, and to enjoy the intellectual treat of Listening to the words of a man of whom England may well be proud — a man whose life has been spent in reading the great book of nature, for the purpose of enriching his fellow men Avith a knowledge of its truths — a man whose name is known and honoured in every corner of this planet to which a knowledge of science has penetrated — and, let me add, a man whose name will live in the grateful memory of mankind as long as the records of such noble work are preserved. At the last Meeting of the Association I had the pleasure of proposing that Dr. Joule be elected President for the Bradford Meeting, and our Council succeeded in overcoming his reliLctance and in persuading him to accept that office. Nobly would Joule have discharged the duties of President had his bodily health been equal to the task ; but it became apparent after a while that he could not rely upon sufficient strength to justify him in performing the duties of the Chaii-, and, in obedience to the orders of his physician, he placed his resignation in the hands of the Council about two months ago. When, under these circumstances, the Council did me the great honour of asldng me to accept their nomination to the Presidentship, I felt that their request ought to have with me the weight of a command. For a good many years past Chemistry has been growing at a more and more rapid rate, growing in the number and variety of facts which are added to its domain, and not less remarkably in the clearness and consistency of the ideas by which these facts are explained and systematized. The current literature of chemical research extends each year to the dimensions of a small library ; and mere brief abstracts of the original papers published annually by the Chemical Society, partly aided by a grant from this Association, take up the chief part of a very stout volume. I could not, if I would, give you to-night even an outline of the chief newly discovered compounds and of the various changes which they undergo, describing each of them by its own name (often a very long one) and recording the specific properties which give to each substance its highest scientific interest. But I am sure that you ADDRESS. Ixxi would not wish me to do so if I could ; for we do not meet here to study chemistry ; I conceive that we meet here for the purpose of considering what this wondrous activity in our science means, what is the use of it, and, true to our object as embodied in the name of this Association, to consider what we can do to promote the Advancement of Science. I propose to lay before you some facts bearing on each of these questions, and to submit to you some considerations respecting them. In order to ascertain the meaning of the work which has been going on in chemistry, it will, I think, be desirable for us to consider the leading ideas which have been in the minds of chemists, and which guided their operations. Now, since the father of modern chemistry, the great Dalton, gave to che- mists a firm hold of the idea of Atoms, their labours have been continually guided by that fundamental idea, and have confirmed it by a knowledge of more and more facts, while at the same time steadily adding to our know- ledge of the properties of atoms. Every chemist who is investigating a new compound takes for granted that it must consist of a great number of atom- clusters (called by him molecules), all of them alike, and each molecule con- sisting of a certain number of atoms of at least two kinds. One of his first endeavours is to ascertain how many atoms of each kind there are in each molecule of the compound. I must not attempt to describe to you the various kinds of experiment which he performs for the purpose of getting this infor- mation, how each experiment is carried out with the aid of delicate instru- ments and ingenious contrivances found by long experience to enable him to obtain the most trustworthy and accurate results ; but I want to draw your attention to the reasoning by which he judges of the value of such experi- ments when they agree among themselves, and to the meaning which he at- taches to their result. If the result of his experiments does not nearly agree with any atomic for- mula (that is, if no conceivable cluster of atoms of the kinds known to be in the compound would on analysis give such results as those obtained), the chemist feels sure that his experiments must have been faulty : either the sample of substance which he worked upon contained foreign matter, or his analyses were not made with due care. He sets to work again, and goes on till he arrives at a result which is consistent with his knowledge of the com- bining-properties of atoms. It is hardly necessary to say that even the best experiment is liable to error, and that even a result obtained with the utmost care cannot be expected to afford more than an approximation to the truth. Every good analysis of a pure compound leads to results which approximate to those required by the Atomic Theory ; and chemists trust so thoroughly to the truth of that guide, that they correct the results of such analysis by the aid of it. The chemical idea of atoms serves for two purposes : — 1. It gives a clear and consistent explanation of an immense number of facts discovered by experiment, and enables us to compare them with one another and to classify them. 2. It leads to the anticipation of new facts, by suggesting new compounds which may be made ; at the same time it teaches us that no compounds can exist with their constituents in any other than atomic proportions, and that experiments which imply the existence of any such compounds are faulty. We have the testimony of the great Bcrzelius to the flood of light which the idea of atoms at once threw on the facts respecting combining proportions which had been accumulated before it was made known ; and from that time /2 Ixxii REPORT — 1873. forward its value has rapidly increased as each succeeding year augmented the number ol' facts which it explained. AUow me at this point of my narrative to pause for a moment in order to pay a tribute of respect and gratitude to the memory of one who has recently passed from among us, and who in the time of his full activity was a leader of the discoveries of new facts in the most difficult part of our science. Liebig has been generally known in tliis countrj' through his writings on agricultural chemistry, through his justly popular letters on chemistry, and other writings, by means of which his brilliant intellect and ardent imagination stimulated men to think and to work. Among chemists he was famed for his numerous discoveries of new organic compounds, and their investigation by the aid of improved methods ; but I believe that the greatest service which his genius rendered to science was the establishment of the chemical school of Giessen, the prototype of the numerous chemical schools for which Germany is now so justly celebrated. I think it is not too much to say that the Giessen laboratory, as it existed some thirty years ago, was the most efficient organization for the promotion of chemistry which had ever existed. Picture to yourselves a little community of which each member was fired with enthusiasm for learning by the genius of the great master, and of which the best energies were concentrated on the one object of experimental inves- tigation. The students were for the most part men who had gone through a full curriculum of ordinary studies at some other University, and who were attracted from various parts of the world by the fame of this school of research. Most of the leading workers of the next generation were pupils of Liebig ; and many of them have established similar schools of research. We must not, however, overlook the foct that Liebig's genius and enthusiasm would have been powerless in doing this admirable work, had not the rulers of his Grand-Duchy been enlightened enough to know that it was their duty to supply him with the material aids requisite for its successful accomphshment. Numberless new compounds have been discovered under the guidance of the idea of atoms ; and in proportion as our knowledge of substances and of their properties became more extensive, and our view of their characteristics more accurate and general, were we able to perceive the outlines of their natural arrangement, and to recognize the distinctive characteristics of various classes of substances. I wish I could have the pleasure of describing to you the origin and nature of some of these admirable discoveries, such as homologous series, types, radicals, &c. ; but it is more to our purpose to consider the effect which they have had upon the idea of atoms, an idea which, still in its infancy, was plunged into the intellectual turmoil arising from a variety of novel and original theories suggested respectively by independent workers as best suited for the explanation of the particular phenomena to which their attention was mainly directed. Each of these workers was inclined to attach quite sufficient importance to his own new idea, and to sacrifice for its sake any other one capable of inter- fering with its due development. The father of the atomic theory was no more ; and the little infant had no chance of Hfe, unless from its own sterling merits it were found useful in the work still going on. What then was the result ? Did it perish like an ephemeral creation of human fancy? or did it surN-ive and gain strength by the inquiries of those who questioned Nature and knew how to read her answers ? ADDRESS. Ixxiii Although anticipating my answer to these questions, you will probably be surprised to hear the actual result which I have to record, a result so won- derful that the more I think of it the more I marvel at it. Not only did these various theories contain nothing at variance with the atomic theory ; they were found to be natural and necessary developments of it, and to serve for its application to a variety of phenomena which were unknown to its founder. Among the improvements of our knowledge of atoms which have taken place, I ought to mention the better evaluations of the relative weight of atoms of different kinds, which have been made since Dalton's time. More accurate experiments than those which were then on record have shown us that certain atoms are a little heavier or lighter than was then believed, and the work of perfecting our observations is constantly going on with the aid of better instruments and methods of operation. But, apart from these special corrections, a more sweeping change has taken place, not in consequence of more accurate experiments interpreted in the usual way, but in consequence of a more comprehensive view of the best experi- mental results which had been obtained, and a more consistent interpreta- tion of them. Thus the atomic weight of carbon had been fixed at 6 by Dumas's admirable experiments ; and it was quite conceivable that a still more perfect determination might slightly increase or diminish this number. But those who introduced the more sweeping change asserted in substance that two of these supposed atoms, whatever may be the precise weight of each, always are together and never separate from one another; and they accordingly applied the term atom to that indivisible mass of carbon weighing twice as much as a carbon atom had been supposed to weigh. So also with regard to other elements, it has been shown that many atoms are really twice as heavy as had been supposed, according to the original interpretation of the best experiments. This change was brought about by what I may be permitted to call the operation of stock-taking. Dalton first took stock of our quantitative facts in a business-like manner ; but the amount and variety of our chemical stock increased so enormously after his time, that the second stock-taking absorbed the labours of several men for a good many years. They were men of different countries and very various turns of mind ; but, as I mentioned just now, they found no other fundamental idea to work with than Dalton's ; and the result of their labo\irs has been to confirm the truth of that idea and to extend greatly its application. One of the results of our endeavours to classify substances according to their natural resemblances has been the discovery of distinct family relationships among atoms, each family being distinguished by definite characteristics. Now, among the properties which thus characterize particular families of atoms, there is one of which the knowledge gradually worked out by the labours of an immense number of investigators must be admitted to consti- tute one of the most important additions ever made to our knowledge of these little masses. I will endeavour to explain it to you by a simple example. An atom of chlorine is able to combine with one atom of hydrogen or one atom of potas- sium ; but it cannot combine with two atoms. An atom of oxygen, on the other hand, can combine with two atoms of hydrogen or with two atoms of potassium, or with one atom of hydrogen and one of potassium ; but we cannot get it in combination with one atom of hydrogen or of potassium solely. Again, an atom of nitrogen is known in combination with three atoms of hydrogen ; while an atom of carbon combines with four of hydrogen. Other Ixxiv REPORT — 1873. atoms are classified, from their resemblance to these respectively, as Monads, Dyads, Triads, Tetrads, ifee. The combining value which we thus recognize in the atoms of these several classes has led us naturally to a consideration of the order in which atoms are arranged in a molecule. Thus, in the compound of oxygen with hydro- gen and potassium, each of these latter atoms is directly combined with the oxygen, and the atom of oxygen serves as a connecting link between them. Hydrogen and potassium have never been found capable of uniting directly ■vvith one another ; but when both combined with one atom of oxygen they are in what may be called indirect combination with one another through the medium of that oxygen. One of the great difficulties of chemistry some few years ago was to ex- plain the constitution of isomeric compounds, those compounds whose mole- cules contain atoms of like kinds and in equal numbers, but which differ from one another in their properties. Thus a molecule of common ether contains four atoms of carbon, ten atoms of hydrogen, and one of oxygen. Butylic alcohol, a very different substance, has precisely the same composition. We now know that in the former the atom of oxygen is in the middle of a chain of carbon atoms, whereas in the latter it is at one end of that chain. You might fancy it impossible to decide upon any thing like consistent evi- dence such questions as this ; but I can assure you that the atomic theory, as now used by chemist?, leads frequently to conclusions of this kind, which are confirmed by independent observers, and command general assent. That these conclusions are, as far as they go, true descriptions of natural phe- nomena is shown by the fact that each of them serves in its turn as a step- ping-stone to further discoveries. One other extension of our knowledge of atoms I must briefly mention, one which has as yet received but little attention, yet which wiU, I venture to think, be found serviceable in the study of the forces which bring about chemical change. The original view of the constitution of molecules was statical ; and che- mists only took cognizance of those changes of place among their atoms which result in the disappearance of the moleci;les employed, and the appearance of new molecules formed by their reaction on one another. Thus, when a solution of common salt (sodic chloride) is mixed with a solution of silver nitrate, it is well known that the metallic atoms in these respective com- pounds change places with one another, forming silver chloride and sodic nitrate ; for the silver chloride soon settles to the bottom of the solution in the form of an insoluble powder, while the other product remains dissolved in the liquid. But as long as the solution of salt remained undecomposed, each little molecule in it was supposed to be chemically at rest. A parti- cular atom of sodium which was combined with an atom of chlorine was sup- posed to remain steadily fixed to it. When this inactive solution was mixed with the similarly inactive solution of silver nitrate, the interchange of atoms known to take place between their respective molecules was nominally ex- plained by the force of predisposing affinity. It was, in fact, supposed that the properties of the new compounds existed and produced effects before the compounds themselves had been formed. I had occasion to point out a good many years ago that molecules which appear to be chemically at rest are reacting on one another when in suitable conditions, in the same kind of way as those which are manifestly in a state of chemical change — that, for instance, the molecules of liquid sodic chloride exchange sodium atoms with one another, forming new molecules of the same ADDllESS. IxXV compound undistinguishable from the first, so that, in an aggregate of like molecules, the apparent atomic rest is the result of the interchange of like atoms between contiguous molecules. Such exchanges of atoms take place not only between molecules of iden- tical composition, but also between contiguous molecules containing different elements. For instance, in a mixture of sodic chloride and potassic iodide an interchange of metallic atoms takes place, forming potassic chloride and sodic iodide. The result of the exchange in such a case is to form a couple of new molecules diff^erent from the original couple. But these products are subject to the same general law of atomic exchanges, and their action on one another reproduces a couple of molecules of the materials. Thus a liquid mixture formed from two compounds, contains molecules of four kinds, which we may describe as the two materials and the two products*. The materials are reacting ou one another, forming the products ; and these products are, in their turn, reacting on one another, reproducing the materials. If one of the products of atomic exchange between two molecules is a solid while the otlier remains liquid (as when sodic chloride is mixed with silver nitrate), or if one is gaseous while the other remains liquid, so that the molecules of the one kind cannot react on those of the other kind and re- produce the materials, then the continued reaction of the materials on one another leads to their complete mutual decomposition. Such complete mu- tual decomposition of two salts takes place whenever they react on one another under such conditions that the products cannot react on one another and reproduce the materials; whereas partial decomposition takes place whenever the materials form a homogeneous mixture with the products. Now, if in any such homogeneous mixture more exchanges of atoms take place between the materials than between the products, the number of mole- cules of the products is increased, because more of them are being made than unmade ; and reciprocally, if more exchanges of atoms take place between the products than between the materials, the number of molecules of the materials is increased. The mixture remains of constant composition when there are in the unit of time as many decomposing changes as reproducing changes. Suppose that we were to determine by experiment the proportion between the number of molecules of the materials, and the number of molecules of the products, in a mixture the composition of which remains constant, and that we found, for instance, twice as many of materials as of products ; what would this mean ? Why, if every two couples of materials only effiect in the unit of time as many exchanges as every one couple of products, every couple of materials is only exchanging half as fast as every couple of products. In fact you perceive that a determination of the proportion in which the substances are present in such a mixture will give us a measure of the rela- tive velocities of those particular atomic motions ; and we may thus express our result : — The force of chemical combination is inversely proportional to the number of atomic interchanges. I cannot quit this part of our subject without alluding to the fact that some few chemists of such eminence as to be entitled to the most respectful attention, have of late years expressed an opinion that the idea of atoms is not necessary for the explanation of the changes in the chemical constitution of matter, and have sought as far as possible to exclude from their language an}r allusion to atoms. It would be out of place on this occasion to enter into any discussion of the questions thus raised ; but I think it right to point out : — Ixxvi REPORT — 1873. I. That these objectors have not shown us any inconsistency in the atomic theory, nor in the conclusions to Avhich it leads. II. That neither these nor any other philosophers have been able to ex- plain the facts of ehemistr}^ on the assumption that there are no atoms, but that matter is infinitely divisible. III. That when they interpret their analyses, these chemists allow them- selves neither more nor less latitude than the Atomic Theory allows ; in fact they are unconsciously guided by it. These facts need no comment from me. Our science grows by the acquisition of new facts which have an intel- ligible place among our ideas of the order of nature ; but in proportion as more and more facts are arranged before us in their natural order, in pro- portion as our view of the order of nature becomes clearer and broader, we are able to observe and describe that order more fullj' and more accurately — in fact, to improve our ideas of the order of nature. These more extensive and more accurate ideas suggest new observations, and lead to the discovery of truths ■which would have found no place in the narrower and less accurate sj'^stem. Take away from Chemistry the ideas which connect and explain the multifarious facts observed, and it is no longer a science ; it is nothing more than a confused and useless heap of materials. The answer to our question respecting the meaning of the earnest work which is going on in our science must, I think, now be plain to you. Chemists are examining the combining-properties of atoms, and getting clear ideas of the constitution of matter. Admitting, then, for the present, that such is the meaning of chemical work, we have to consider the more important question of its use ; and I think you will agree with me that, in order to judge soundly whether and in what manner such a pursuit is useful, we have to consider its effect upon Man. What habits of mind does it engender ? What powers does it de- velope ? Does it develope good and noble qualities and aspirations, and tend to make men more able and more anxious to do good to their felloAV men ? Or is it a mere idle amusement, bearing no permanent fruits of improvement ? You wiU, I think, answer these questions yourselves if I can succeed in describing to you some of the chief qualities which experience has shown to be requisite for the successful pursuit of Chemistry, and which are neces- sarily cultivated by those who qualify themselves for such a career. One of the first requirements on the part of an investigator is accuracy in observing the phenomena with which he deals. He must not only see the precise particulars of a process as they present themselves to his observation ; he must also observe the order in which these particular appearances present themselves under the conditions of each experiment. No less essential is accuracy of memory. An experimental inquirer must remember accurately a number of facts ; and he needs to remember their mutual relations, so that one of them when present to his mind may recall those others which ought to be considered with it. In fact he cultivates the habit of remembering facts mainly by their place in nature. Accuracy in manual operations is required in all experimental inquiries ; and many of them aiford scope for very considerable skill and dexterity. These elementary qualities are well known to be requisite for success in experimental science, and to be developed by careful practice of its methods ; but some higher qualities are quite as necessary as these in all but the most rudimentary manipulations, and are developed in a remarkabl* degree by the higher work of science. ADDRESS. Ixxvii Thus it is of importance to notice that a singularly good training in the accurate use of words is afforded by experimental Chemistry. Every one who is about to enter on an inquiry, whether he be a first-year's student who wants to find the constituents of a common salt, or whether he be the most skilled and experienced of Chemists, seeks beforehand to get such in- formation from the records of previous observations as may be most useful for his purpose. This information he obtains through the medium of words ; and any failure on his part to understand the precise meaning of the words conveying the information requisite for his guidance is liable to lead him astray. Those elementary exercises in analytical chemistry, in which brief directions to the students alternate with their experiments and their reports of experiments made and conclusions drawn, afford a singularly effective training in the habit of attending accurately to the meaning of words used by others, and of selecting words capable of conveying without ambiguity the precise meaning intended. Any inaccuracy in the student's apprehension of the directions given, or in the selection of words to describe his obser- vations and conclusions, is at once detected, when the result to which he ought to have arrived is known beforehand to the teacher. Accuracy of reasoning is no less effectively promoted by the work of ex- perimental chemistry. It is no small facility to ns that the meaning of the words which we use to denote properties of matter and operations can be learnt by actual observation. Moreover each proposition comprised in che- mical reasonings conveys some distinct statement susceptible of verification by similar means ; and the validity of each conclusion can be tested, not only by examining whether or not it follows of necessity from true premisses, but also by subjecting it to the independent test of special experiment. Chemists have frequent occasion to employ arguments which indicate a probability of some truth ; and the anticipations based upon them serve as guides to experimental inquiry by suggesting crucial tests. But they distin- guish most caroiuUy such hypotheses from demonstrated facts. Thus a pale green solution, stated to contain a pure metallic salt, is found to possess some properties which belong to Salts of Iron. ]!^othing else pos- sesses these properties except Salts of Nickel ; and they manifest a slight dif- ference from Iron Salts in one of the properties observed. The analyst could not see any appearance of that peculiarity which distin- guishes Nickel Salts ; so he concludes that he has probably got Iron in his solution, brt almost certainly either Iron or Nickel. He then makes an ex- periment which will, he knows, give an entirely different result with Iron Salts and Nickel Salts ; and he gets very distinctly the result which indicates Iron. Having found in the green liquid properties which the presence of Iron could alone impart, he considers it highly probable that Iron is present. But he does not stop there ; for, although the facts before him seem to admit of no other interpretation, he knows that, from insufficient knowledge or attention, mistakes are sometimes made in very simple matters. The analyst therefore tries as many other experiments as are known to distinguish Iron Salts from all others ; and if any one of these leads distinctly to a result at variance with his provisional conclusion, he goes over the whole inquiry again, in order to find where his mistake was. Such inquiries are practised largely by students of chemistry, in order to fix in their minds, by frequent use, a know- ledge of the fundamental properties of the common elements, in order to learn by practice the art of making experiments, and, above all, in order to acquire the habit of judging accurately of evidence in natural phenomena. 1873. n Ixxviii REPORT — 1873. Such a student is often surprised at being told that it is not enough for him to conduct his experiments to such a point that every conclusion except one is contrary to the evidence before him- — that he must then try every confirma- tory test which he can of the substance believed to be present, and ascertain that the sample in his hands agrees, as far as ho can see, in all properties Avith the known substance of which he believes it to be a specimen. Those who tread the path of original inquiry, and add to human know- ledge by their experiments, are bound to practise this habit Avith the most scrupulous fidelity and care, or many and grave would be the mistakes they would make. Thus a Chemist thinks it probable that he might prepare some well-known organic body of the aromatic family by a new process. He sets to work and obtains a substance agreeing in appearance, in empirical composition, in molecular weight, and in many other properties with the compound which he had in view. He is, however, not satisfied that his product is a sample of that compound until he has examined carefully whether it possesses all the properties which are known to belong to the substance in question. And many a time is his caution rewarded by the discovery of some distinct dif- ference of melting-point, or of crystalline form, &c., which jn-ovcs that he has made a new compound isomeric with the one which he expected to make. It seemed probable, from the agreement of the two substances in many particulars, that they might be found to agree in all, and might be considered to be the same compound ; but complete proof of that conclusion consists iu showing that the new substance agrees with all that we know of the old one. In the most vai'ious ways chemists seek to extend their knowledge of tbe uniformity of nature ; and their reasonings by analogy from particulars to particulars suggest the working hypotheses M'hich lead to new observations. Before, however, proceeding to test the trutli of his hypothesis by experi- ment, the chemist passes in review, as well as he can, all the general know- ledge which has any bearing on it, in order to find agreement or disagree- ment between his hypothesis and the ideas estabhshed by past experience. Sometimes he sees that his hypothesis is at variance with some general law in which he has full confidence, aud he throws it aside as disproved by that law. On other occasions he finds that it follows of necessity from some known law ; and he then proceeds to verify it by experiment, with a confident anticipation of the result. In many cases the hypothesis does not present sufficiently distinct agreement or disagreement with the ideas established by previoiis investigations to justify either the rejection of it or a confident belief in its truth; for it often happens that the results of experience of similar phenomena are not embodied in a sufficiently definite or trustworthy statement to have any other effect than that of giving probability or the contrary to the hypothesis. Another habit of mind which is indispensable for success in experimental chemistry, and which is taught by the practice of its various operations, is that of truthfulness. The very object of all our endeavours is to get true ideas of the natural processes of chemical action ; for in proportion as our ideas are true do they give us the power of directing these in'ocesses. In fact our ideas are useful only so far as they are true ; and he must indeed be blind to interest and to duty who could wish to swerve from the path of truth. But if any one were weak enough to make the attempt, he would find his way barred by innu- merable obstacles. Every addition to our science is a matter of immediate interest and im- ADDRESS. Ixxix portance to those who are working in the same direction. They verify in various ways the statements of the first discoverer, and seldom fail to uoticc further particulars, and to correct any little errors of detail into which he may have fallen. They soon make it a stepping-stone to further disco- veries. Any thing like wilful misrepresentation is inevitably detected and made known. It must not, however, he supposed that the investigator drifts imcon- sciously into the habit of truthfulness for want of temptation to be un- truthful, or even that error presents itself to his mind in a grotesque and repulsive garb, so as to enlist from the iirst his feelings against it j for I can assure you that the precise contrary of these things happens. Error comes before him usually in the very garb of truth ; and his utmost skill and attention are needed to decide whether or not it is entitled to retain that garb. You will easily see how this happens if you reflect that each working hypothesis employed by an investigator is an uuproven proposition, which bears such resemblance to truth as to give rise to hopes that it may really be true. The investigator trusts it provisionally to tlie extent of trying one or more experiments, of which it claims to predict the specific result. Even though it guide him correctly for a while, he considers it still on trial until it has been tested by every process which ingenuity can suggest for the pur- pose of detecting a fault. Most errors which an experimentalist has to do with are really imperfect truths, which have done good service in their time by guiding the course of discovery. The great object of scientific work is to replace these imper- fect truths by more exact and comprehensive statements of the order of nature. Whoever has once got knowledge from nature herself by truthful reason- ing and experiment, must be dull indeed if he does not feel that he has ac- quired a new and noble power, and if he does not long to exercise it further, and make new conquests from the realm of darkness by the aid of known truths. The habit of systematically searching for truth by the aid of known truths, and of testing the validity of each step by constant reference to nature, has now been practised for a sufficiently long time to enable us to judge of some of its results. Every true idea of the order of nature is an instrument of thought. It can only be obtained by truthful investigation ; and it can only be used effec- tively in obedience to the same laws. But the first idea which is formed of any thing occurring in nature affords only a partial representation of the actual reality, by recording what is seen of it from a particular point of view. By examining a thing from different points of view we get different ideas of it ; and when we compare these ideas accurately with one another, recollect- ing how each one was obtained, we find that they really supplement each other. We try to form in our minds a distinct image of a thing capable of pro- ducing these various appearances ; and when we have succeeded in doing so, we look at it from the different points of view from which the natural object had been examined, and find that the ideas so obtained meet at the central image. It usually happens that an accurate examination of the mutual bearings of these ideas on the central image suggests additions to them, and correction of some particulars in them. Thus it is that true ideas of a natural phenomenon confirm and strengthen r/2 ■ IXXX KEPORT — 1873. one another; and he who aids directly the development of one o them is sure to promote indirectly the consolidation of others. Each onward step in the search for truth has made us stronger for the work ; and when we look back upon what has been done by the efforts of so many workers simply but steadily directed by truth towards further truth, we see that they have achieved, for the benefit of the human race, the con- quest of a systematic body of truths which encourages men to similar efforts while affording them the most effectual aid and guidance. This lesson of the inherent vitality of truth, which is taught us so clearly by the history of our science, is well worthy of the consideration of those who, seeing that iniquity and falsehood so frequently triumph for a while in the struggle for existence, are inclined to take a desponding view of human affairs, and almost to despair of the ultimate predominance of truth and goodness. I beheve it would be impossible at the present time to form an adequate idea of the vast consequences which will follow from the national adop- tion of systematic measures for allowing our knowledge of truth to develope itself freely, through the labours of those who are wUling and able to devote themselves to its service, so as to strengthen more and more the belief and trust of mankind in its guidance, in small matters as Avell as in the highest and most important considerations. T am desirous of describing briefly the more important of those measures ; but first let me mention another habit of mind which naturally follows from the effective pursuit of truth, — a habit which might be described in general terms as the application to other matters of the truthfulness imparted by science. The words which the great German poet put into the mouth of Mephisto- pheles when describing himself to Faust, afford perhaps the most concise and forcible statement of what we may call the anti-scientific spirit : — „ Icli bin der Geist der stets verneint, Dem alles, was entsteht, zuwider 1st." The true spirit of science is certainlj' affirmative, not negative ; for, as I men- tioned just now, its history teaches us that the development of our knowledge usually takes place through two or more simultaneous ideas of the same phe- nomenon, quite different from one another, both of which ultimately prove to be parts of some more general truth ; so that a confident belief in one of those ideas docs not involve or justify a denial of the others. I could give you many remarkable illustrations of this law from among ideas familiar to Chemists. But I want you to consider with me its bearing on the habit of mind called toleration, of which the development in modern times is perhaps one of the most hopeful indications of moral improvement in man. In working at our science we simply try to find out what is true ; for although no usefulness is to be found at first in most of our results, we know well that every extension of our knowledge of truth is sure to prove useful in manifold ways. So regular an attendant is usefulness upon truth in our work, that wc get accustomed to expect them always to go together, and to believe that there must be some amount of truth wherever there is manifest usefulness. The history of human ideas, so far as it is written in the records of the progress of science, abounds with instances of men contributing powerfully to the development of important general ideas, by their accurate and conscien- tious experiments, while at the same time professing an actual disbelief in ADDRESS. Ixxxi those ideas. Those records must indeed have been a dead letter to any one who could stand carping at the intellectual crotchets of a good and honest worker, instead of giving him all brotherly help in furtherance of his work. To one who knows the particulars of our science thoroughly, and who knows also what a variety of ideas have been resorted to in working oiit the whole body of truths of which the science is composed, there are few more impressive and elevating subjects of contemplation than the unity in the clear and bold outline of that noble structure. I hope that you will not suppose, from my references to Chemistry as pro- moting the development of these habits and powers of mind, that I wish to claim for that particular branch of science any exclusive merit of the kind ; for I can assure you that nothing can be further from my intention. I conceived that you would wish me to speak of that department of science which I have had occasion to study more particularly ; but much that I have said of it might be said with equal truth of other studies, while some of its merits may be claimed in a higher degree by other branches of science. On the other hand, those highest lessons which I have illustrated by chemistry are best learnt by those whose intellectual horizon includes other provinces of knowledge. Chemistry presents peculiar advantages for educational purposes in the combination of breadth and accuracy in the training which it affords ; and I am inclined to think that in this respect it is at present unequalled. There is reason to believe that it will play an important part in general education, and render valuable services to it in conjunction with other scientific and with literary studies. I trust that the facts which I have submitted to your consideration may suffice to show you how fallacious is that materialistic idea of Physical Science which represents it as leading away from the study of man's noblest faculties, and from a sympathy with his most elevated aspirations, towards mere inani- mate matter. The material work of science is directed by ideas towards the attainment of further ideas. Each step in science is an addition to our ideas, or an improvement of them. A science is but a body of ideas respecting the order of nature. Each idea which forms part of Physical Science has been derived from ob- servation of nature, and has been tested again and again in the most various ways by reference to nature ; but this very soundness of our materials enables us to raise upon the rock of truth a loftier structure of ideas than could be erected on any other foundation by the aid of uncertain ma- terials. The study of science is the stiidy of man's most accurate and perfect intel- lectual labours ; and he who would know the powers of the human mind must go to science for his materials. Like other powers of the mind, the imagination is powerfully exercised, and at the same time disciplined, by scientific work. Every investigator has frequent occasion to call forth in his mind a distinct image of something in nature which could produce the appearances which he witnesses, or to frame a proposition embodying some observed relation ; and in each case the image or the proposition is required to be true to the materials from which it is formed. There is perhaps no more perfect elementary illustration of the ac- curate and useful employment of the imagination than the process of forming in the language of symbols, from concrete data, one of these admirable general propositions called equations ; on the other hand, the contemplation of the order and harmony of nature as disclosed to us by science supplies the Ixxxii REPOKT — 1873. imagination with materials of surpassing grandeur and brilliancy, while at the same time affording the widest scope for its efforts. The foregoing considerations respecting the meaning and use of scientific work wiU, I trust, afford us aid in considering what measures ought to be taken in order to promote its advancement, and what we can do to further the adoption of such measures. Like any other natural phenomenon, the growth of knowledge in the human mind is favoured and promoted by certain circumstances, impeded or arrested by others ; and it is for us to ascertain from experience what those circumstances respectively are, and how the favourable ones can be best com- bined to the exclusion of the others. The best and noblest things in this world are the result of gradual growth, by the free action of natural forces ; and the proper function of legislation is to systematize the conditions most favourable to the free action which is desired. I shall consider the words " Advancement of Science " as referring to the develoijment and extension of our systematic knowledge of natural phenomena by investigation and research. The first thing wanted for the work of advancing science is a supply of well-qualified workers. The second thing is to place and keep them under the conditions most favourable to their efficient activity. The most suitable men must be found while still young, and trained to the work. Now I know only one reaUy effectual way of finding the youths who are best endowed by nature for the purpose ; and that is to systematize and develope the natural conditions which accidentally concur in particular cases, and enable youths to rise from the crowd. The first of these is that a young man gets a desire for knowledge by seeing the value and beauty of some which ho has acquired. When he has got this desire, he exerts himself to increase his store ; and every difficulty surmounted increases his love of the pursuit, and strengthens his determination to go on. His exertions are seen by some more experienced man, wlio helps him to place himself under circumstances favourable to further progress. He then has op]5ortunities of seeing original inquiries conducted, perhaps even of aid- ing in them ; and he longs to prove that he also can work out new truths, and make some permanent addition to human knowledge. If his circumstances enable him to prosecute such work, and he succeeds in making some new ob- servations worthy of publication, he is at once known by them to the com- munity of scientific men, and employed among them. We want, then, a system which shall give to the young favourable oppor- tunities of acquiring a clear and, as far as it goes, a thorough knowledge of some few truths of nature such as they can understand and enjoy ^ — which shall afford opportunity of further and further instruction to those who have best profited by that which has been given to them, and are anxious to obtain more — which shall enable the best students to see what original investigation is, and, if possible, to assist in carrying out some research — and, finally, which shall supply to each student who has tlie power and the will to conduct researches, all material conditions which are requisite for the j)urpose. But investigators, once found, ought to be placed in the circumstances most favourable to their efficient activity. The first and most fundamental condition for this is, that their desire for the acquidtion of knowledge be kept alive and fostered. Tlioy must not merely retain the hold which they have acquired on the general body of their ADDRESS. Ixxxiii science ; they ought to strengthen and extend that hold, by acquiring a more complete and accurate knowledge of its doctrines and methods ; in a word, they ought to be more thorough students than during their state of preli- minary training. They must bo able to live by their work, without diverting any of their energies to other pursuits ; and they must feel security against want, in the event of illness or in their old age. They must be supplied with intelligent and trained assistants to aid in the conduct of their researches, and whatever buildings, apparatus, and ma- terials may be required for conducting those researches effectively. The desired system must therefore provide arrangements favourable to the maintenance and development of the true student-spirit in investigators, while providing them with permanent means of subsistence, sufficient to enable them to feel secure and tranquil in working at science alone, yet not sufficient to neutralize their motives for exertion ; and at the same time it must give them all external aids, in proportion to their wants and powers of making good use of them. Now I propose to describe the outline of such a system, framed for the sole purpose of promoting research, and then to consider what other results would follow from its working. If it should appear possible to establish a system for the efficient advance- ment of science, which would be productive of direct good to the community in other important ways, I think you will agree with me that we ought to do all that we can to promote its adoption. Let the most intelligent and studious children from every primary school be sent, free of expense, to the most accessible secondary school for one year ; let the best of these be selected and allowed to continue for a second year, and so oji, until the elite of them have learnt all that is to be there learnt to advantage. Let the best pupils from the secondary schools be sent to a col- lege of their own selection, and there subjected to a similar process of annual weeding ; and, finally, let those who get satisfactorily to the end of a college curriculum be supplied with an allowance sufficient for their maintenance for a year, on condition of their devoting their undivided energies to research, under the inspection of competent college authorities, while allowed such aids and facilities as the college can supply, with the addition of money-grants for special purposes. Let aU who do well during this first year be allowed similar advantages for a second, and even a third year. Each young investigator thus trained must exert himself to obtain some apjM)intment, which may enable him to do the most useful and creditable work of which he is capable, while combining the conditions most favourable to his own improvement. Let there be in every college as manj' Professorships and Assistantships in each branch of science as are needed for the efficient conduct of the work there going on, and let every Professor and Assistant have such salary and such funds for apparatus &c. as may enable him to devote all his powers to the duties of his post, under conditions favourable to the success of those duties ; but let each Professor receive also a proportion of the fees paid by his pupils, so that it may be his direct interest to do his work with the utmost attainable efficiency, and attract more pupils. Let every college and school be governed by an independent body of men, striving to increase its usefulness and reputation, by sympathy with the labours of the working staff, by material aid to them when needed, and by getting the very best man they can, from their own or any other college, to supply each vacancy as it arises. Ixxxiv REPORT — 1873. Ill addition to colleges, •whicli are and always have been the chief institu- tions for the advancement of learning, establishments for the observation of special phenomena are frequently needed, and will doubtless be found de- sirable in aid of a general system for the advancement of science. Now, if a system fulfilling the conditions which I have thus briefly sketched out were once properly established on a sufficient scale, it ought to develop and improve itself by the very process of its working ; and it behoves us, in judging of the sj'stem, to consider how such development and improvement would come about. The thing most needed at the present time for the advancement of science is a supply of teachers devoted to that object— men so earnestly striving for more knowledge and better knowledge as to be model students, stimulating and encouraging those around them by their example as much as by their teaching. Young men do not prepare themselves in any numbers for such a career : — I. Because the chief influences which surround them at school and at college are not calculated to awaken in them a desire to obtain excellence of such kind. II. Because they could not expect by means of such qualities to reach a l)osition which would afford a competent subsistence. Let these conditions be reversed, to the extent that existing teachers have powerful inducements to make their students love the study of science for its own sake, with just confidence that they will be able to earn a livelihood if they succeed in qualifying themselves to advance science, and the whole thing is changed. The first batch of young investigators will be dispersed among schools and colleges according to their powers and acquirements, and will at once improve their influence upon the pupils, and enable them to send up a second batch better trained than the first. This improvement will go on increasing, if the natural forces which promote it are allowed free play ; and the youth of each successive generation will have better and more fre- quent oj)portunities of awakening to a love of learning, better help and guidance in their efforts to acquire and use the glorious inheritance of know- ledge which had been left them, better and more numerous living examples of men devoting their whole lives to the extension of the domain of truth, and seeking their highest reward in the consciousuess that their exertions have benefited their fellow men, and are appreciated by them. A young man who is duly qualified for the work of teaching the investi- gation of some particular branch of science, and who wishes to devote him- self to it, will become a member of an association of men selected for their known devotion to learning, and for their ability to teach the methods of investigation in their respective subjects. Around this central group is arranged a frequently changing body of youths, who trust to them for en- couragement and guidance in their respective studies. Our young investigator finds it necessary to study again more carefully many parts of his subject, and to examine accurately the evidence of various conclusions which he had formerly adopted, in order that he may be able to lead the minds of his pupils by easy and natural yet secure steps to the dis- covery of the general truths which are within their reach. He goes over his branch of science again and again from the foundation upwards, striving each time to present its essential particulars more clearly and more forcibly, arranging them in the order best calculated to stimulate an inquiring mind to reflect upon their meaning, and to direct its efforts effectively lo the dis- covery of the general ideas which are to be derived from them. He is en- ADDRESS. IXXXV couragcd in these cflforts by the sympathy of his colleagues, and often aided by suggestions derived from their experience in teaching other branches of science, or by information respecting doctrines or methods which throw a light upon those of his own subject. No known conditions are so well calculated to give a young investigator the closest and strongest grasp of his subject of which he is capable as those in which he is placed while thus earnestly teaching it in a college ; and in- asmiieh as a thorough mastery of known truths is needed by every one who would work to advantage at the discovery of new truths of that kind, it will, in most cases, be an object of ambition to the ablest young investigators to get an opportunity of going through the work of teaching in a college, in order to improve themselves to the utmost for the work of original research. There is, however, another advantage to them in having such work to do; for the best way to ascertain at any one time what additions may be made to a science, is to examine the facts which have been discovered last, and to consider how far they confirm and extend the established ideas of the science, how far they militate against those ideas. An investigating teacher is con- stantly weaving new facts into the bodj^ of his science, and forming antici- pations of new truths by considering the relation of these new facts to the old ones. "When our investigator has thus got a thoroiigh mastery of his science and new ideas for its extension, he ought to have the opportimity of turning his improved powers to account by devoting more of his time to original research ; in fact he ought to teach research by example more than hitherto, and less by elementary exercises upon known facts. If he has discharged the duties of his first post with manifest efliciency, he will be promoted, either in his own or some other college, to a chair affording more leisure and facility for original research by his own hands and by those of his assistants and pupils. Some investigators may find it desirable to give up after a while all teaching of previously published truths, and confine themselves to guiding the original researches of advanced pupils, while stimulating them by the example of their own discoveries. But most of them wiU probably prefer to do elemen- tary teaching work from time to time, for the sake of the opportunity of going over the groundwork of their science, with a knowledge of the new facts and enlarged ideas recently established. Now it must be observed that such a system as the above, once developed to its proper proportions, so as to send annually to secondary schools many thousands of poor children who would otherwise never enjoy such advantages, and so as to train to original investigation a corresponding proportion of them, would not only provide more young investigators than would be needed for systematic teaching functions, but would also give a partial training of the same kind to many whose abilities proved to be insufficient, or whose tastes were not congenial to such pursuit. Some would be tempted by an advan- tageous opening in an industrial pursuit or in the public service to break ojff their studies before completion, and others would find, after completing their training, a position of that kind more desirable or more attainable than a purely scientific appointment. Not only would much good of other kinds be accomplished by this circumstance, but we may say with confidence that the system could not work with full advantage for its own special purpose of promoting the advancement of science if it did not diffuse a knowledge of the truths and methods of science beyond the ciixle of teachers. There is an urgent need of accurate scientific knowledge for the direction of manufacturing processes, and there could not be a greater mistake than to IxXXvi REPORT — 1873. suppose tliat such knowledge need not go beyond tlie elementary truths of science. In every branch of manufacture improvements are made from time to time, by the introduction of new or modified processes which had been discovered by means of investigations as arduous as those conducted for purely scientific purposes, and involving as great powers and accomplish- ments on the part of those who conducted them. Any manufacturer of the present day who does not make efficient arrange- ments for gradually perfecting and improving his processes ought to make at once enough money to retire ; for so many are moving onwards in this and other countries, that he would soon be left behind. It would be well worth while to establish such a system of scientific educa- tion for the sake of training men to the habits of mind which are required for the improvement of the manufacturing arts ; and I have no doubt that the espensc of Avorking tlie system would be repaid a hundred times over by the increase of wealth of the community ; but I only mention this as a secondarjr advantage of national education. A system of the kind could not expand to due dimensions, nor could it, once fully established, maintain itself in full activity, without intelligent sj'mpathy from the community ; and accordingly its more active-minded members must be taught some good examples of the processes and results of scientific inquiry, before they can be expected to take much interest in the results achieved by inquirei's, and to do their share of the work requisite for tlie success of the system, I need hardly remind you that there are plenty of other strong reasons why some such knowledge of the truths of nature, and of the means by which they are found out, should be difi^used as widely as possible throughout the community. You perceive that in such educational system each teacher must trust to his own exertions for success and advancement ; and he wiU do so if he is sure that his results will be kno^m and compared impartially with those attained by others. Each governing body must duly maintain the efficiency of their school or college, if its siipport depend in some degree on the evi- dences of that efficiency ; and they will try to improve their school if they know that every imx^rovement will be seen and duly appreciated. The keystone of the whole structure is the action of the State in distri- buting funds carefully among schools and colleges proportionally to the evi- dence of their doing good work, which could not be continued without such aid. I am inclined to think that the State ought, as far as possible, to confine its educational grants to the purpose of maintaining and continuing good work which is actually being done, and rarely if ever to initiate educational experiments : first, because it is desirable to encourage private exertions and donations for the establishment of schools and colleges upon new systems, or in new localities, by giving the public fuU assui'ance that if any new institution establishes its right to existence, by doing good work for a while, it will not be allowed to die off for want of support ; and, secondly, because the judicial impartiality required in the administration of public funds, on the basis of results of work, is hardly compatible with an advocacy of any particular means of attaining such results. On the other hand, experience has shown that special endowments, which tie up funds in perpetuity for a definite pm'pose, commonly fail to attain their object under the altered circumstances which spring up in later generations, and not unfrequently detract from the efficiency of the institutions to which they are attached, by being used for objects other than those which it is their proper function to promote. ADDRESS. IxXXvii "When there is felt to be a real want of any new institution for the promo- tion of learning, men are usually wUling enough to devote time and money to the purpose of establishing it and giving it a fair trial. It is desirable that they should leave the State to judge of their experiment by its results, and to maintain it or not, according to the evidences of its usefulness. No institution ought, for its own sake, to have such permanent endowments as might deprive its members of motives for exertion. The State could not, however, discharge these judicial functions without accurate and trustworthy evidence of the educational work done at the various schools and of its success. Por this purpose a record must be kept by or under the direction of every teacher of the weekly progress of each pupU, showing what he has done and how he has done it. Official inspectors would have to see to these records being kept upon a uniform scale, so that their results might be comparable. The habit of keeping such records conduces powerfully to the efficiency of teachers ; and, for the sake of the due develop- ment of the teaching system, it ought to prevail generally. Having such full and accui'ate means of knowing what opportunities of improvement jrapils have enjoyed and what use they have made of those opportunities, Govern- ment ought to stimulate their exertions and test their progress by periodical examinations. It is of the utmost importance to allow any new and improved system of instruction to develope itself freely, by the exertions of those who are willing to undertake the labour and risk of trying it on a practical scale ; and the pupils who acquire upon sivAi new system a command of any branch of science, ought to have a fair opportunity of showing what they have achieved and how they have achieved it. An able and impartial examiner, knowing the new systems in use, will encourage each candidate to work out his results in the manner in which he has been taught to work out results of the kind. Examinations thus impartiallj'^ conducted with a view of testing the suc- cess of teachers iu the work which they are endeavouring to do, have a far higher value, and consequent authority, than those which are conducted in ignorance or disregard of the process of training to which the candidates have been subjected ; and we may safely say that the examination system will not attain its full usefulness until it is thus worked in intimate connexion with a system of teaching. In order to give every one employed in the educational system the utmost interest in maintaining and increasing his efficiency, it is essential that a due measure of publicity be given to the chief results of their respective labours. Schools and colleges ought, to a considerable extent, to be supported b)' the fees paid by pupUs for the instruction received ; and every Professor being in part dependent upon the fees of his pupils Avill have a direct interest in attracting more pupils to his classes or laboratories. The fame of important original investigations of his own or his pupils, published in the scientific journals, is one of the natural means by which a distinguished Professor attracts disciples, and the success of his pupils in after life is another. His prospects of promotion will depend mainly on the opinion formed of his powers from such materials as these by the governing bodies of colleges and bj- the public ; for if each college is dependent for success upon the efficiency of its teaching staff, its governing body must do their best to till up every vacancy as it arises by the ajipointment of the ablest and most successful Professor whom they can get ; and any college which does not succeed in obtaining the services of able men will soon lose reputation, and fall off in numbers. Ixxxviii KEPORT — 1873. There are, however, further advantages to the working of the system to be derived from full publicity of aU its more important proceedings. It will supply materials for the formation of a sound public opinion respecting the proceedings of the authorities in their various spheres of action. A claim for money might be made upon Government by the rulers of some college iipon inadequate grounds ; or a just and proper claim of the kind might be disre- garded by Government. Neither of these things wiU be likely to happen very often if the aj^plications, together with the evidence bearing on them, are open to public scrutiny and criticism ; and when they do occasionally happen, there will be a natural remedy for them. If I have succeeded in making clear to you the leading principles of the plan to be adopted for the advancement of science, including, as it necessa- rily must do, national education generally, you will, I think, agree with me that, from the very magnitude and variety of the interests involved in its action, such system must of necessity be under the supreme control of Government. Science wiU never take its proper place among the chief ele- ments of national greatness and advancement until it is acknowledged as such by that embodiment of the national wUl which we call the Government. Nor can the various institutions for its advancement develope duly their useful- ness until the chaos in which they are now plunged gives place to such order as it is the proper function of Government to establish and maintain. But Government has already taken, and is continuing to take, action in various matters affecting elementaiy popular education and higher scientific education, and it would be difficult to arrest such action, even if it were thought desirable to do so. The onlj' practical ^question to be considered is how the action of Government can be systematized so as to give fi'ee plaj' to the natural forces which have to do the work. By establishing official examinations for appointments and for degrees, Government exerts a powerful influence on the teaching in schools and colleges, without taking cognizance, except in some few cases, of the systems of teaching which prevail in them. Again, they give grants of public money from time to time in aid of colleges or universities, or for the establishment of a high school under their own auspices. Sometimes they endow a Professor- ship. In taking each measure of the kind they are doubtless influenced by evidence that it is in itself a good thing, calculated to promote the advance- ment of learning. But a thing which is good in itself may produce evil effects in relation to others, or good effects incommensurate with its cost. Thus examinations afford most valuable aid to educational work when carried on in conjunction with earnest teachers ; yet when established in the absence of a good system of education, they are liable to give rise to a one-sided train- ing contrived with a special view of getting young men through the exami- nations. If no properly educated young men were found for a particular de- partment of the public service, and an examination of aU candidates for such appointments were to be established for the purpose of improving the system of training, candidates would consider their power of answering such ques- tions as appeared likely to be set as the condition of their obtaining the ap- pointments, and they svould look out for men able and willing to train them to that particular work in as direct and effective a manner as possible. The demand for such instruction would soon be supplied. Some teachers would undertake to give instruction for the mere purpose of enabling candidates to get through the examination; and by the continued habit of such work would gradually come to look upon the examiners as malignant beings who keep youths out of ofiice, and whose vigilance ought to be evaded by such means ADDRESS. Ixxxix as experience might show to be most effective for the purpose. Once this kind of direct examination-teaching has taken root, and is known to produce the desired effect of getting young men through the examinations, its exist- ence encourages the tendency on the part of the candidates to look merely to the examination as the end and aim of their study ; and a class of teachers is developed whose exertions are essentially antagonistic to those of the examiners. There are, no doubt, teachers with a sufficiently clear apprehension of their duty, and sufficient authority, to convince some of the candidates that the proper object of their study should be to increase their power of usefulness in the career for which they are preparing themselves, by thoroughly master- ing up to a prescribed point certain branches of knowledge ; and that until they had honestly taken the means to do this and believed they had done it effectually, they ought not to go up for examination nor to wish to commence their career. But it is desirable that all teachers be placed under such circumstances that it may become their interest as well as their duty to cooperate to the utmost of their powers in the object for which the examiners are working. For this purpose their records of the work done under their guidance by each pupil ought to be carefully inspected by the examiners before framing their questions, and ought to be accepted as affording the chief evidence of the respective merits of the pupils. This is not the place for considering how the general funds for an effective system of national education can best be raised, nor how existing educational endowments can best be used in aid of those funds. It is well known that some colleges of Oxford and Cambridge are possessed of rich endowments, and that many distinguished members of those universities are desirous that the annual proceeds of those endowments should be distributed upon some system better calculated to promote the advancement of learning than that which generally prevails. Indeed we may confidently hope that, true to their glorious traditions, those colleges will be led, by the high- minded and enlightened counsels of their members, to rely upon improving usefulness in the advancement of learning as the only secure and worthy basis of their action in the use of their funds, so that they may take a leading part in such system of national education as may be moulded out of the present chaos. But the foundations of a national system of education ought to be laid independently of the present arrangements at Oxford and Cambridge, for we may be sure that the more progress the system makes the more easy will become the necessary reforms in the older universities and colleges. It is clearly undesirable that Government should longer delay obtaining such full and accurate knowledge of the existing national resources for educational purposes, and of the manner in which they are respectively utilized, as may enable them to judge of the comparative prospects of use- fulness presented by the various modes of distributing educational grants. They ought to know what has been done and what is doing in the various public educational establishments before they can judge which of them would be likely to make the best use of a grant of public money. We have official authority for expecting such impartial administration of educational grants; and it cannot be doubted that before long due means will be taken to supply the preliminary conditions. You are no doubt aware that a lloyal Commission was appointed some y' Xe REPORT— 1873. time ago in consequence of representations made to Government by the Britishi Association on this subject, and it is understood that their instruc- tions are so framed as to direct their particular attention to the manner in which Government may best distribute educational grants. The Commission is moreover composed of most distinguished men, and we have every reason to anticipate from their labours a result worthy of the nation and of the momentous occasion. In speaking of public educational establishments, I refer to those which by their constitution are devoted to the advancement of learning without pecuniary profit to their respective governing bodies. The annual expen- diture requisite for keeinng up a national system of popular education will necessarily be considerable from the first, and will become greater from year to year ; but once Englishmen are fully alive to the paramount importance of the object, and see that its attainment is within their reach, we may be sure that its expense will be no impediment. England would not deserve to reap the glorious fruits of the harvest of knowledge if she grudged the necessary outlay for seed and tillage, were it even ten times greater than it will be. It is no use attempting to establish a national system on any other than a truly national basis. Private and corporate funds inevitably get diverted from popular use, after a few generations, to the use of the influ- ential and rich. A. national system must steadily keep in view the improve- ment of the poor, and distribute public funds each year in the manner best calculated to give to the youths of the poorest classes full opportunities of improvement proportional to their capacities, so that they may qualify them- selves for the utmost usefulness to their country of which they are capable, " The best possible security for the proper administration of the system will be found in the full and speedy publicity of all the particulars of its working. It has been frequently remarked that a great proportion of English in- vestigators are men of independent means, who not only seek no advance- ment as a reward of their labours, but often sacrifice those opportunities of improving their worldly position which then- abilities and influence open up to them, for the sake of quietly advancing human knowledge. Eich and powerful men have very great temptations to turn away from science, so that those who devote their time and money to its service prove to us how true and pure a love of science exists in this country, and how Englishmen will cultivate it when it is in their power to do so. Now and then a youth from the poorer classes is enabled by fortunate accidents and the aid of a friendly hand to climb to a position of scientific activity, and to give us, as Faraday did, a sample of the intellectual powers which lie fallow in the great mass of the people. Now, the practical conclusion to which I want to lead you is, that it rests with you, who represent the national desire for the advancement of science, to take the only measures which can now be taken towards the establish- ment of a system of education worthy of this country and adapted to the requirements of science. In the present stage of the business the first thing to be done is to arouse public attention by all practicable means to the im- portance of the want, and to get people gradually to agree to some definite and practicable plan of action. You will, I think, find that the best way to promote such agreement is to make people consider the natural forces which have to be systematized by legislation, with a view of enabling them to work freely for the desired purpose. When the conditions essential to any national system come to be duly appreciated by those interested in the ADBRESS. XCl cause of education, means will soon be found to carry out the necessary legislative enactments. The highest offices in the State are on our present system filled by men ■who, ■whatever their political opinions and party ties, almost infallibly agree in their disinterested desire to signalize their respective terms of office by doing any good in their po^wer. Convince them that a measure desired by the leaders of public opinion is in itself good and useful, and you are sure to carry it. And, on the other hand, England is not ■wanting in men both able and Avilliug to come forward as the champions of any great cause, and to devote their best po^wers to its service. I may ■v\'ell say this at Bradford after the results achieved by your Member in the Elementary Education Act. • Objections ■will of course be raised to any system on the score of difficulty and expense, more especially to a complete and good system. Difficult of realization it certainly must be, for it will need the devoted and indefatigable exertions of many an able and high-minded man for many a long year. Only show how such exertions can be made to produce great and abiding results, and they will not be wanting. And as for expense, you will surely agree with me that the more money is distributed in such frugal and effective manner, the better for the real greatness of our country. "What nobler privilege is attached to the j^ossession of money than that of doing good to our fellow men ? and who would grudge giving freely from his surplus, or even depriving himself of some comforts, for the sake of pre- paring the rising generation for a life of the utmost usefulness and consequent happiness ? I confidently trust that the time will come when the chief item in the annual budget of the Chancellor of the Exchequer will be the vote for National Education. And when in some later age our nation shall have passed away, when a more true civilization has grown up and has formed new centres for its throbbing life, when there are but broken arches to tell of our bridges and crumbling ruins to mark the sites of our great cathedrals ■ — then will the greatest and noblest of England's works stand more jDerfect and more beautiful than ever ; then will some man survey the results of Old England's labours in the discovery of imperishable truths and laws of nature, and see that her energy and wealth were accompanied by some nobler attributes — that while Englishmen were strong and ambitious enough to gi'asp power, they were true enough to use it for its only worthy purpose, that of doiug good 'to others. I must not, however, trespass longer upon your time and your kind at>- tcntion. !My subject would carry me on, yet I must stop without having half done justice to it. If I have succeeded in convincing you that a National system of Educa- tion is now necessary and possible, and in persuading you to do what you respectively can to prepare the way for it, I shall feel that the first step is made towards that great result. REPORTS ON THE STATE OE SCIENCE. Heport of the Committee, consisting of Professor Cayley, F.R.S., Professor Stokes, F.R.S., Professor Sir W. Thomson, F.R.S., Professor H. J. S. Smith, F.R.S., and J. W. L. Glaisher^ B.A., F.R.A.S. (Reporter), on Mathematical Tables. § 1. General Statement of the Objects of the Committee. The purposes for which the Committee was appointed were twofold, viz. (1) to form as complete a catalogue as possible of existing mathematical tables, and (2) to reprint or calculate tables which were necessary for the progress of the mathematical sciences. These two objects, although so far connected, that it Avas absolutely essential before any tables were calculated or reprinted to be certain that such tables were not already in existence or easily accessible, were in other respects quite different ; and the Committee have therefore decided to keep them distinct. The reasons in favour of the adoption of this course are ob- viously very strong, as a new table would be out of place in a Report which in other respects was merely a detailed catalogue. A further argument against the publication of the tables in the Reports of the Association, is the great objection to needlessly scattering tables. Tables of a kindred nature collected together, are of far more value than the same could be if dispersed in several volumes of a periodical ; and if the tables of the Com- mittee were published annually as calculated, it would happen not only that they would have to be sought in several volumes, and their utihty in conse- quence considerably impaired, but sometimes even portions of the same table would be separated. The Committee have therefore considered that they would best carry out the second object for which they were appointed, by publishing their tables separately and independently of the Annual Reports of the Association. The form cliosen for this publication is a quarto of the same size as that of the Philosophical Transactions, this size being necessary for the uniformity of the tables, as a large page is required in order to contain the values of the function tabulated, together with its first, second, and third differences, which, when given, should range with the former on the same page. Before the 1873. jj 2 . HEPORT— 1873. appointment of the Committee, certain tables of hyperbolic antilogarithms or exponentials (viz. e^ and e~^) and of hyperbolic sines and cosines had been commenced by Mr. J. W. L. Glaisher ; and these the Committee determined to print and stereotype on their completion. They are now in the press. A mass of ealcnlations has been made for the tabulation of Eessel's functions, for real and imaginary values ; and it is intended to complete these tables, and then to undertake calculations connected with the Elliptic Functions. As yet no tables have been reprinted by the Committee ; and it clearly would not be possible to decide which most reqiiired reproduction, uutil the Report was considerably advanced beyond its present stage. All the tables printed by the Committee, whether calculated or reprinted, are to be stereotyped ; and it is intended that they shall ultimately form a volume ; but the tables relating to each function will be published and circu- lated separately as calculated, the stereotype-plates remaining in the posses- sion of the Committee for future use. The first object of the Committee was rendered necessary by the fact that the mathematical tables that have been formed, are scattered all over the world in the various mathematical and scientific journals, transactions of societies, &c., so that it is extremely difficult to ascertain what tables have been already calculated in any particular branch of science. Another reason is that tables formed for some particular purpose, and published under a title of special application, are often of equal importance in other investigations ; so that great inconvenience is sometimes felt for the want of a table which already exists under another name and having reference to a different subject ; or it may even be recalculated. The difficulty of knowing exactly the work already done in any subject is one which is common to all parts of science ; but the inconvenience resulting from the nature of a work being obscured by its name is to a great extent peculiar to this subject, or at all events is more painfully felt in connexion with it. A familiar instance of a function occurring in several distinct subjects is the integral le-^V.r, which is of importance in the determination of the probable error in the method of Least Squares, Astronomical Refractions, and the theory of Heat; and good instances of the manner in which the nature of a table can be obscured by its name are afforded by nautical collections, where under such headings as " Table to find the latitude by double altitudes of the sun and the elapsed time," or " Table of logarithmic risings," &c., are given log cosecants, log versed sines, (Src. A catalogue, therefore, in which the tables were carefully described from their contents seemed very desirable ; and this the Committee hope to be able to accomplish by their Reports. It is intended to include all numerical tables that can be regarded as belonging to mathematical science, or which are of interest in connexion therewith ; but none will be noticed in which the tabular results or data are derived from observation or experiment, or merely concern special subjects that arc not generally classed under the head of mathematics. Thus the great majority of astronomical tables, including catalogues of stars, tables of refraction, tables depending on the figure of the earth, &c., will be ex- cluded, as the data for the formation of such tables are derived from observa- tion. ^ The same remark applies to all chemical tables, tables of specific gravity, of weights and measures, for the determination of the longitude at sea, mortality tables, &c. Life-assurance and annuity tables, and all commercial tables will also be excluded. With regard to these last, however, although all tables such as ready reckoners and common interest tables will in general be omitted, ON MATHEMATICAL TABLES. 3 any one that is of value in relation to mathematics as a science will be in- cluded, although it may have been calculated for merely commercial purposes and published under a name that would apparently exclude it from this Report. ]\[auy tables of compound interest are valuable when viewed as tables of powers; and many navigation tables calculated merelj^ for the use of the sailor, and pub- lished under titles that would imply that they were of a merely technical cha- racter, are in reality trigonometrical tables under a disguised form. From the above remarks it will be found in most cases very easy to decide whether a table is included in the scope of this Report or not. A few of course come on the boundary ; and then there is some little difficulty in drawing the line fairly. Of this kind are tables for the expression of hours and minutes as decimals of a day, &c. ; most of these it has been thought better to include. It was necessary as a preliminary to form a classification of mathematical (numerical) tables ; and the following classification was drawn up by Prof. Cayley and adopted by the Committee. A. Auxiliary for non-logarithmic computations. 1. Multii:)lication. 2. Quarter-squares. 3. Squares, cubes, and higher powers, and reciprocals. B. Logarithmic and circular. 4. Logarithms (Briggian) and antilogarithms (do.) ; addition and sub- traction logarithms, &c. 5. Circular functions (sines, cosines, (fee), natural, and lengths of circular arcs. 6. Circular functions (sines, cosines, &c.), logarithmic. C. Exponential. 7. Hyperbolic logarithms. 8. Do. antilogarithms (e^) and h . Itan (45°-\-i(j)), and hyperbolic sines, cosines, «fcc., natural and logarithmic. D. Algebraic constants. 9. Accurate integer or fractional values. Bernoulli's Nos., A'' 0'"^, &c. Binomial coefficients. 10. Decimal values auxiliary to the calculation of scries. E. 11. Transcendental constants, e, tt, y, &c., and their powers and functions. F. Aritbmological. 1 2. Divisors and prime numbers. Prime roots. The Canon arithmeticus &c.' 13. The Pellian equation, 14. Partitions. 15. Quadratic forms a--\-h^, &c., and partition of numbers into scjuarcs, cubes, and biquadratcs. 10. Binary, ternary, &c. quadratic and higher forms. 17. Complex theories. G. Transcendental functions. 18. Elliptic. 19. Gamma. 20. Sine-integral, cosine-integral, and exponential-integral, 21. Bcsscl's and allied functions. 22. Planetary coefficients for given -,. • 23. Logarithmic transcendental. 24. Miscellaneous. b2 4 REPOR'T — 18/3. Several of those classes need some little explanation. Thus D 9 and 10 are intended to include the same class of constants, the only difference being that in 9 accurate values are given, while in 10 they are only approximate ; thus, for example, the accurate Bernoulli's numbers as vulgar fractions, and the decimal values of the same to (say) ten places are placed in different classes, as the former are of theoretical interest, while the latter are only of use in calcu- lation. It is not necessary to enter into further detail with respect to the classification, as in point of fact it is only very partially followed in the Report ; the final index, however, will be constructed as much in accordance with it as possible. The only perfect method by which all the tables on the above subjects could be found with any certainty, is to examine all the volumes of the mathema- tical and philosophical journals and transactions, given in the list prefixed to the Eoyal Society's Catalogue of Scientific papers — a most laborious work, as it requires every page in all these periodicals to be looked at, and any nu- merical tables noted and subsequently examined, while if included in the scojie of the Committee's work they must further be described. The mere turning over the pages of several thousand volumes is a work of some labour, and the completion of the lleport must occupy the Committee for several years. The Avork is also of such a nature that it would not be possible to obtain even an approach to completeness in any one class till very considerable progress had been made with the preliminary examination. This, however, is not the case to any great extent with the groups A and B, or with C 7 or the first part of F 12, as tables in these classes are gene- rally to be found in separate books, and not in the memoirs of societies, or journals. It was possible, therefore, to make progress in the above classes immediately ; and the portion of the Report now presented to the Association, practically contains a catalogue of tables which form separate books. The three broad divisions into which mathematical tables divide themselves practically are found to be : — I. Subsidiary tables, which are rather of value as a means of performing calcidations than of interest in themselves : e. g. multiplication tables, logarithms, &c. They generally form separate books. II. Tables of continuous functions, generally definite integrals. III. Tables in the theory of numbers. Divisions II. and III. contain conclusive (in opposition to siibsidiari/) tables. A fuller description of the contents &c. of Division I. will be found in § 2. ^ It is hoped next year to report on Division II., and the next year on Division III. It will be necessary afterwards to add supplements to diflPerent classes, and notably to the present portion of the Report, which has no claim at all to be regarded as complete, but is published on the distinct understand- ing that it is by no means exhaustive with regard to the subjects treated in it : a supplementary Report on the same subject will be subsequently added; and it is hoped that thus it will be rendered complete (see § 2). .§ 2. General Intro<lmtion to the present Bejiort, and Explanation of Its Arrangement and Use. Art. 1. The present Report is intended to include all general tables, viz. tables that are of general application in all branches of mathematics, and are therefore useful wherever calculations have to be performed. The most simple instances arc multiplication tables, common logarithms of numbers, ON MATHEMATICAL TABLES. 5" and trigonometrical functions, -whicli form the basis of, and are the means by -which all other calculations are made, llegarded from this point of view, tliis division may be said to contain auxifiary or subsidiary tables, viz. such as are not per se of any very great intrinsic interest (multiplication tables are a good instance), but which are nevertheless of such paramount import- ance that, without their aid, the calculation of other tables would be too laborious to be practicable. As before remarked, one reason why these tables may well form a division by themselves is, that, being intended for calcula- tions of all kinds, they arc usually published separately, and have not to be sought among the transactions of societies and other periodicals. The num- ber of tables in this class is of course many times greater than are all the other classes put together ; but then, on the other hand, they admit of more brief description, as scarcely any explanation is needed of the functions tabulated, or of the purposes for which the calculation or publication was undertaken. In the present Eeport not above five or six tables printed in periodical publications are noticed ; while it is probable that in the Reports on the other classes there will not be a much greater number that will have appeared as separate and independent books. Art. 2. The object of the lleport is to enable any one by means of it to find out with ease what tables have been computed on any of the twenty- five subjects (see § 3) to which it relates, and where they are to be found ; and the desire to form a catalogue that shall give a systematic and practical account of the numerical tables in existence that bear upon each of the subjects included has been steadily kept in view ; in fact little else has been aimed at. Still, as in the search for and examination of so many books of tables (the Repoi-t contains an account of more than 230) a good many works of considerable historical or bibliographical interest came to light, it was ]iot thought desirable to suppress all notice of them. The majority of seven- teenth-century works included are described, on account either of their rarity or because they serve to illustrate the history and progress of the subject. Of this kind are Napier's ' Canon Mirificus ' (1614), containing the first an- nouncement of logarithms, Ltidolf's 'Tetragonometria' (1 690), (fee. ; and when such works have been included, their full titles have been given in § 5, with suitable bibliographical accuracy. It would be a mistake, however, to suppose that all the tables of the seventeenth century have been superseded ; VLAca's ' Arithmetica,' 1028, is the most convenient ten-figure table of logarithms that exists (it has only been reprinted once, and not in so useful a form) ; and no natural canon published subsequently can bear comparison with Pitiscus, 1613. In performing mathematical calculations, we have had repeated occa- sion to use both Vlacq and Pitiscus. Uksinus's 'Napierian Canon' (1624) is the largest in existence. The points in which the Keport is least complete are the descriptions of common tables of the eighteenth century, and of com- paratively modern Italian, Spanish, &c. tables of logarithms. Tlie former class we have purposely omitted, though avo have examined many, as they are neither of value intrinsically nor historically ; a good many are biiefly noticed by De Morgan ; and the latter we have not been able to see : several titles Avill be found in the Babbage Catalogue. Art. 3. The most valuable detailed list of tables hitherto published is the article Tables written by De Morgan for Knight's ' English Cyclopasdia ' (1861). This article first appeared in the ' Penny Cyclopaedia' (1842), but it was carefully revised and largely augmented by its author before its re- printing in the ' English Cyclopedia.' In this article arc contained notices of 457 tables, many of which, however, are outside the scope of this lleport. 6 REPORT — 1873. "We have had occasion to make great use of this article ; and whenever De Morgan's name is cited without reference to any work of his, it is always to be understood that it is this article'Avhich is referred to. Other works which Ave have used, but which contain information almost wholly of a bibliogra- phical or historical nature, are : — (1) ' Historia Matheseos Universa3 a mundo condito ad seculum P. C. N. XVI. . . . accedit . . . historia Arithmetices ad nostra tcmpora,' autore Jo. Christoph. Heilbronner. Lipsias, . . . 1742, 1 vol. 4to. The ' Liber quartus sistens Historiam Arithmetices ' is at the end of the book, and occupies pp. 723-924. (2) ' Geschichte der Mathcmatik,' von Abraham Gotthelf Kastner. Gbt- tingen. (4 vols. 8vo, 179G-1800.) It forms the seventh 'Abtheilung' of the ' Geschichte der Kiinste und Wissonschaftcn ' (57 vols.). The tables are contained in vol. iii. (3) ' Bibliotheca Mathcmatica,' auctore Frid. Guil. Aug. Murhard. Lipsise, 1797-1804 (also German title, ' Litteratur der mathematischcn Wissen- schafteu'). 4 vols. 8vo. ' Mathematischo Tafcln' is the heading of the fourth division of vol. ii., and occupies pp. 181-201 ; they are divided into two classes, the first containing logarithmic and trigonometrical tables, and the second the rest; works that Murhard has had in his own hands are marked with an asterisk. (4) ' Bibliotheca Mathematica sive Criticus Librorum Mathematicorum, .... commode dispositus ab J. Eoggio.' Sectio I. ' Libros Arithmetices ct Geometricos complectens.' Tubingaj, .... 1830 (also with German title- page). This work we have found very useful. A great number of logarithmic and trigonometrical tables arc carefully described in Div. IV. ' Elementar- Geometrie' (B.), pp. 367-410. It is right to add that the titles of tables are to be found in all portions of the work, and are by no means restricted to the arithmetical divisions. We believe that no more than the ' Sectio I.' was ever published. The following is a continuation of Rogg : — (5) ' Bibliotheca Mathematica. Catalogue of Books in every branch of Mathematics .... which have been published in Germany and other coun- tries from the year 1830 to the middle of 1854.' Edited by L. A. Sohnke, . . . Avith a complete index of contents. Leipzig and London, 1854. 1 vol. 8vo. (6) ' Bibliographie Astronomique, avcc I'histoire de 1' Astronomic. , . . Par Jerome De La Lande ... A Paris. ... An XI. = 1803. 1 vol. 4to. A sepa- rate index to the general tables is given on pp. 960, 961. (7) ' Litteratur der Mathcmatik, Natur- imd GcAverbs-Kunde mit Inbegriff der Kriegskunst,' , . . von J. S. Erseh. ' Xeue fortgesetztc Ausgabe,' A'on F. W. Schweigger-Seidel. ' Aus der ncuen Ausgabe des Haudbuchs der Deuts- chen Litteratur besonders abgedruckt.' Leipzig, 1828. 1 vol. 8vo. (8) ' Biographisch-literarisches Handworterbuch zur Geschichte der exactcn Wissonschaftcn . . . gesammelt' von J. G. Poggendoif. Leipzig, 1803. 2 vols. 8vo. (9) 'R. p. Claudii Francisci Milliet Dechales Camberieusis e Societatc Jesu Cursusseu Muiidus Mathematicus.' . . . Lugduui, 1690. 4 vols. fol. The first volume opens with a ' tractatus Proemialis de progressu Mathe- seos et illustribus Mathematicis ; ' and pp. 28-37 are devoted to arithmetical bibliography. AVe may state that a previous edition of 1674, in 3 vols, fol., does not contain the ' De progressu.' ON MATHEMATICAL TABLES. 7 We may also mentiou Do Morgan's ' Arithmetical Books from the inveutiou of printing to the ^jresent day,' London, 1847, 8vo, the introduction of which contains useful bibliographical information about the description of books, and Peacock's " History of Arithmetic " in the ' Eueyclopffidia Metropolitana.' There is one bibliographical work, viz. Scheibel's ' Einleitung zur mathe- matischen Biicherkenntuiss.' Neue Auflage. 3 vols. Svo, Breslau, 1781 (as given in the Babbage Catalogue), which is continually referred to by Murhard, Rogg, &c., though we have never been able to see a copy in any library to which we have had access, or procure one otherwise. _ De Morgan says, "Scheibel (additions) may be considered as partly repetition, partly extension, of Heilbronner. He is one of those bibliographers who collect from various sources the names and dates of more editions than those who know catalogues wiU readily believe in." It is imnecessary here to mention works on general bibliography, such as Hain, Ebert, Watt, &c., Avhich are well known; we maj', however, parti- cularly notice 'Tresor de livres rares et precieux ou Nouveau dictionnaire bibliographique,' par Jean George Theodore Graesse, Dresde [also Geneva, London, and Paris], 1859-1867 (7 vols, including supplement), which might be of use, though we have found the mathematical works it contains very inaccurately described ; but this is a fault common to all works of general bibliography. Montucla, ' Histoire des Mathematiques,' we have not found valuable ; but we may call attention to the accurate information given by Delambre in his ' Histoire de I'Astronomie Moderne,' t. i. Paris, 1821 ; and also in his other histories. Reuss's ' Eepertorium Commentationum a.societatibus litterariis editarum,' GottingEB, 1801-1821, 16 vols. 4to, is a work very similar in its plan to the Royal Society's Catalogue of Scientific Papers, except that it is an iiidea; rerum instead of an ijidex auctorum. The mathematics is contained in vol. vii., the arithmetic occupying pp. 2-31 of that volume. On p. 30 are refer- ences to descriptions of calculating and other arithmetical machines. We have found Nos. XIX. and XX. (on trigonometrical and logarithmic tables) of Hutton's ' Mathematical Tracts,' London, 3 vols. Svo, 1812, very useful. Art. 4. The mode of arrangement of this Report (which properly occu- pies § 3, § 4, and § 5), and the reasons that have led to its adoption, are as follows : — If every table were published separately and formed a work by itself, the obvious course would be to divide them into a certain number of classes according to their contents, to prefix to each class a brief intro- duction and explanation, and then to give a detailed description, in chrono- logical order, of the tables included under it. This is, in fact, the course that has been pursued with regard to separate tables (i. e. works containing either a single table or only tables that come under the same class) ; § 3 is divided into 25 articles, each article being devoted to one subject: — art. 1, multiplication tables ; art. 2, tables of proportional parts, *fec. (for the con- tents of all the articles, see the commencement of § 3). Each article begins with a general account, partly historical, of the subject included in it; and then follow the' descriptions of the separate tables ou that subject. But the majority of works noticed are collections, and iuclude tables that are com- prised under several articles ; thus Hutton's tables contain Biiggian and hyperbolic logarithms of numbers, a natural and logarithmic canon, &c. &c., each of which belongs to a different article. Two courses were therefore open for the treatment of such works : — (1) to describe them under the article S PEroRT — 1873. having rcftTciicc to the first or largest table in the work, and insert cross references under each of the articles concerned with the other tables in-, eluded in the work ; or {2) to describe all collections of tables in a section. by themselves, and pivo references to each of the tables thej' contain under the appropriate article in § 3. The second course was clearly the more proper, for three reasons — (1) because it was free from the arbitrary element involved in the choice of the leading table, which would be required in the first method, (2) because it M'as undesirable to overload the articles of § 3 with descriptions of tables not belonging to them, and (3) because reference to the works would be greatly facilitated by placing them in an article by themselves ; § 4 therefore contains all woi'ks the contents of which do not belong wholly to one of the articles in § 3, or, in other words, which con- tain at least two tables, the subjects of which are included in different articles of § 3. As the works in § 4 will thus have to be continually re- feiTcd to separately, they are arranged alphabeticalh', not chronologicallj', § 5 is a complete list of all the works containing tables that are described in this Eeport ; and to facilitate its use as an index, a reference is attached to the section, or section and article, in which the work is described. To take an example of the manner in which the Report is intended to be used. Supposing it were required to know what tables there were of log versed sines ; the reader would turn to the beginning of § 3, and, looking down the list of articles, see that, coming under the head of "logarithmic trigonometrical functions," such tables belonged to art. 15. He would ac- cordingly turn to art. 15, and read or glance through the introductory remarks to that article, and the works described there ; not finding any book containing log versed sines alone described in the article, he would conclude that no separate table of the kind had come under the notice of the reporter ; he would then look at the references to § 4 ; and if he wished for detailed information with regard to any of those tables, he would examine the de- scriptions in that section. Any one, on the other hand, desiring to know the contents of any particular work would seek it in § 5 ; if it occurred there, a reference would be found added either to § 4, or to § 3 and the article iu which it is described. No difRcuhy will be experienced in finding the descrip- tion if it be remembered that all the works are cited by the author's name and the date ; and that while in § 4 they are arranged alphabetically, in the articles of § 3 the arrangement is chronological. The date is throughout appended to the author's name in citing a work, in order to identify the work in § 5 (the date given being always that assigned to the work in § 5) ; there is also the further advantage, that any one who requires information only with regard to modern tables, still procurable from the bookseller, need not waste time in seeking the detailed descriptions of works published in the seventeenth and eighteenth centuries. It may be mentioned that a few works that ch contain tables of more than one kind, arc nevertheless included in § 3 : this happens when the smaller tables are insignificant compared with those under which the work is classed ; references are then appended also in the articles to which the smaller tables belong. An asterisk prefixed to an author's name (thus * Voisin or * Voisin) in- dicates that the description of the work of his referred to has not been derived from inspection. In ever}' ease where there is no asterisk, the description has been written by the reporter with the book itself before him. Art. 5. In all eases where the author of a collection of tables has num- bered or marked them himself, his numbering or marking has been followed ON MATHEMATICAL TABLES. i) in this llcport, except in very exceptional circumstances. "Where, however, tlic table's are not numbered or otherwise denoted, they have been marked [T. I.], [T. II.], &c., as it was necessary to have the means of referring to them, invariably, therefore, where the number of the table is not included in square brackets, it is to be understood that it is the author's own number. Thus T. VII. in any particular work implies that the table in question is numbered A'll. in that Avork, while [T. VII.] implies either that the table has no number, or that the classification in the work is different from that adopted in this Ileport. Whenever logarithms arc mentioned Avithout tho epithet hyperbolic or Napierian, common or Eriggian logarithms (viz. to base lU) are intended. In some cases, where there might bo some doubt, the adjective " common " is introduced. By hyperbolic logarithms are always meant logarithms to the base e (2-71828 . . . ); and these are never called Napicfiaii, this word being reserved for logarithms of exactly the same kind as those introduced by Napier (see § 3, art. 1 7). Such a sentence as " Five- figure logarithms to lOOO," is always to be understood as meaning " logarithms of numbers from unity to 1000, at intervals of unity to five decimal places ;" viz., wlien the lower limit of a table is not expressed, it is always to be taken as unity ; and when the intervals are not mentioned, they are always unity'. The term "places" is used throughout for " decimal places " or " decimals," a number " to 3 places " meaning a number given to 3 2''^(^ces of decimals (not 'djir/ares). The only exception made to this rule is in the description of tables of common logarithms ; the words " seven-figure logarithms, six-figure logarithms," <fec., have become by usage so completely recognized as meaning logarithms to seven places, to six places, &c., that it did not seem worth while disturbing the established mode of expression, as it could lead to no error. The contents of old works have been described in the language and nota- tion of the present day, and not in the manner adopted by their authors ; any peculiarities of notation &c. in a table, however, are pointed out. It was long universal, and is still very common, to describe trigonometrical tables as being computed to a certain radius ; these are translated into the language of decimals ; thus a table " to radius 10,000,000 " is described as a table " to seven places," and so on. As a rule the characteristics of the logarithms have been ignored in describing a table ; i. e. it has not been stated whether the characteristic was given or no, or, if given, what was the understanding on which it was added. In many tables, contained in works intended for a special purpose (as in collections of nautical tables, &c.), arbitrary numbers are added to or subtracted from the characteristics to facilitate their use in working some particular formula ; to have included details of this kind would have taken much room, and been really superfluous, as in most eases all that is required to be known in the description of a table of logarithms, is the range of the table, and the number of places to which the mantissa} are given. We m!iy here mention that an ambiguity occurs in the description of propor- tional-part tables ; thus a *' table of proportional parts to tenths " may mean cither that the proportional parts are given for one, two, three, &c. tenths of the difference, or else that the numbers that form the proportional-part table are given to one place of decimals. The former is tlie meaning generally in- tended ; and it would be better if in this ease the words " to tenths " were replaced by " for every tenth." A good many tables had been' described before the ambiguity was noticed; but it is believed the context will generally show the true meaning ; when the words to tenilis, to hundredths, &c. arc italicized, the latter interpreta- tion (viz. results given to onO; two, &c. decimal places) is to be assigned. 10 REPORT 1873. Art. 6. To the particular editions of the ■works described no importance is to be attributed. It would obviously have been impossible to alwaj^s tix upon the first or last edition as the one to be described ; in fact we had no choice ; we took what we could get. The list in § 5 always contains portions of the titlepage of the same edition of the work that is described in § 3 or § 4 of the Eeport ; the particular edition chosen was usually determined by the accidental circumstance of its being the first that was examined, anj^ informa- tion that was subsequently obtained about other editions being added at the end of the description of the contents of the woric in § 3 or § 4. It would have been better to have always taken as the standard the last edition pro- curable, and pointed out whereia it differed from its predecessors ; but this would have required much rewriting of particular portions, and considerably increased the labour of preparation, with a very small increase of regularity in the arrangement of the llcport, but with no corresponding increase in its value. Art. 7. In every case where a table has been described from inspection, all the tables themselves have been examined, and not merely their titlepages, tables of contents, &c. This was of course absolutely necessary in very many instances, as it is comparatively rare that any thing more than a general notion of the contents of a collection of tables can be gathered from the author's explanations ; but in any case it was essential if the Ileport was to have any value for accuracy, because the titles assigned by their authors were sometimes misleading, if not absolutely erroneous ; and frequently, even if the more im- portant tables had headings or descriptions prefixed, the smaller ones (which are often more worthy of notice on account of their rarity or mathematical value) were passed over. It must here be remarked that it is never safe to take a description of a table from its author or editor, as it is not a very uncommon thing to give as the contents of a table, not that which can be found from it at once, but what can be obtained from the table by means of additional Avork, such as an interpolation. Thus, under the heading " Table of logarithms to eight decimals " is sometimes given a table to five places, and a formula from which to calculate the remaining three. • Another case in point is Steinbeeger's table, described in this Report, the titlepage of which describes it as giving the logarithms of all numbers to 1,000,000, when in point of fact it only extends to 10,000- — the jusiification for the title being that two more figures can be interpolated for. It is not to be supposed, when svich misstatements occur, that the author of the table has any desire to mislead, as they usually result from ignorance ; but it is a matter of regret, when it has become customary (and most properly so) that a table should be described on its title as giving onlj' what can be taken out of it without additional calculation, that this rule should sometimes be vio- lated and a designation given that is, to say the least, misleading. AVe have also met with such instances as the following :- — The title of a book is given ill a bookseller's catalogue as (sav) " Table of divisors of numbers from 1 to 10,000,000 ;" but ^ the following"' words (say), " Part I. from 1 to 150,000" (when perhaps no more was ever published), are left out — an omission of rather an important character as regards the contents and value of the table. Cases of this kind show how imperatively necessary it is to examine the table itself; and whenever the description of a table is taken from an adver- tisement, bookseller's catalogue, or other second-hand source, there is great liability to error. Art. 8. The names of authors occurring in the text have been printed in small capitals when the work of theirs alluded to is described in this Eeport, ON MATHEMATICAL TABLES. 11 otherwise in ordinary roman type : thus we should write " the table was copied from ' Bkiggs's ' Arithmctica ' of 1624," because an account of Eriggs's work is given in the Eeport ; but we should write " the sines were taken from Vieta s ' Canon ' 1579," because Vieta's work is not described. _ This rule is attended to always whenever an author's name is mentioned in juxtaposition with his work, and it wiU be found to save unnecessary trouble in searching for works not noticed in the Eeport. Of course all rules are sometimes diffi- cult to carry out ; and in cases such as when the author's name and work are separated from one another, or the name occurs frequently in a paragraph by itself, but really in connexion with some work not expressly named each time, &c., we have attempted to carry out the spirit of the rule and no more. An author's name is enclosed in square brackets (thus [Pell] or [Pell]) when his name does not occur on the titlepage of the work of his referred to. Art, 9. The words 8vo, 4to, &c. are used in § 5 to signify works of octavo, quarto, &c. size, without reference to the number of pages to the sheet. They are merely intended to give a rough idea of the size and shape of the work, which is better done by using them in a general sense than by attaching to them their technical meanings. The words " large " or "small " have been prefixed when the size was markedly different from what is usual. It must be remembered that two hundred years ago all the sizes were much smaller thau at present, so that the usual quarto page of 1650 is smaller than an octavo page of our day, though the shape is of course more square. Old works are generally described as they would have been at the time ; but it sometimes may have happened that a true quarto of old date is here given as octavo, &c. : this caution is necessary for those who might use §5 bibliographically. Whenever, in transcribing portions of works in § 5, words have been omitted from the titlepage, dots have been inserted to mark the omissions. We may mention that we have used the word reprint in its proper sense ; viz. we have not spoken of a reprint except when the type was reset. Art. 10. In the preparation of this Eeport extensive use has been made of the libraries of the British Museum, the Eoyal Society, the University of Cam- bridge, the Eoyal Observatory, Trinity CoUege (Cambridge), and the Eoyal Astronomical Society, in one or other of which the majority of the works noticed are contained. We have also, through the kindness of Professor Henrici, been enabled to consult the Graves Library at University College, London, which contains an almost imrivalled collection of old mathematical works ; but as they are not yet arranged, it is not possible to find any par- ticular work without great expenditure of time and labour. The De-Morgan library at the London University is also still in process of arrangement, and is therefore inaccessible for the present. By the kindness of Mr. Tucker, who forwarded us an early copy of the sale-catalogue of the late Mr. Babbagc's library, we have been enabled to extract several titles from it, and identify works of the titles of which we had only imperfect descriptions ; but we have not been able to see any of the books themselves. It must not be understood that the Eeport contains notices of all the books of mathematical tables contained in the libraries mentioned at the beginning of this article. Por in- stance, the Eoyal Society's catalogue contains the titles of several works that should be included but which we have not yet examined ; and of course no one can know what tables there are in such Ebraries as those of the British Museum or the Cambridge University, where there is no catalogue of subjects. For the omissions we could have rectified we must plead in excuse the already great extent of the Eeport, and consequent necessity of drawing the line somewhere. Of coiu'sc many of the works noticed o,xq either in our own 12 REPORT —1873. possession or were lent by friends ; and wc must acknowledge the kind assist- ance rendered b)^ Mr. C. W. Mcrrifield, F.R.S., of whose mathematical library we hope to make more nsc in a future Eeport. Art. 11. The lleport is avowedly very imperfect; it contains probably not one half of the works that have as good a right to be noticed as those that are iuchided. This defect will be remedied by the publication of an Appen- dix or additional lleport on the same subject, probably after the appearance of the Eeports on the other divisions. As it would be clearly impossible to have made this lleport perfect (and had it been possible, it would have occu- pied more space than could be given to it), an Appendix giving the results of the examinations of the memoirs, transactions, &c. in reference to this class of tables would have had in any case to be added after the com- pletion of the other divisions ; and on this account it seemed unnecessary to take especial pains to procure works that were clearly of no very great importance, or to insert imperfect second-hand accounts of tables that would in all probability be met with in the course of the formation of the subse- quent Reports. Invariably, however, whenever a reference was found to a table that seemed of importance, no pains have been spared in the endeavour to obtain and examine a copy ; in the event of these efforts being fruitless, a notice of the work compiled from other accounts has been given, with an intimation of the source whence the information was derived ; but only three or four works arc included that have not come under the eye of the reporter. It is probable that there may have been published recent works on the continent no coj^j' of which is contained in any of the public libraries of this country ; and on this account it will probably be found very difficult to make the list perfect. The present Report is, however, so far complete that the Committee think they may ask mathematicians or comimters who are ac- quainted with any works not included in it or in De Morgan, to inform them of the fact. It is only in this way that completeness can bo obtained, as although, by an examination of the transactions &c. to M^hich references are given at the beginning of the Royal Society's catalogue, the completion of the accounts of tables contained in memoirs &c. would be merely a matter of time and labour on the part of the members of the Committee, the discovery and de- scription of books printed in out-of-lhe-way places, or for private circulation, can only be effected by the cooperation of mathematicians who may happen to possess copies*. The Report, however, as it now stands, will be found to contain more information about tables than is to be found anywhere else ; in fact, except De Morgan's list (referred to in art. 3 of this section), we know no place where any attempt is made to cover the ground included in this Report ; and though De Morgan has referred to more works than are described here in detail (even when commercial tables are excluded), it must be borne in mind that his descriptions are too short and general to be of great value, that more than a third of his accounts arc compiled from sources other than the original works, and that he has made no attempt to do more than roughly classify the works (not the tables) ; in fact a more detailed description or classification was excluded by the plan of his article, Avhich notwithstanding gives a great deal of information in a very small space. Art. 12. By an oversight (which was not discovered till it was too late to remedy it) wc have excluded from the Report traverse tables, viz. Differencc- of-latitude and Departure tables, which under the head of multii^les of sines and cosines ought to have been noticed. Such tables are of general use in « It is requested that communication.^: may be addressed to BJr. J. W. L. Glaishcr, Trinity College, Cambridge. ON MATHEMATICAL TAIJLES. 13 all niatbcmaties, as tliey arc iti reality merely tables for the solution of right- angled triangles ; we have noticed one such table (MASS.VLOur, § 3, art. 10), •which was constructed for mining- (not nautical) purposes. AVo hope to repair the omission by Jippending a separate list of traverse tables to a future Keport. Art. 13. A very important incidental gain that it was hoped would be afforded by the present lleport, was the opportunity of correcting errors in loga- rithmic and other tables by giving references to the places in which errata-lists had been published. In the introductions or prefaces to works containing tables, it is usual to give a list of the errors that have been found during their preparation in previous tables ; and as few possessors of a work can be acquainted with the publications that have appeared subsequently, it was thought that by referring, under each title, to the works or periodicals in which lists of errata in it had appeared, an important service would be rendered. It was soon evident, however, that it was impossible to deal adequately with the subject of errors in this manner. Many of the important collections have been through very numerous editions ; and it was not always stated in which editions the errors M'crc found ; and when the edition was stated, it was doubtful (without examination) whether the errata-list in question had come under the eye of the editor, and the errors been corrected in subsequent editions, or not. In the case of stereotyped tables, successive tirages are more and more accurate ; and in regard to collections of such tables published long- ago, as, for example, Callct (first published in 1783, though since reset), it seems useless to waste space by giving references to the numerous errata-lists that have been published, some of which must necessarily relate only to the earlier tirages, and must have been corrected long ago. This is the case with all the chief tables, and only in particular instances, when circumstances rendered it probable that the errata-lists would be of use, have references been given to them. As, however, this state of affairs is very unsatisfactory, it is hoped that in a subsequent Report a complete list of errors in later editions of the most-used mathematical tables, still unsuperseded, may be given ; but it is ne- cessary first to be satisfied that the errata given are not erroneous themselves. Many of the chief modern lists of errata arc noticed in this lleport, and also others that it seemed desirable to give references to at once ; but we have made no effort to deal with the matter in a complete manner. It is much to to be regretted that it is not iisual for editors of a new edition of a table to give a list of the errors that occurred in former editions, and have been corrected in that edition. It is only fair for the purchaser of a new edition of a work to be informed wherein it difters from its predecessors ; but imfortunately the object of the editor and publisher is to sell as many copies of the new edition, not to render the old as valuable as the new. It is proper to add, however, that usually, when tables are published by a mathematician for the advance- ment of science, and not by a bookseller and editor for the sake of profit, an exception is made to this rule, and errata are freely acknowledged. A renuirk made by De Moi'gan with reference to mathematical books in general, viz. that the absence of a list of errata means, not that there are no errors, but merely that they have not been found out, is more applicable to tables than to any other class of work, in spite of the care usually bestowed on them ; and an error in a table is far more fatal than an error in any other class of work, as there is no context (as far as the user is concerned) to show imme- diately that the result taken from the table is erroneous. The subject of errors will particularly occupy the attention of the Committee in a future lleport. o 14 REPORT — 18/3. Art. 14. The whole of the work required in the preparation of the Eeport has been carefully performed ; and we believe that not many inaccuracies will be found. Every work noticed, except only three or four, has been described from actual inspection ; and the account has invariably been written with tlie book before us. Every one, however, who has had any experience of biblio- graphical work knows how impossible it is to be always accurate ; the work has often to bo performed in public libraries open only for a few hours in the day, so that any one who has not an unlimited number of days at his command, must sometimes work under pressure. Omissions are thus made, which, when discovered during the revision six months afterwards, cannot be rectified without great loss of time, even if it be remembered what library it was that contained the work in question. The references from one part of the Eeport to another will also, it is believed, be found correct ; but as the whole plan and arrangement have been altered in the course of the year over which the preparation of the Ileport has lasted, it is possible that some of the old refer- ences may remain still uncorrected. If this should be found to be the case, not much difficulty can ever be experienced in seeing what is meant with the aid of the list of articles at the beginning of § 3, and the list of works in § 5 ; also if any misprints (such as T. IT. for T. III. &c.) should escape notice in the correction of the proofs, the reader will be enabled to correct these with- out much waste of time. Lists of errata and corrections, should such bo needed, will be given in subsequent Reports. ^Yhenever we have made a statement on some otlier authority than that of our own observation, we have invariably stated it, though we are aware that we thus lay ourselves ojien to the imputation of not having verified facts of the accuracy of which we might have assured ourselves ; but, as De Morgan has observed, the possibility of writing a history entirely from personal observation of the originals has not yet been demonstrated. § 3. Separate Tables, arranged accordinr/ to the nature of tlieir contents ; with Introductory liemarlcs on each of the several Jcinds of Tahles inclnded in the present Bejiort. This section is divided into twenty-five articles, the subject matter of which is as follows : — Art. 1. Multiplication tables. 2. Tables of proportional parts. 3. Tables of quarter squares. 4. Tables of squares, cubes, square roots, and cube roots, 5. Tables of powers higher than cubes. G. Tables for the expression of vulgar fractions as decimals. 7. Tables of reciprocals. 8. Tables of divisors (factor tables), and tables of primes, 9. Sexagesimal and sexcentenary tables. 10. Tables of natural trigonometrical functions, 11. Lengths (or longitudes) of circular arcs. 12. Tables for the expression of hours, minutes, &c. as decimals of a day, and for the conversion of time into space, and vice versa. 1 3. Tables of (Briggian) logarithms of numbers, 14. Tables of antilogarithms. 15. Tables of (Briggian) logarithmic trigonometrical functions. 16. Tables of hyperbolic logarithms (viz. logarithms to base 2-71828 . , .). 17. Napierian logarithms (not to base 2-71828 . . ,). ON MATHEMATICAL TABLES. 15 Art. 18. Logistic and proportional logarithms. 19. Tables of Gaussian logaritlims. 20. Tables to convert Briggian into hyperbolic logarithms, and vice versa, 21. Interpolation tables. 22. Mensuration tables. 23. Dual logarithms. 24. Mathematical constants. 25. Miscellaneous tables, figurato numbers, &c. Art. 1. MnU'ii^llcation Tables. The use of the multiplication table is so essential a part of the history of Numeration and Arithmetic, that for information witli regard to its introduc- tion and application -^e must refer to Peacock's ' History of Arithmetic ' in the ' Encyclopaedia Metropolitana,' to De Morgan's ' Arithmetical Books ' (London, 1847), as well as to Heilbronner, Delambre, &c. (see § 2, art. 3), to Leslie's * Philosophy of Arithmetic,' and perhaps to Barlow's * Theory of jS'umbers' (London, 1811), in most of which references to other works Avill be found. There is abundant evidence that, till comparatively recent times (say the beginning of the eighteenth century), multiplication was regarded as a most laborious operation ; this is testified not only indirectly by the very simple examples given in old arithmetics, but explicitly by Decker in his ' Eerste Deel vande Nieuwe Telkonst ' (see Phil. Mag. Suppl. Number, Dec. 1872). The great popularity of Napier's bones, and the eagerness with which they were received all over Europe, show how great an assistance the simplest contrivance for reducing the labour of multiplications was considered to be. It would be interesting to know how- mucli of the multiplication computers were in the habit of committing to memory, as the bones would be no great help to any one who knew it as far as nine times nine. In this Report, however, we are only concerned with extended multiplication tables (viz. such as are to be used as tables, and were not intended to be committed to memory). The earliest printed table of multiplication we have seen re- ferred to is Thomas Finck's ' Tabulae Multiplicationis et Divisionis, seorsim ctiam Moneta3 Danicas accommodatae,' Hafnia;, 1604 (which title De Morgan obtained from Prof. "Werlauff, Royal Librarian at Copenhagen) ; but the work, from its title, must have been rather a ready reckoner than a proper scientific table. The earliest largo table, which, strange to say, is still as exten- sive as aiiy (it has been equalled, but not surpassed, by Ckelie, 1864), is Heeavaet AB Hohenbukg's ' Tabulae Arithmetical 7rpoi7da(j>aipeaeios Universales,' 1010, described at length below. Of double-entry tables, Creli-e's ' Rechentafeln,' 1804, is the most useful, and the most used, for general purposes. The other important tables are chiefly for multiplication by a single digit. A multiplication table is usually of double entry, the two arguments being the two factors ; and when so arranged, it is frequently called a " Pythagorean Table.'' The great amount of room occupied by Pythagorean tables (no table so arranged could extend to 1000 x 10,000, and be of practicable size) has directed attention to modes of arrangement by which multiplication can be performed by a table of single entry ; the most important of these are tables of quarter-squares, which are described in § 3, art. 3, where are also added some remarks on multiplication tables of single entry. See also Dilling, described below. It is almost unnecessary to add that, when not more than seven or ten figures arc required, multiplication can be performed at once by logarithms, which (though not the best method for two factors when either a Pythagorean 16 KEPOiiT— 1873. or quarter-square table of suitable extent is at hand) have the advantage that by their means any number of factors can be multiplied together at once. Geuson's table, 1798,13 for multiplications of a somewhat different kind from the rest. Crelle, in the introduction to his ' Itcchentafeln ' (1820), mentions a work, ' Tables do Multiplication, i\ I'usagc do MM. les gcomL'trcs, de Mm. les ingenieurs verificateurs du Cadastre, etc' sec. edit. Paris, Chez Valacc, 1812, which he says extends to 500 x 500, and occupies 500 quarto pages ; while, he adds, his own work, -which is four times the extent, occupies only 1800 octavo pages. For the full titles of Picarte's ' Tables de MultipHcation ' and ' Tableau Pithagorique,' see under Picaete (1861), in § 3, art. 7. Closely connected with multiplication tables ai-e so-called ProportionaJ-parts tables (described in the next article) ; and very frequently in the latter the last figure is not contracted, so that by a mere chauge of the position of the decimal point they become tables of multiples. Herwart ab Hohenburg, 1610. Multiplication table, from 2x 1 to 1000 X 1000. The thousand multiples of any one of the numbers are con- tained on the same page, so that (as the number 1 is omitted) there are 999 pages of tables. By a strange oversight, the numbering begins with 1 on the first page of the table instead of 2, so that the multiples of n are found on page n — \ : this is inconvenient, as the number of the page alone appears on it, so that (say) to find a multiple of 898 we seek the page headed 897. Each page contains 100 lines, numbered in the left-hand column 1, 2, 3, ... ; and besides this column of arguments there are ten columns headed 0, 100, . . . 900. The first figure of the multiplier is therefore found at the top of the column, and the last two in the left-hand column (on p. 3 it will be noticed 200 and 300 are interchanged at the top of the columns). There being more than 1000 pages of thick paper, the book, as De Morgan Jias observed, forms a folio of almost unique thickness. Also, as the pages con- tain 100 lines, pretty Avell leaded, the size of the book is very large ; so that Leslie (Philosophy of Arithmetic, 2nd edit. 1820, p. 246) was quite right in calling it " a very ponderous folio." De Morgan saj^s"the book is exces- sively rare ; a copy sold by auction a few years ago was the onlj' one we «ver saw." Ktistner (' Geschichte,' t. iii. p. 8) quotes the remark of Heilbronner (who gives the title of the work, ' Hist. Math.' p. 801), " Docet in his tabulis sine abaco mulliplicationcm atque divisionem perficere," &c., and adds that Heil- bronner could not have seen the work, or he would have described it ; he remembers to have read that it was like a great multiplication table. The title is given by Murhard, and marked with an asterisk to show that he had seen a copy. Hogg gives the title very imperfectly ; and it is clear the work has not been in his hands. There is a complete copy in the Britisli Museum, and a copy in the Graves Library ; but the latter is imperfect, the pages 12-25, 120-145, and 468-517 having been lost, and their places supplied with blank paper. On account of the rarity of the work, and the great in- terest attaching to it from the time when it was published, we have thought it worth while to give tlie title in full in § 5. The clearness of the type and the extent of the table (which has not been surpassed, and only equalled by Crelle, 1864), taken in connexion with its early date (fou]- years before Napier's ' Canon Mirificus '), give the work a peculiar interest. De Morgan writes : — " it is truly remarkable that while the difficulties of trigonometrical ON MATHEMATICAL TABLES. 17 calculations were stimulating the iiivcntiou of logarillims, they Averc also giving rise to this the earliest work of extended tabulated multiplication. Herwart passes for the author ; but nothing indicates more than that the manuscript was found in his possession." We have seen the statement that while Napier solved triangles by logarithms, Herwart did so by prosthaphce- rcsis, and others of the like kind, the inference being that Herwart invented a method which has been superseded by logarithms ; this (if the present work is the source of the statement) is incorrect, Herwart's table being merely useful in facilitating the multiplications required in the formulte. There are in the Eritish Museum three other works of Herwart ab Hohen- burg : viz., ' Thesaurus Hieroglyphicorum e museo Joanuis Georgii Herwait ab Hohenburg . . .' (Obi. fol. Munich ?, 1610 ?) ; ' Novaj, verse et exacte ad cal- culum . , . Chronologise e museo . . .' Small 4to, 1612; and 'Ludovicus Quartus Imperator defensus . . . ab Joanne Georgio Herwarto' &c. 4to. Munich, 1618 (the middle one of which is given in Lalande's Bib. Ast.). "We have looked at these three books in the hope that some mention might be made in them of the table, or some information given about Herwart's Museum ; but they appear to contain nothing of the kind. We have seen also the titles of several other works of Herwart's, and references to where particulars of his life are to be found ; so that, considering the attention so large a work as his table must have received from contemporary mathematicians, we still have hopes of being able to bring to light some information with regard to its calciilator, • his objects, &c. It should be stated that Herwart ab Hohenburg is spoken of quite as fre- quently by the name of Hohenburg as by that of HerAvart. The author of the anonymous table (17*J3) described below, states that many errors were found in Herwabt, and that Schiibler (whose table we have not seen) was much more correct. Riley, 1775. The first nine multiples of all numbers from 1 to 5280. The multiples of the same number are placed one under the other, the factors 1, 2 ... 9 being three times repeated on the page, which contains ten columns of results and twenty-seven lines. The preface is signed Geo. Riley and T. O'i?. Macmahon. Tliere is an ad- vertisement of Eiley's " historical playing-cards" &c. at the end, and of several works by Macmahon. On the relation of this book to another, " printed for J. Plummer" (anonymous) in the same year, see De Morgan. Anonymous, 1793. ^Multiplication table exhibiting products from 2x13 to 100 X 1000, arranged so that there are 100 multiples (in two columns) of four numbers on each page, which therefore contains eight columns. Gruson, 1798. The first part of this book contains a number of tables, the description of any one of which will explain the arrangement. Take the table 36 : it has ten columns, headed 0, 1, 2, . . . , 9 (as have all the other tables), and 36 lines, numbered 0, 1, 2, . . . , 35 ; we find in column 6 and line 21 (say) 237=6 x 36-|-21. The use of the table is as follows : — suppose it required to find the number of inches in 6 yards 21 inches ; 36 in. =1 yd., we find table 36, column 6, line 21, and have the result given in inches. There are tables for all numbers from 1 to 100, and for primes from 100 to 400, the number of lines in each table being equal to the number of the table. The use of the tables in performing ordinary divisions and multipli- cations when there are four or more figures in the divisor or dividend, &c. is fully explained by the author in the introduction. When used for division, the table gives the quotient and the remainder. There is also given a table of all simple divisions of numbers (not divisible 1873. ' c 18 REPORT 1873. by 2, 3, or 5) to 10,500. A short aud graudiloqucnt dedication to the French Institute is prefixed. Eogg gives also a German title, ' Pinacothek, oder Sammlung allgemein- niitzlicher Tafeln fiir Jedermann' &c. Gruson, 1799. A table of products to 9 x 10,000. The pages, which arc very large (containing 125 lines), are divided into two by a vertical line, each half page containing ten columns, giving the numbers and their first nine multiples : the first half of the first page thus ends at 9 x 124, the second half at 9 x 249 ; and there are 1992 tabular results to the page. The table has only one tenth of the range of Bketschneider's ; but the result is given at once ; however, the large size of the page (almost, if not quite, the largest we have seen for a table) is a great disadvantage. There are two pages of explanation &c. The title describes the table as extending to 100,000, the above being only the first part, "We do not know whether any more was published, but think probably not. Eogg mentions no more. At the end of the introduction three errors occurring in some copies are given. Martin, 1801. This is a large collection of tables on money-changing, rentes, weights and measures, &c. The only part of the book that needs notice here is Chapter XI., which contains a multiplication table giving the first nine multiples of the numbers from 101 to 1052 (19 pp.). Dilling, 1820. In the use of a table of logarithms to multiply numbers together, the logarithms used are of no vahie in themselves, being got rid of . before the final result. If, therefore, letters a, 6, c, ... be used instead, we have no occasion to know the values of any one of them, but only the way in which they are related to one another. The present table is constructed for numbers up to 1000 on this principle ; within this range there are about 170 l^rimes, the logarithms of which have to be denoted by separate symbols, a,h, . .. , z, ffj, 6j, . . . , &c. ; the powers of 2 are denoted by numbers ; thus log (2^)=2, log (2')=3, &c. ; and the logarithms of any number to 1000 can be easily expressed in not more than four terms; thus log 84=2 + rt-|-c. There is also a table of antilogarithms arranged according to the last letter involved; thus log 21=«,-|-o, log 15=a-|-6, the sum =2« + Z»4-c ; and entering the antilogarithmic table at c, we find 315 the product. We can thus only multiply numbers whose pi'oduct is less than 1000 ; and a table of products of the same size would certainly have been more useful. The table can of course be used for division, square roots, &c., but only if the result is integral, so that it is little more than a matter of curiosity. Tliis table was intended, however, only as a specimen, to be followed by a larger one to 10,000. We believe the continuation was not published ; and Eogg refers to no Other work of Dilling, The work, although nominally a table of logarithms, is included in this article, as it is reaUy a multiplication table. It is the only table we have met with involving a principle which at one time would have been of value with respect to multiplication, viz. to resolve the numbers into their prime factors, and multiply them by adding their factors. Thus 21 =3 X 7, 15 = 3x5, and their product 315=3' X 5 X 7 ; if therefore we had a table giving the prime factors of all numbers from 1 to 1000, arranged in order, and another table of like extent giving the numbers corresponding to the same products of factors, arranged with the largest factor first, and the others in descending order, so as to facilitate the entry, we could perform multiplication (where the product does not exceed 1000) by addition only. In the construction of such a table it would soon be found convenient to replace the two and throe ON MATIIEM^.TICAL TABLES. 19 figure primes by letters, to save room, and, in fact, to use letters tliroiigli- out — and further to simplify the printing by writing «■* as 4a, &c., which would do equally well ; we then have Dilling's tables, which have not the smallest connexion with logarithms. Such a table might once have been found useful ; but the slightest consideration shows that (except as a factor table) it would be all but valueless now. The space a large table of the kind would occupy, the impossibility of arranging the antifactor table so as to admit of easy entry, and the great convenience of existing tables (both Pythagorean and logarithmic) are alone sufficient to prove this. Crelle, 1836. This table occupies 1000 pages, and gives the product of a number of seven figures by 1, 2, ... , 9, by a double operation, very much in the same manner as Bretschneider's does for a number of five : viz., each page is divided into two tables ; thus, to multiply 9382477 by 7, we turn to page 825, and enter the right-hand table at line 77, column 7, where we find 77339 ; we then enter the left-hand table on the same page, at line 93, column 7, and find 050, so that the product required is 05077339.^ We think for numbers seven figures long the table effects a considerable saving of time, as it is as easy to use as BRExsciiNEinEK s for five figures. It would take some little practice to use the table rapidly in all cases, as of course the mode of entry, &e. must be varied according as the number consists of seven, six, five, &c. figures ; but the value of a table is measured not by the trouble lequired to learn to use it, but by the time saved by means of it after the computer has learnt its use. Bretschneider, 1841. This table is for the multiplication of any number up to 100,000 by a single digit. On each page there are two tables, the upper of which occupies ten lines, and the lower fifty. An example will show the method of using the table. Suppose it required to multiply 50878 by 7, then the table is entered on the page headed 0800 (the headings run from to 99, with two ciphers added to each). Facing 78 in the lower table we find *146 ; and in the upper table facing 568, in the column for 7, we find 397; the product required is therefore 398146, the third figure being increased because the 146 was marked by an asterisk. The arguments in the upper table, on the page headed 0800, are 08,168,208 . . . 908 (twice repeated for the two cases when succeeding numbers are less and greater than 50), and also 1, 2 ... 9, as the table is of double entry. The arrangement of the table is thus very ingenious ; but, as De Morgan has remarked, multiplication by a single digit is so simple an operation that it is questionable how far a table is serviceable when its use requires three distinct points to be attended to. The introduction (10 pages) gives a complete explanation of how the table can be used when the number of figures is greater than five. Having made some use of the table for this purpose, we do not think any time is saved by it ; at all events, not imtil the computer has had much practice in using it. Crelle, 1804. This magnificent table gives products up to 1000 x 1000, arranged in a most convenient and elegant manner, one consequence of which is that all the multiples of any number appear on the same page. It is also very easy to get used to the arrangement of the table, which is as useful for divisions as multiplications. It can be used for multiplying numbers which contain more than three figures, by performing the operation, three figures at a time ; but it requires some practice to do this readily ; and a similar remark applies to the extraction of square roots. There is one great ineouvenieuco that every computer must feel in using the work, viz. that the multiples of numbers ending in arc omitted, so that, c 2 30 REPORT — 1873. for example, we pass from 39 to 41. It is quite true that the columns for 40 are the same as those for 4 with the addition of a ; but the awkward- ness of turning to opposite ends of the book for (say) 889 and 890, and then having to add a to the latter, is very great. It is a pity that a desire to save a few pages should have been allowed to impair the utility (and it docs so 'most seriously) of so fine a table. The matter is referred to in the preface, where it is said that Crelle, " after mature reflection," decided to omit these numbers. The original edition was published in 1820, and consisted of two thick octavo volumes, the first proceeding as far as 500x1000, and the secord completing the table to 1000 x 1000. The inconven'cnce refeiTcd to above is felt more strongly in this than in the one-vo\imc edition, as frequently the numbei-s ending in have to be sought in a different volume from the others. Both editions are, we believe, very accurate. There are 3 pp. of errata (pp. xvii-xix) at the beginning of the edition of 1820. De Morgan gives 1857 as the date of Bremiker's reprint, and says he has heard that other copies bear the date 1859, and have no editor's name. Laundy, 1865. The first nine multiples of all numbers from 1 to 100,000, given by a double arrangement : viz., if it is required to multiply 15395 by 8, we enter the table on p. 4 (as 395 is intermediate to 300 and 400) at 15, and in column 8 find 122 ; we enter another table on the same page at 395, and in column 8 find 160; the product is therefore 123160. We take this number instead of 122160 because in the column headed 8, first used, there appears the note [375]*, the meaniug of which is that if the last three figures of the number exceed 375 (they are 395 in the above example) the third figure is to be increased bj'^ unity. The table is thus seen to be the same in lyrlnciple as ERETScnNEiDER, but not quite so convenient. There are the same objections to this as to the latter table. The present table occupies 10 pp. 4to, and Beetschneider's 99 pp. 8vo. Mr. Laundy remarks in his preface that Crelle's ' Erloichtcrungs-Tafel,' 1836, although one hundred times as largo as his, "must not bo estimated as presenting advantages proportionate to its vast difference of extent." In this we scarcely agree ; for it is only when the numbers are six or seven figures long that one begins to feel the advantages of a table for so simple an operation as multiplication by a single digit, and Crelle's table would not take much longer to use than the present. The following is a list of references to § 4 : — MaJtlpUcatloa Tables.— Dovsoy, 1747, T. XXXVIII. to 9 x 999<).; Hutrox, 1781 [T. I.] to 100 X 1000 ; Callet, 1853 [T. VIII.] ; SchrGx, 1800, T. III. ; Paekuurst, 1871, T. XXVI., XXXIII., and XXXIV.; see also Leslie, 1820, § 3, art. 3, and Wtjcheree, 1796, T. II. (§ 3, art. 6.) Art. 2. Tables of Proportional Paris. By a table of the proportional parts of any number x is usually under- stood, a table giving -^j^x, -f^A\ . . . -fj^x true to the nearest unit. Of course the assumption of 10 as a divisor is conventional, and any table giving X 2x (a 1),^^ -J — , . , . ^^ '- would equally bo called a proportional-part table. Ordi- (^ Gi CI nary proportional-part tables (viz. in which rt = 10) are given at the sides of the pages in all good seven -figure tables of logarithms that extend from 10,000 to 100,000. The difference between consecutive logaritlims at the commencement of the tables (viz. at 10,000) is 434, and at the end is there- fore 43 ; so that a seven-figure table of the above extent gives the proportional ON MATHEMATICAL TABLES. 21 parts of all numbers from 43 to 434 (note that near the commencement of the table, viz. from diff. 434 to diff. 340, the proportional parts are only given for every other difference in some tables ; whether a table gives the proportional parts of all the differences or not is generally noted in § 4), ISeveral scven-fignre tables extend to 108,000 ; and for the last 8000 the dif- fei'ences decrease from 434 to 403. Tables in which «=60 often accompany canons of trigonometrical functions that give the results for every minute, for convenience of interpolating for seconds; such must be sought from the descriptions of trigonometrical tables in § 3, arts. 10 and 15, and in § 4 ; we have also seen tables for which rt=30, where the functions are tabulated for every two minutes or two seconds. There are several tables to which proportional parts of the differences to hundredths (viz. in which «=:100) are attached, e.g. Gray (§ 3, art. 19), FiLiPowsKi (§ 4), and Pineto (§ 3, art. 13) ; but the ranges of the differences arc generally so small that it is not worth while giving references. In PiNETO, for instance, the range of the differences is only from 4295 to 4343 (in this work multiples are given, the last two figures being separated by a comma). The only separate table of proportional parts, properly so called, that we have seen, is Bremiker, 1843 (' Tafel der Proportionaltheile '). Proportional parts to hundredths (viz. miiltiples from 1 to 100, with the last figure omitted, and the last but one corrected) of all numbers from 70 to 099. A very useful table, chiefly intended for use in interpolating for the sixth and seventh figures in logarithmic calculations. T. III. of ScnEON (§ 4) (which is there called an Interpolation Table) is a large table of proportional parts. It is to be noticed that all multiplication tables are, or rather can be used as proportional-part tables. A table of multiples, with the last figure omitted, and the last but one corrected (which can be done at sight), is a proportional- part table to tenths ; and if the last two figures are omitted, and the last remaining figure corrected, to hundredths (see therefore § 3, arts. 1 and 3). It is proper here to allude to slide-rules and other mechanical appliances for working proportions &c. A card intended to do the work of a very large slide-rule is described in § 4 (Eveeett) ; and some information and references about slide-rules of different shapes M'ill be found in a paper " On a New Proportion Table," by Prof. Everett, in the Phil. Mag. for Nov. 1806. The following are references to works described in § 4 : — T(djJes of Fro2>ortio7ial Parts.— iiir J. Mooke, 1681 [T. II.]; DucoM, 1820, T. XX.; Lynn, 1827, T. Z; Callet, 1853 [T. YIII.]; Schkon, 1860, T. Ill, Art. 3. Tables of Quarter Squares. Tables of quarter squares have for their object to facilitate the performance of multiplications ; and the principle on which their utility depends is con- tained in the formula ab = i(a+hy-l{a-hy-, so that with such a table to multiply two numbers we subtract the quarter square of the difference from that of their sum ; the multiplication is there- fore replaced by an addition, a subtraction, two single entries of the tables, and a final subtraction — a very considerable saving if the numbers be high. The work is more than with a product table, where a double entry gives the result at once ; but the quarter squares occupy much less space, and can Z2 REPOBT — 1873. therefore be tabulated to a mucb. greater extent without inconvenience. In tables of quarter squares the fraction ^ which occurs when the number is odd is invariably left out ; this gives rise to no difficulty, as the sum and difference of two numbers must be both odd or both even. A product can, of course, be obtained by logarithms with about the same facility as by a table of quarter squares ; but the latter is preferable when all the figures of the result are required. LuDOLF, 1690 (see § 3, art. 4), in the preface to his 'Tetragonometria,' explains the method of quarter squares completely, and shows how his table is to be used for the purposes of multiplication. The earliest tabic oi quarter squares De Morgan had heard of was Voisin, 1817 ; but Centnerschwer (see below) refers to one by BUrger of the same date, the full title of which we have quoted from Bogg. Crelle, in the preface to the first edition of his ' Eechentafeln ' (1820, p. XV.), speaks of " Quadrat-Tafelu nach Laplace und Gergonne, mittelst welchcr sicli Producte fiudcn lassen," &c. The allusion to Laplace doubtless refers to the memoir in the ' Journal Polytechuique,' noticed further on in this article ; but we cannot give the reference to Gergonne. The largest table of quarter squares that has been constructed is that published by the late Mr. Laundy, which extends as far as the quarter square of 100,000 ; it would be desirable, however, to have a table of double this extent (viz. to 200,000), which would perform at once nnilti])lications of five figures by five figures (Mr. Laundy's table is only directly available when the sum of the ni;mbers to be multiplied is also of five figures). The late General Shortrede constructed such a table, we believe, in India, but unfortunately abandoned the idea of publishing it on his return to England, where he found so much of the field already covered by Laundy's tables. De Morgan, writing when it was anticipated that Shortrede's table would be pubhshed, suggested that it would be convenient that the second half should appear first ; and we should much like to see the publication of a quarter- square table of the numbers from 100,000 to 200,000. Mr. Laundy, in the preface to his ' Table of Quarter Squares ' (p. vi), says that Galbraith, in his ' General Tables,' 2nd edit. 1836, Avhich Avcrc intended as a supplement to the second edition of his ' Mathematical and Astronomical Tables,' gives a table (T. xxxiv.) of quarter squares of numbers from 1 to 8149. This book is neither in the British Museum nor the Cambridge Uni- versity Library. The second edition of his ' Mathematical and Astronomical Tables ' (1834) contains no such table. There is, however, no doubt about the existence of the work, as the Babbage Catalogue contains the title " Galbraith, "\Y., N"ew and concise General Tables for computing the Obhquity of the Ecliptic, &c. Edinburgh, 1836." In 1854, Prof. Sjdvester having seen a paper in Gergonne in which the method was referred to, and not being aware that tables of quarter squares for facilitating multiplications had been published, suggested the calculation of such tables, in two papers — " jSFote on a Formula by aid of which, and of a tabic of single entry, the continued product of any set of numbers . . . may be effected by additions and subtractions only without the use of Logarithms " (Philosophical Magazine, S. 4. vol. vii. p. 430), and "On Muhiplication by aid of a Table of Single Entry " (Assurance Magazine, vol. iv. p. 236). Both these papers were probably written together ; but there is added to the former a postscript, in which reference is made to Voisin and Shortrede's manuscript. Prof. Sylvester gives a generalization of the formula for ah as the difference of two squares, in which the product a^ a., . . . «,, is expressed as the sum of ON MATHEMATICAL TABLES. 23 nth. powers of a^, a^, . , . «„, connected by additive or subtraetive signs. For the product of three quantities the formula is abc=^{{a+b + cf-(a + b-cy-(c + a-hf-{b + c-ayi And at the end of the ' Philosophical-Magazine ' paper, Prof. Sylvester has added some remarks on how a table to give triple products should bo arranged. At the end of a memoir, " Sur divers points d' Analyse," Laplace has given a section " Sur la Eeduction des Fonctions en Tables " (Journal de I'Ecole Polytechnique, Cah. xv. t. viii. pp. 258-265, 1809), in which he has briefly discussed the question of multiplication by a table of single entry. His aual}-sis leads him to the method of logarithms, quarter squares, and also to the formula siua sin&=r|{cos(rt— 6)— cos(a + 6)}, by which multiplication can be performed by means of a table of sines and cosines. On this he remarks, " Cette maniere ingenieuse de faire servir des tables de sinus a la multiplication des nombres, fut imaginee et employe'e un siecle environ avaut I'iuveution des logarithmes." It is worth notice that the quarter-square formula is deduced at once from sin rt sin 6 =g { cos (« — i) — cos (a + b)},'by expanding the trigonometrical func- tions and equating the terms of two dimensions ; similarly from sin a sin b sin c = j{sin (« + c— 6) + sin(a + 6— c) + sin(6-|-c— a) — sin (a + 6 + c)}, by equating the terms of three dimensions, we obtain abc=-^{(a-\-b-^cy — &c. }, as written down above, and so on, the general law being easily seen. We may remark that there is an important distinction between the trigonometrical formulas and the algebraical deductions from them, viz. that by the latter to multiply two factors we require a table of squares, to multiply three a table of cubes, and so on ; {. e. each different number of factors requii'es a sepa- rate table; while one and the same table of sines and cosines wiU serve to multi])ly any number of factors. This latter property is shared by tables of logarithms of numbers, the use of which is of course in every way preferable ; still it is interesting to note the inferiority that theoretically attaches to the algebraical compared with the trigonometrical formulo3. Other remarks on the subject of multiplication by tables are to be found in § 3, art. 1. It is almost imnecessary to remark that a table of squares may be used instead of one of quarter squares if the semisum and semidifference of the numbers to be multiplied be taken as factors. Tables of squares and cubes are described in the next section. *Voisin, 1817. Quarter squares of numbers from unity to 20,000. We have taken the title from the introduction to Mr. Latjndt's ' Quarter Squares' (1856). De Morgan also so describes the work. We have seen no copy; but there is one in the Graves Library, although we were unable to find it : it will be described from inspection in the supplement to this Eeport. Leslie, 1820. On pp. 249-250 there is a table of quarter squares of numbers from 1 to 2000, reprinted from Voisijt, 1817, whose work Leslie met with at Paris in 1819. There is also given, facing p. 208, a large folding sheet, containing an enlarged multiplication table, exhibiting products from 11x11 to 99x9 9, the table being of triangular form. There are also, on the same sheet, two smaller tables, the fii'st giving squares, cubes, square roots (to seven places), cube roots (to six places), and reciprocals (to seven places) of numbers from 1 to 100, and the second being a small m.ultiplication table from 2 x 2 to 25 x 25. In the first edition (1817, pp. 240) the quarter- square table does not appear ; and in the folding sheet (which follows the 24 EEroRT — 1873. preface) the Bmaller multi])lication table is not added ; squares and cubes only arc given in the other small table. Centnerschwer, lb25. [T. I.] A table of quarter squares to 20,000 ; viz. -r, is tabulated from .r=l to .r=20,000, the fraction I, -which occurs -when -± X is odd, being omitted. The last two figures of the quarter square, whicli only depend on the last two figures of the number, are given once for all on two slips bound up to face pp. 2 & 41. TuU rules are given as to how to use the table as a table of squares ; and three small tables are added, by means of which the square of any number of five figures can be found tolerably easily. The arguments are printed in red. [T. II.] Square roots of numbers from 1 to 1000 to six places. There is a long and full introduction prefixed. In his prefiice Centnerschwer states that after his work was in the press, lie received from C'relle a table, by J. A. P. Burger, entitled " Tafeln zur Erleichtcrung in Eechnungen," Karlsruhe, 1817, in wbich the author claims to be inventor of the method, while Centnerschwer states it was known to LuDOLF (1690), and even Euclid. That Ludolf was the inventor of the method is true ; and there is attached to his work a table of squares to 100,000 (see Ludolf, § 3, art. 4). The full title of Biirger's work, which we have not been successful in ob- taining a sight of, is (after Hogg) as follows : — " Tafeln zur Erleichtcrung in llechnungen fiir den allgemeinen Gebrauch eingerichtet. Deren ausserst ein- fach gegebene Ilcgeln, nach Avelchen man das Product zweier Zahlen ohne Mul- tiplication finden, auch sie sehr vortheilhaft bei Ausziehung der Quadrat- iind Cubicwurzel anwendcn kann, sich auf den binomischen Lehrsatz griinden. Nebst Anhaug iiber meine im vorigen Jahr erschienene Paralleltheorie. Carlsruhe, 1817. 4to." The book last referred to was entitled "VoUstiindige Thcorie der Parallellinien &c. Carlsruhe, 1817 ; 2nd edit. 1821," as given by Hogg under Elcmentar-Geometrie. Merpaut, 1832. The premeire partle gives the arWmome (/. e. quarter square) of all numbers from 1 to 40,000, so arranged that the first three figures of the argument are sought at the head of the table, the fourth figure at the head of one of the vertical columns, in which, in the line with the final (fifth) figure in the left-hand column, is given the quarter square required. The quarter squares are printed in groups of three figures, the second group being under the first, &c. A specimen of this table is given by LArNnr (1850, p. V of his Introduction). The deuxibne ]partie gives the reciprocals of all numbers from 1 to 10,000 to nine figures. The author seems not to have been aware of the existence of any of the previous works on the subject of quarter squares. Laundy, 1S5G. Quarter squares of all numbers from unity to 100,000, the fraction |, which occurs when the number is odd, being, as usual, omitted. The arrangement is rs in a seven-figure logarithm table ; viz. the first four figures are found in the left-hand column, and the fifth in the top row ; the three or four figures common to the block of figures are also separated as in logarithmic tables, and the change in the fourth or fifth figure is denoted by an asterisk prefixed to all the quarter squares affected : at the extreme left of each page is a column of corresponding degrees, minutes, and seconds (thus, corresponding to 43510 we have 12° 5' 10" = 43510"). At the bottom of the page arc differences (contracted by the omission of the last two figures) ON MATHEMATICAL TABLES. 25 and proportional parts. The figures are very clear ; and there is a full intro- duction, with explanations of the use, &c. of the tables. Mr. Laundy was induced to construct his tabic by Prof. Sjdvester's paper in vol. iv. of the 'Assurance Magazine,' referred to above ; and a description of the mode of construction &c. of the table (most of which is also incor- porated in the introduction to it) is given in vol. vi. of the ' Assurance Magazine.' Art. 4. Tables of Squares, Cubes, Square roots, and Cube roots. Tables of squares (or square roots of square numbers) are of nearly as great antiquity as multiplication tables, and would, wc think, be found to be rather common in early manuscripts on arithmetic. They are, as a role, but slightly noticed in histories of the subject (see references in § 3, art. 1), partly because the latter are very meagre, and very many manuscripts remain still unexamined, and partly because it is rather the province of a history to de- scribe the improvement of processes. The perfection of the methods of ex- tracting the square root of numbers not complete squares, however, occupies a conspicuous place. In the MSS. Gg. ii. 33 of the Cambridge University Library, are two frag- ments, one of Theodorus Meletiniotes, the second of Isaac Argyrus (botb much of the same date, time of John Palaeologus, 1360) (concerning the first, see Vin- cent, Manuscrit de la Bibliotheque Imperiale, xix. pt. 2. p. 6). The fragment is a portion of the first book, and contains rules and small tables for multi- plication, fractional computation &c. The tract of Isaac Argyrus is entitled " tov 'Apyvpou evpeffis rwy Terpayw- riKiJii}' TrXevpdJi' tiLv fxt) prjrwi' uptdjjiiiijy. At the end there is a table of the square roots of all integral numbers from 1 to 120, in sexagesimal notation. The table is prepared as if for three places of sexagesimals ; but usually two only are perfect. Errors (probably due to the copyist) are frequent. Before the table is a description of the method of its use, including an explanation of the method of proportional parts. De Morgan speaks of two early (printed) tables in Pacioli's ' Summa,' 1494, and by Cosmo Bartoli, 1564, extending respectively to the squares of 100 and 661. The tables which we have examined are described below; but there are several of some extent, which De Morgan refers to, that we have not seen, viz. : — Guldinus, 1635, squares and cubes to those of 10,000 ; AY. Hunt, 1687, squares to that of 10,000 ; and J. P. Biichner's ' Tabula Radicum,' Nuremberg, 1701, which gives squares and cubes up to 'that of 12,000 (full title given in Eogg). Lambeet (Introd. ad Suppl. &c. 1798) says that Biichner's table is " j^lena errorum." Eogg gives the title " Bobert, K. W., Tafeln der Quadratzahlen aller natiii-lichen Zahlen von 1-25,200 ; der Kubik- zahlenvon 1-1200; der Quadrat- u.Cubicwurzeln von 1-1000. Neu berechnet, Leipzig, 1812 ;" and the title occurs in the Eoy. Soc. Lib. Cat. (though the book is not to be found in the Library). De Morgan mentions " Schiert, 'Tafeln,' &c. Eohn om Eheim, 1827," as giving squares to 10,000, which is no doubt a misprint for " Schiereck, J. F., Tafeln aller Quadrate von 1 bis 10,000. 4to. Koln am Ithein, 1827," which occurs in the Babbage Catalogue, and also in Eogg. From the title of another work of Schiercck's given in the former catalogue, it appears that the table of squares also appeared as an appendix to his ' Handbuch fiir Geometer,' published in the same year. De Morgan speaks of Ludolf's ' Tetragonometria,' 1090, which gives squares up to that of 100,000, " as being the largest in existence, and very 26 REPOiiT — 1873. little known." This is true ; but Kulik, 1848, is of the same extent, and also gives cubes vp to that of 100,000, thus giving the largest table of squares, and by far the largest table of cubes in the same work, and in a compact and convenient form : of this work also it may be said that it is very little known. Httttois-, 1781 (§ 4), gives squares to that of 25,400, and cubes to that of 10,000 ; but for most purposes Baelow (stereo. 1840), which gives squares, cubes, and square roots and cube roots (and reciprocals) of numbers to 1000, and is very accurate, is the best. We have not seen any square-root or cube- root table of greater extent. Extensive tables of quarter squares have been published, which are de- scribed in § 3, art. 3 ; and some tables of squares, as Eaa de Bruno, were constructed with the view of being used in applying the method of least squares. It is scarcely necessary to remark that logarithms find one of tlie most valuable applications in the extraction of roots. Multiplications &c. can bo performed generally without their aid with a little more trouble ; for finding square and cube roots they are extremely useful ; but for the extraction of higher roots there exists no other method admitting of convenient application. Maginus, 1592. The ' Tabula Tetragonica ' is introduced by the words " sequitur tabula numerorum quadratorum cum suis radicibus nunc primum ab auctore supputata, ac in lucem aBdita," and occupies leaves 41-04. It gives the squares of all numbers from 1 to 100,100. We have seen the ' Tabula Tetragonica ' quoted as an independent work ; and De Morgan says that it was published separately, with headings and explanations in Italian instead of Latin. In the copy before us Tavola is misprinted for Tahvla on pp. 41 and 43 back (only the leaves are numbered). The work contains sines, tangents, and secants also. Magini was, we suppose, the vernacular name of the author, and Maginus the same Latinized. We have somewhere seen Magini and Maginus spoken of as if they were different persons. Alstedius, 1649. In part 3. pp. 254-260, Alsted gives a tabic of squares and cubes of numbers from 1 to 1000. Alsted's is the first Cyclopaedia, in the sense that we now understand the word. [Moore, Sir Jonas, 1650?] Squares and cubes of numbers from 1 to 1000, fourth powers from 1 to 300, fifth and sixth powers from 1 to 200. In the book before us (Brit. Mus.) this tract (which has a separate pagina- tion) is bound up at the end, after Moore's 'Arithmetick (and Algebra), Contemplationes Geometricoe, and Conical Sections.' De Morgan says that power tables, exactly the same as these, were given in Jonas Moore's ' Arith- metic ' of 1650, and reprinted in the edition of 1660 ; so that probably the tract noticed here usually formed part of the 'Arithmetick.' [Pell], 1672. Squares of numbers from 1 to 10,000 (pp. 29). This is followed by the 6 one-figure endings, the 22 two-figure endings, the 159 three-figure endings, and the 1044 four-figure endings, which square numbefs admit of. They are given at length, and also in a synoptical form. The last page in the Eoy. Soc. copy is signed John Pell. (In the Itoyal Society's Li- brary Catalogue this table is entered under Fell, the signature at the end in the Society's copy having been struck out so as to render the first letter uncertain.) In the Brit. Mus. is a copy without any name (so that perhaps Pell's name was supplied in the Eoy. Soc. copy only in manuscript). ' Dr. Poll's Tables,' however, is written in it, and no doubt can exist about its authorship. ON MATHEMATICAL TABLES. 27 Ludolf, 1690. Squares of all numbers from ixnity to 100,000, arranged in columns, so that the first three or four figures of the root are to be found at the top of the column, while the final ones are given in the left-hand column of the page. The table is well printed and clear, and, except Kulik, 1848, which is of the same extent, is the largest table of squares that has been published, and occupies about 420 pages. Some errata in it are given at the end of the introduction (150 pp. in length), in which aU possible uses of the table are explained. Lambert (Introd. ad Supplementa, 1798) speaks of the numbers in the table as " satis accurati." In chapter v. (pp. 48-86) (' Do Tabularum usu sou Praxi circa Multiplicationem et Divisionem ') the use of the table as one of quarter squares (see § 3, art. 3) is fully explained ; as squares are given in the table, the sum and difference have to be divided by 2. llules and examples are also added as to how to proceed when the semisum exceeds the limits of the table by any amount ; and the processes &c. arc explained with such fulness as to prove that all the credit of first perceiving the utility of the method and calciilating the necessary table is due to Ludolf. The work is said to be very scarce ; biit we have seen several copies ; there is one in the Library of Trinity College, Cambridge, and another in the Graves Library. Heilbronner (under Herwart ab Hohenbtjrg) mentions Ludolf (Hist. Math. p. 827), and (referring doubtless to the method of quarter squares) says that he invented a method of performing multiplications and divisions without the Pythagorfcan abacus, " quae prolixe ab lUustr. Wolfio in scinen Anfangs- Griinden et suis Elementis Matheseos exjjonitur.'' Seguin, 1786. At the end of the book is given a table of the squares and cubes of numbers from unity to 10,000. The figures have heads and tails, and are very clear. De Morgan states that the table was reprinted at about the beginning of the century, and that it was this table which convinced him of the superiority of the numerals with heads and tails, aud led him in the reprint of Lalande's table, 1839, to adopt this figure — an example Avhich has since been very frequently followed. As De Morgan does not appear to have seen it, it is possible that the ori- ginal table was not reprinted, but only published separately, as the figures in the table attached to Seguin answer De Morgan's description very well. Barlow's tables (the stereotyped edition of 1840). Squares, cubes, square roots, cube roots, and reciprocals to 10,000. The square roots and cube roots are to seven places, and the reciprocals to seven significant figures, viz. nine places to 1000, and above this ten. The work is a reprint of the more im- portant tables in Barlow, 1814 (described in § 4) ; it was suggested by De Morgan, who wrote the preface (2 pp.), and edited by Mr. Farley, of the Nautical-Almanac Ofiice, who also examined carefully Barlow's tables. A list of ninety errors found in the latter is given on the page following the preface. This reprint is, we believe, very nearly, if not quite, free from error ; it is clearly printed and much nsed, We have also an edition, 1866, from the plates of 1840. Kulik, 1848. The principal table occupies pp. 1-401, and gives the squares aud cubes of all numbers //-ohi 1 to 100,000. There is a compression resembling that in Ckelle's ' Eechentafeln ;' viz. the last four figures of the square and cube are printed but once in each line, these figures being the same for all squares and cubes in the same line across the double page. The arrangement will be rendered clear b)' the description of a page — say, that corresponding to 92, There are ten columns headed 92, 192, 292. . . .092, 28 KEPORT — 1873. each containing two vertical rows of numbers, the one corresponding to W, iukI the other to N'; the lines are numbered 0,1,2. . . .49 (and on the next double page 50 ... . 99). If, then, we wish to find the cube of 79217, we take the figures 49711300i31 from column 792, lino 17, and add the last four figures 1313 (which conclude the cube of 9217 in the same line) ; so that the cube required is 497113061311313. Certain figures, common to the whole or part of a column, are printed at the top, and the change in the column is denoted by an asterisk. This is the largest table of cubes in ex- istence, and (excei>t Ludolf, Avhich is of the same extent) is also the largest table of squares. The printing is clear, and the book not bulky ; so that the table can be readily used. At the end are eleven subsidiary tables. T. 1 (Perioden (jcmder Sumnienden) consists of columns marked 4, 6, 8 .... 48 at the top, and 96, 94 .... 52 at the bottom, each containing the " complete period " of the number in question ; thus for 42 we have 42, 84, 26, 68, 10, &c. (these numbers being the last two figm'cs of a series of terms in arith- metical progression, 42 being the common difference) ; and these are given till the period is completed, i. e. till 42 occurs again. This may be at the end of 25 or 50 additions ; if the former, the periods are given commencing with 1 , 2, 3 (as well as 0) ; if the latter, with 1 or 2 only, as the case may be; the periods for .r and 100 — x are of course the same, only in reverse order. Tlao use of the table as a means of verifying the table of squares is obvious. T. 2. Primes which are the sum of two squares (these being given also) up to 10,529. T. 3. Odd numbers which are the difference of two cubes (these being given also) to 12,097. T. 4. Odd numbers which are the sum of two cubes (these being given also) to 18,907. T. 5-9. Four-figure additive and subtractive congriicnt endings for numbers ending in 3 and 7, or 1 and 9, etc. : the more detailed description of these tables belongs to the theory of numbers, which will form a j^art of a subse- quent Report. T. 10. The 1044 four-figure endings for squares, and the figures in which the corresponding numbers must end. T. 11. First hundred multiples of tt and tt"^ to twelve places. There is appended to the tables a vcrj- full description of their object and use. Bruno, Faa de, 1809. T. I. of this work (pp. 28) contains squares of numbers from 0-000 to 12-000, at intervals of -001 to four places (stereo- typed), intended for use in connexion with the method of least squares. The following are references to § 4 : — Tables of Sqiiairs and Cuhes, or hoth Squares and Cubes. — ScnuLZE, 1778 [T. IX.] and [T. X.] ; Hutton, 1781 [T. II.] and [T. III.] ; Vega, 1797, VoL II. T. IV. ; Lambert, 1798, T. XXXV. and XXXVI. ; Barlow, 1814, T. I. ; Schmidt, 1821 [T. V.] (with subsidiary tables) ; Hantschl, 1827, T. VIII. ; *Salomon, 1827, T. I. ; Geuson, 1832, T. II. and III. ; Hulsse's Vega, 1840, T. IX. C. ; Trotter, 1841 [T. VI.] ; Muller, 1844 [T. III.] ; MiNsiNGER, 1845 [T. II.] ; KonLER, 1848, T. V. and VI. ; Willich, 1853, T. XXI. ; Beardmore, 1862, T. 35 ; IIankine, 1866, T. I. and II. ; Wackeebarth, 1867, T. VI. ; PARKHrRST, 1871, T. XXVI. and XXXII., and XXXIV. (multiples of squares); Peters, 1871 [T. VI.]. See also Taylor, 17S0 [T. IV.] (§ 3, art. 9). Tables of hiquarc Roots and Cube Boots.— I) odsos, 1747, T. XIX. ; ScHULZK, 1778 [T. XI.] and [T. XII.]; Maseres, 1795 (two tables); ON MATHEMATICAL TABLES. 29 Vkoa, 1797, Vol. ir. T. IV. ; Hantscul, 1827, T. YIII. ; *Salomon, 1827, T. I.; GutisoN,11832, T. IV. and V.; Hulsse's Vega, 1840, T. VIII.; Trottek, 1841 [T. VI.] ; Minsinger, 1845 [T. II.] ; Koiilek, 1848, T. VII. ; WiLLicn, 1853, T. XXI. ; Bearduore, 1862, T. 35 ; *ScnLOMiLCH [18G5?] ; Eankine, 1866, T. I, A ; Waceerbarth, 1867, T. VII. Sec also Centner- scnwER, 1825 [T. II.] (§ 3, art. 3). And for Squares (for method of least squares), Muller, 1844 [T. III.]. Endings of Squares. — (Three-figure endings') Lambert, 1798, T. IV. Art. 5. Tables of Poivers higher than Cubes. We know of no work coutaiuing powers of numbers (except squares and cubes) only. Both Hutton, 1781, and Barlow, 1814, give the first ten powers of the first hundred numbers ; but we have seen no more extensive table of this kind. Shanes (§ 4) gives every twelfth power of 2 as far as 2"-' ; and, according to De Morgan, John Hill's 'Arithmetic,' 1745, has all powers of 2 up to 2"^ Tables of compound intei-est are, in fact, merely power tables, as the amount of <£M at the end of n years at r per cent, is M( 1 + | . In "' ^ \ 100/ interest tables r has usually values from 1 to 8 or 10 at intervals of | or | for ditfercnt periods of years ; but they could not be of much use, except for the purpose for which they are calculated. A good table of powers is still a desideratum, as the need for it is often felt in mathematical calculations. Very many functions are expansible in an ascending (convergent) scries of the form A^-{- A^x + A^.v^ -\- &c., and a de- scending series (generally semiconvergont) of the form B^4-Bj.i-~^4-B2.r~- + itc. The former is usually very convenient for calculation when x is small, and the latter when x is large ; but between the two, for values of :c included between certain limits above unitj', there will be an interval where neither series is suitable — the ascending series because the terms x, x"^,. . . . (x >1) increase so fast that n must be taken very large (t. e. very many terms must be included) before A„ is so small that A„.r" can be neglected, and the de- scending series because it begins to diverge before it has yielded as many decimals as are required. For these intermediate values the former series (if there is no continued fraction available) must be used ; and then the terms begin by increasing, often so rapidly, if x bo moderately large, that it may be necessary to calculate some of them to fifteen or twenty figures to obtain a correct value for the function to only seven or eight decimals. In these cases, so long as ten figures only are wanted, logarithms are employed ; but when more are required recourse must be had to simple arithmetic ; and it is then that a power table is so much needed. Mr. J. "W. L. Glaisher has had formed in duplicate a table giving the first twelve powers of the first thousand numbers, which, after the calculation has been made independently a third time, will be stereotyped and published, probably in the course of 1873 ; it is hoped that it will help to make the tabulation of mathematical functions somewhat less laborious and difficult. The following tables on the subject of this article are described in § 4 : — Tables of Poivers higher than Cubes. — Dodson, 1747, T. XXI. (powers of 2) and T. XXII. ; Schulze, 1778 [T. VIIL] ; Htjtton, 1781 [T. IV.] ; Vega, 1797, Vol. II. T. II. (powers of" 2, 3, and 5); Vega, 1797, Vol. II. T. IV. ; Lambert, 1798, T. VIT.-IX. (powers of 2, 3, and 5) and T. XL. ; Barlow, 1814, T. II. and III. ; Hijlsse's Vega, 1840, T. VI. (powers of 2, 3, 5) and T. IX. A, B, D, E ; Kohler, 1848, T, II. (powers of 2, 3, and 5) and T. IV. ; Shanks, 1853 (powers of 2 to 2"') ; Beakdmoee, 1862, T. 35 ; 30 REPORT— 1873. Eaneine, 1866, T. 2. See also Sir Jonas Moobe [1650?], § 3, art. 4; Taylor, 1781 [T. IV.] (§ 3, art. 9). Tables for the solution of Cubic Equations, Vv/.. +(x — aj^). — Lambert, 1798, T. XXIX. ; Barlow, 1814, T. IV. Art. 6. Tables for the expression of vulgar fractions as decimals. The only separate tables we have seen are Wuoherer and Goobwyn's works described at length below. The Babbago Catalogue contains the title of an anonymous book, " Tafeln zur Verwandlung aller Briichc von j^-^ bis ■^roli^' ^^^"^ "^°^ Tu'uo ^^^ TtMmro ^^ fiinf- bis siebenziftrige Decimalbriiche, 4to, Oldenbiirg, 1842," of which De Morgan says "it gives every fraction less than unity whose denominator does not exceed three figures, nor its nu- merator two, to seven places of decimals. It is arranged by numerators ; that is, all fractions of one numerator are upon one double page." Eecipro- cals would property be included in this article ; but from their more frequent use they have been jjlaccd in an article by themselves (§ 3, art. 7) ; Picabte's table in that article gives multiples of reciprocals. We must especially mention the " Tafel zur Verwandlung gemeiner Briiche mit Nennern aus dem ersten Tausend in Decimalbriiche," Avhicli occupies pp. 412-434 of vol. ii. of ' Carl Friedrich Gauss Werke,' Gottingen, 4to, 1863, and which somewhat resembles Goodwyn's tables described below. In it, among other things, the reciprocal of every prime less than 1000 is given completeh/ (i. e. till the figures circulate). Had we met with the table eai'lier we should have given a full description ; but we merely confine our- selves hei'e to giving the reference, reserving a more detailed explanation for a future Eeport. Wucherer, 1796, The decimal fractions (to five places) for all vulgar fractions, whose numerators and denominators are both less than 50 and prime to one another, arranged according to denominators ; so that all having the same denominator are given together : tlius the order is ... . -jL., -fif' TT'- • • -i?' tV' t\- • • -5 the arguments being only given in their lowest terms. After ^^ the system is changed, and the decimals are given for vulgar fractions whose numerators are less than 11 only; thus we have -Jj,, tV' TiJ- • ■ •'k?)' ftV' 7)"t- • • -^3 consecutive arguments (tlie arguments not being necessarily in their lowest terms) ; and the denominators proceed from 50 to 999. [T. II.]. Sea-arjesimal-Bi'ikhe, viz. sexagesimal multiplication table to 60 X 60 ; thus, as 5 X 29" = 145" = 2' 25", the table gives 2.25 as the tabular result for the joint-entry 5 and 29. There are seven other tables (II.-VIII.) for the conversion of money into decimals of other money, for the coins of different countries ; the English table will serve as an example. There are given as arguments ^-^, -jf^, ^„ |fi^ (/. e. Id., 2d., 'M., &c.), and as tabular results the corresponding decimal fraction to ten places (/. e. of £1), and also the shillings and pence ; thus for ^4^ there are given -3041666666, and 6s. Id. Tlie Ileichs-Geld and Pfennig table is practically the same ; the denomi- nators are in all cases 240, or 960, or submultiples of the latter. Regarded ma- thematically the English table gives nearly as much as all the rest, as for denominators above 240 only a few numerators are taken. There are also tables of interest, present value, &e., to a great many places. The value of tt is given on the last page to 306 places; thus, if the diameter = lOOUO. . . .(306 ciphers), then tt = 31415 (307 figures), the ciphers and figures being written ON MATHEMATICAL TABLES, 31 at length — a curious mode of statement at the end of a book occupied with decimal fractions. Goodwyn's Tables, 1816-1S23. It is convenient to describe Good- ■wyn's four works (the titles of which are given at length in § 5) together, as they all relate to the same subject. The Tahidar Series of Decimal Quotients (1823) forms a handsome table of 153 pages, and gives to eight places the decimal corresponding to every vulgar fraction less than ^^j, whose numerator and denominator are both not greater than 1000. The arguments are not arranged according to their numerators or denominators, but according to their magnitude, so that the tabular results exhibit a steady increase from -001 (= xu\rLr)to -09989909 (= JJL). The author intended the table to include all fractions whose numerators and deno- minators were both less than 1000 without restriction ; and at the end of the book is printed " End of Part I. ; " but no more was ever published. The arrangement of the arguments in order of magnitude is not very good, as it requires the first two figures of the decimal to be known in order to know where to look for it in the table ; the table would be more useful if it were re- quired to find a vulgar fraction (with not more than three figures in numerator or denominator) nearly equal to a given decimal*; but this is not a trans- formation that is often wanted. "When the decimal circulates and its period is completed within the first eight figures, points are placed over the first and last figures of the period, if not, of course only over the first; and by means of the same author's table of ' Circles ' described boloM", the period can be easily completed, and the whole decimal fraction found. The fractions which form the arguments are given in their lowest terms. The Table of Circles (1823) gives all the periods of the circulating decimals that can arise from the division of any integer by another integer less than 1024. Thus for 13 we find -076923 and -1 53846, which are the only periods in which the fraction I- can circulate. The periods for denominator 2" 5"' x are evidently the same as those for denominator x; and arguments of this form are therefore omitted ; but a table is given at the end (pp. 110 and 111), showing whether for any denominator loss than 1024 the decimal ( 1 ) terminates, and is therefore not included in the table, (2) is in the table as it stands, or (3) is in the table but has to be sought under a different argument (these last being numbers of the form 2» 5'" .r). A third table (p. 112) also gives the' number of places after the sejyaratrix (decimal point) at which the period commences. The principal table occupies 107 ])p. Some of the numbers are very long, (e. g., for 1021 there are 1020 figui-es in the period), and are printed in lines of difterent lengths, giving a very odd ai^pearance to many of the pagesf. A table at the end contains all numbers of the form 2" 5"' that are less than * It is proper to note, however, that the table was no doubt calculated for this purpose ; the author considered his 'Table of Circles ' as giving decimals to vulgar fractions, and in- tended this table to give vulgar fractions to decimals (see the introduction to the second part of the ' Centenary ' 1816) ; the ' Tabular Series ' (181C) is complementary to the ' Cen- tenary ;' but not so the ' Tabular Series' (1823) to the ' Table of Circles ' (1823), as the latter only gives the periods. t If the period of a decimal consists of an even number of figures, it is well known that the figures in the last half ai-e the complements to nine of tlie figures in the first lialf; and tlie periods have been printed so that the comiilementary figures should bo under one another. When the period is odd, there is always anollicr period of eoraplementary figures, and the two are printed one under the other ; these facts account for wliat at first siglit appears a capricious arrangement of the figures. 32 REPORT — 1873. 1,000,000, arranged in order of magnitude, with the values of n and m, and also the values of the reciprocals of the numbers (expressed as decimals) and the total number of the proper vulgar fractions in their lowest terms which can arise for any of the arguments as denominator. An example of the use of the tables is given at the end of the book. The First Centenary Sfc. [1816] contains the factors of all numbers to 100, and the complete periods of their reciprocals or multiples of theii- reciprocals, also the first six figures of every decimal fraction equivalent to a vulgar frac- tion whose denominator is equal to the argument. The following is a spe- cimen of one of the tables : 34 2.17 •70588235 294117G4 33 •970588 1 31 •911764 3 29 •852941 5 27 •794117 7 25 •735294 9 23 •676470 11 21 •617647 13 19 •558823 15 The explanation is veiy simple: we have || = -970588, and the other figiires of the period are 23529411764; fj- = •911764, and the other figures are 70588235294, &c. If the numerator is in the third column m-c take the complement of the result {i. c. subtract each figure from 9) ; thus J^ = •029411, and the other figures of the period are 76470588235'. The even numbers are omitted, as the fractions are not in their lowest terms ; thus ^^ = -}--f-, and must be sought under argument 17. [This table was published separately by Goodwyu for private circulation. There is no date on the title- page*; but the address is Avritten from Blaekheath, and dated March 5, 1816.] There is added a tabular series of complete decimal quotients of fractions whose numerator is not greater than 50 and denominater not greater than 100 (the heading of the table incorrectl)" says, " iieither numerator nor de- nominator greater than 100 "), arranged as in the ' Tabular Series' &c.,1823 ; it is followed by an auxiliary table for completing such quotients as consist of too many places to allow all the digits of their periods to appear in the principal table. There is an appendix on Circulates &c. The ' Tabular Series' (1816 and 1823) are interesting as exhibiting in the order of magnitude all fractions whose numerators and denominators are both less than 100 up to ^, and whose numerators and denominators are both less than 1000 up to Afl_. In the preface to the latter table the author gives as a fact he has observed, that * It IS by no means improbable that the titlepage has been torn out from the onlv copy we have seen, viz. that in the Eoyal Society's Library. ON MATHEMATICAL TABLES. 33 " In ail}' three consecutive vulgar fractions in the table, if the numerators of the extremes and the denominators be added together, the sum will form the numerator and denominator of a fraction equal to the mean." That this is the case with all fractions, ranged in order, whose numerators and denomi- nators are integers less than given integers, is a theorem discovered by Cauchy and published by him in his ' Exercices.' It has been thought worth while to describe Goodwyn's works at some length, as they are almost unique of their kind, and are rarely to be met with. De Morgan states that " Mr. Goodwyn's manuscripts, an enormous mass of similar calculations, came into the possession of Dr. Olinthus Gregory, and were purchased by the Eoyal Society at the sale of his books in 1842."' There is no mention of them, however, in the lloyal Society's Catalogue of MSS. ; and nothing is known of them at the Society. They may possibly be brought to light in the rearrangement of the manuscripts consequent upon the approaching change of rooms. Art. 7. Tables of Mecijirocals. The most extensive table is Oakes, 1865. Reciprocals from 1 to 100,000. This table gives seven figures of the reciprocal, and is arranged as in tables of seven-figure logarithms ; viz. the first four figures are found in the column at the left-hand side of the page, the fifth figures run along the top line, and the sixth and seventh are inter- polated for by proportional parts. The reciprocal of a number of five figures is therefore taken out at once, and the process of taking out a recipi'ocal is exactly similar to that of taking out a logarithm. From 10,000 to 22,500 the differences and proportional parts (being numerous) are placed on the lower half of the page, the differences being also placed at the side of each line ; but above 22,500 the differences and proportional parts are placed at the side of the page as in tables of logarithms. The figures have heads and tails ; and the change in the third figure of the reciprocal is made evident by prefixing an asterisk to the succeeding numbers in the line. The table is the result of an original calculation, and was con- structed by means of the obvious theorem that the difference of two recipro- cals, divided by the difference of the corresponding numbers, is the reciprocal of the product of those numbers. The reciprocals of the highei' numbers, however, were calculated by differences, which difterences were found by logarithms. Various checks were applied ; and the whole was virtually re- computed on the Arithmometer of M. Thomas de Colmar. The significant figures of the reciprocals alone are tabulated, decimal points and ciphers being omitted, for the same reason that characteristics are left out in loga- rithmic tables. In T. I. of Baelow (§ 4) reciprocals are given of numbers from 1 to 10,000 ; and this table also appears in the stereotype reprint of 1840 (see § 3, art. 4) : the latter is the most generally used table of reciprocals, and is of sufficient extent for most purposes ; it is also reputed to be very accurate, and is perhaps free from error. It must be added that Goodwyn's ' Table of Circles,' and ' Tabular Series,' &c., 1823 (§ 3, art. 6), give reciprocals of numbers less than 1024 complete; viz. the whole period is given, even where it exceeds a thousand figures. See also the reference to Gauss, vol. ii., near the beginning of the last article (§ 3, art. 6). As most nearly connected with a table of reciprocals (it gives not only 1873. D 34. REPOKT — 1873. the reciprocals, but also multiples of them), wo here describe Picarte's ' La Division reduite a une Addition.' Ficarte [18G1], The principal table occnpies pp. 15-104, and gives, to ten significant figures, the reciprocals of all numbers from 1000 to 10,000, and also the first nine multiples of the latter (which are therefore given to 10 or 11 sig- nificant figures). It is easy to see how this table reduces Division to Addition. The arguments run down the left-hand column of the page ; and there are niuo other columns for the multiples ; each page contains 100 lines ; so that there are 10,300 figures to the page. Owing, however, to its size, and to the smallness and clearness of the figures, there is no confusion, the lines being well leaded. The great table is preceded by two smaller ones, the first of which (pp. 6, 7) gives the figures from the ninth to the fourteenth (inclusive) of the logarithms of the numbers from 101,000 to 100,409 at intervals of unity (downwards), with first, second, and third differences ; and the second (pp. 10, 11) gives ten-figure logarithms of numbers to 1000 ; and from 100,000 to 101,000 at in- tervals of unity (with differences). There is also some explanation &c. about the manner of calculating logarithms by interpolation, &c. The author remarks on the increasing rarity of ten-figure tables of logarithms, referring, of course, to Vlacq and Vega. The whole work was submitted by its author to the French Academy, and reported on favourably by a Commit- tee consisting of MM. Mathieu, Hermite, and Bienayme. The report (made to the Academy Feb. 14, 1859) is printed at the beginning of the work. M. Eamon Picarto describes himself as Member of the University of Chili ; and the Chilian Government subscribed for 300 copies of the work. There is no date ; but the "privilege" is dated ISTov. 1860, and the book was re- ceived at the British Museum, April 29, 1861, so that the date we have assigned is no doubt correct. On the cover of the book are advertised the following tables by the same author, which we have not seen : — " Tables de multiplication, contenant les produits par 1, 2, 3 .... 9 et toutes les quantites au-dessous de 10,000, 1 vol. in-l8 jesus." " Tableau Pithagoriqiie, etendu jusqu'a 100 par 100, sous une nouveUe forme qui a permis de suppriraer la moitie dcs produits." It is scarcely necessary to remark that any trigonometrical table giving sines and cosecants, cosines and secants, or tangents and cotangents, may bo used (and sometimes with advantage) as a table of reciprocals. The extreme facility with which reciprocals can be found by logarithms has prevented tables of the fonner from being used or appreciated as much as they deserve. The following is the list of references to § 4 : — Tables of Reciprocals.— }S.xmmiS, 1795; Baelow, 1814, T. I. (to 10,000) ; Teottee, 1841 [T. YIII.] ; Willich, 1853, T. XXI. ; Beabdmoee, 1862, T. 35 ; ScHLoMiLCH [1865 ?] ; Eankine. 1866, T. I. and I. A ; Wackeebaeth, 1867, T. IX.; Paekhtjest, 1871, T. XXV.; see also Meepact, 1832 (§ 3, art. 3) ; Baelow (1840) (§ 3, art. 4). Art. 8. Tables of Divisors (Factor tables), and Tables of Primes. If a number is given, and it is required to determine whether it be prime, and if not what are its factors, there is no other way of effecting this ex- cept by the simple and laborious process of dividing it by every prime less than its square root, or until one is found that divides it without remainder*. The construction of a tabic of divisors is on the other hand very simple, as it * Wilson's theorem (viz. that 1 . 2 . 3. ...(«- 1) + 1 is or is not divisible by n, according as n is or is not prime) theoretically affords a criterion ; b-at the labour of applying it would be far greater thun the direct procedure bv trial. ON MATHEMATICAL TABLES. 35 is merely necessary to form the miiltiples of 2, 3, 5 . . up to the extent of the table, the numbers that do not occur being of course primes. The manner in which the formation of these multiples is best effected, and other practi- cal details, are explained by Bueckhaedt in his preface to the second million. The following is a Hst of tables of divisors and of primes, abridged from an elaborate account prefixed to Cheriiac : — 1657. Francis Schooten : table of primes to 9997. 1668. Pell (in Branker's translation of Rhonius's ' Algebra,' iniblished at London) : least divisors of odd numbers not ending in 5 to 100,U00. 1728. Poetius. An ' anatome' of numbers to 10,000. 1746. KEtJGEE. Primes to 100,999. 1767. Anjema. All divisors (simple and compound) of numbers to 10,000. 1770. Lambeet. Least divisors of numbers to 102,000 (multiples of 2, 3, and 5 omitted). 1772. Marci. Extension of Lambert's table by the addition of primes to 400,000. 1785. Neumann. Simple divisors (Pell only gave the least) of numbers to 100,100 (multiples of 2, 3, 5 omitted). 1797. Vega. Simple factors to 102,000, and 'primes to 400,000 (seo Yega, ' Tabula;,' 1797, Vol. II. T. I.). 1804. Krause. Factor table to 100,000. From the above list Chernac has omitted Rahn (1659), giving factors to 24,000, and Pigri (1758) to 10,000, which are described below. A more important omission. is that of Felkel, whose table is noticed at length further on. The titles of Anjema's, Neumann's, and Krause's works are given in the Babbage Catalogue as follows : — " Anjema (Henricus), Tabula divisorum omnium numerorum naturalium ab 1 usque ad 10000. 4to, Lugd. Bat. 1707 ; " " Neumann (Jobann), TabcUen der Prim-Zahlen- und der Factoren der Zahlen, welche unter 100100, und durch 2, 3, oderS nicht theilbar sind ; herausgegoben durch J. N. 4to, Dessau, 1785 ; " and " Krause (Karl C. F.), Factoren- und Primzahlen-Tafel von 1 bis 100000 neu berechnet. Fol. Leipzig, 1804." The same catalogue also contains the title, " Snell (F. W. D.), Ueber cine neue und bequeme Art, die Faktorentafeln einzurichten, nebst einer Kup- fcrtafel der einfachen Faktoren von 1 bis 30000. 4to. Giessen und Darm- stadt, 1800." The following are accounts of tables we have seen : — Rahn, 1659. On pp. 37-48 is given a table of divisors; viz. the least divisor of every number, not divisible by 2 or 5, is tabulated from 1 to 24,000, the primes being marked with a p. Pigri, 1758. All the simple factors (so that if multiplied together they give the number) are given of all numbers from 1 to 10,000. When the number is a power, letters are used instead of numbers (a = 2,b = 3,c = 5, &c., as explained on p. 11 of the book) ; thus, answering to 25 Ave have cc, to 27 hhh, to 225 bh, cc, &c. Kriiger, 1746. At the end of the ' Algebra ' is a list of primes to 100,999, arranged consecutively in pages of six columns, and occupying 47 pp. The titlepage runs ' Primzahlcn von 1 bis 1000000' ; but the limit is as above stated ; and there is no possibility that the copy before us is incomplete, as the last page is a short one, and there is no printing on the back. 1)2 36 REPoiiT — 1873. The primes of each hundred are separated, which for some purposes would be an advantage. LiMBERT states (Introd. ad ' Supplemental &c., 1798) that Kruger received this table from Peter Jaeger. Felkel, 1776. Table of all simple factors of numbers to 144,000, the tabular results being obtained from three tables. Thus Table A gives primes to 20,.353 ; these occupy one page, along the top line of which run the Greek letters o, /3 . . . . and down the left-hand column four alphabets consecutivel}-, viz. small italic, small German, capital italic, and capital Gorman (there being 100 lines); and any prime given on this page is henceforth in the book denoted by its coordinates, so to speak : thus 9839 would be printed yu^i, &e. The principal table occupies 24 pp. ; and then Table B occupies one page at the end. Suppose it required to find the factors of 1.38, .593. The middle table is entered at 138 and Table E at 593. In tlie latter we find as result " g line 20," so that we know that the compartment under ^ in the 20th line of the block 138, refers to the number in question. In this compartment is printed e, g, (3x, which, interpreted by Table A, gives 7, 13, and 1523 as the factors. There are a few details that have been omitted in this description ; the last three figures are written in the compartment wherever there is room for them. On the titlepage is a large engraving of a student (no doubt a portrait of Felkel) turning in contempt from a disordered cabinet of military books to another neatly arranged, containing Euler, Newton, Maclaurin, Bernoulli, Boscovich, &c., and holding in his hand the works of Lambert ; with mottoes " Bella odi, Pacem diligo, vera soqiior," &c. above. It will be seen that this table is entirely superseded by Chernac and Burckhardt. In the arrangement of the latter the table would only have occupied 16 much smaller pages, and its use would have required no explauation ; but on account of the rarity of the work, it has been thought worth while to describe at some length what is certainly the most remarkable-looking table we have seen. De Morgan states that " Murhard mentions the first part of a table (by A. Pelkel) of the factors of all numbers not divisible by 2, 3 or 5 from 1 to a hundred millions, Vienna (1776)." On referring to Murhard we find such is the case, " 100,000,000 " being an obvious misprint for " 10,000,000 ; " wo have seen Murhard's error reproduced by other writers. Of Felkel's table Gauss (in the letter prefixed to Base's Seventh Million) says : " Felkel hatte die Tafel im Mauuscripte bis 2 Millionen fertig und der Druck war bis 408,000 fortgeschritten, daun aber sistirt, und die ganze Anflage wurde vernichtet bis auf wenige Exeraplare des bis 336,000 geheuden Theils, woven die hicsige Bibliothek eines besitzt." The copy of Felkel in the Eoyal Society's Library, which extends to 144,000, is that which has been described above. Felkel's table is also referred to by Hobert and Ideler in the introduction to their work (see § 4). Felkel was editor of the Latin edition (Lisbon, 179S) of Lambert's 'Zusiitze' (the ' Supplemcnta' &c., see § 4) ; and he has there given, in the ' Introductio Interpretis ' and at the end, some account of his life and the work he accomplished and hoped to accomplish with regard to the theory of numbers. He commenced the study of mathematics when of a somewhat advanced age ; and he speaks in the warmest terms of Lambert, with whom he was in cor- respondence, and from whom he derived much assistance. This accounts for Lambert being the book open before the student in the engraving described above. In a note on p. siv of the Introductio to the ' Supplemental he (Felkel) ON MATHEMATICAL TABLES, 37 says : " Non solum iiiveni formam omnes divisores nuraerorum excepto maxi- mo, ab 1 usque 1,008,000 in spatio 42 plagularum rcprffiscntandi, vcruni etiam rcipsa opus spatio 16 meusium usque ad 2,016,000 coiifeci, annoque 1785 .... ad 5,000,000 usque continuavi." (See also p. yII of the ' Introductio lu- tcrpretis '). Since writing the above description of Felkel, I have examined (in the Graves Library) a far more complete copy, which contains probably all that Tclkel ever printed. There are three parts (bound together). The first is the same as that described above, and extends to 144,000 ; the second part (with fresh pagination) extends from 144,001 to 336,000 (pp. 2-03) ; we then have 'Tabula Factorum pars III exhibens factores numerorum ab 336,001 usque 408,000,' occupying pp. 65-87. The table thus gives factors as far as 408,000. The words " 336,001 usque 408,000 " have clearly ori- ginally stood " 144,001 usque 366,000 ;" but the latter numbers have been stamped out and the former printed over them. There is a note in the work in the handwriting of Mr. Graves's librarian, which, referring to Gauss's remark quoted above, proceeds : — " This copy contains 3 parts and gives the factors of all numbers up to 408,000 ; such a copy is perhaps unique." Gauss stated that all the copies were destroyed except a few, which extended to 330,000 ; so that there can be no doubt that tlie Graves copy, extending to 408,000, must be, to say the least, excessively rare. It should be added that the title and preface to the Graves copy are in Latin, while the lloyal Society's copy has them in German (Poggendorff also quotes the title in German with date 1777) ; the preface is dated April 1, 1777, although the titlepage bears date 1776. In the Graves copy some errata in Part I. are given. For several reasons Felkel's connexion with numerical tables is a curious one, and the record of his life would be interesting. "VVe have seen (in some work of reference) a number of mechanical contrivances assigned to him as their inventor. Chernac, 1811. In a thick quarto are given all the simple divisors of numbers from 1 to 1,020,000 (multiples of 2, 3, and 5 being excluded). This book was found by Burckhardt (who subsequently published the same table, the least divisor only being given) to be very acciu-ate ; he detected only 38 errors (he has given them in the preface to his first million), of which only 9 are due to the author, the remaining 29 having been caused by the slipping &c. of type in the printing. Hutton's Phil, and Math. Diet. 1815. In vol. ii. pp. 236-238 (Art. ' Prime Numbers ') is a table giving the least divisor of all numbers from 1 to 10,000, multiples of 2 and 5 being omitted. Burckhardt (First Million), 1817. Least divisors of every number to 1,020,000. The library of the Institute contained a manuscript (calculated by Schenmarck ?) giving the least divisor of numbers to 1,008,000 ; Burck- hardt therefore computed the next 12,000 himself, and compared the manu- script with Cheenac — a laborious work, as when a wrong divisor was given, Burckhardt had to satisfy himself if the number was really prime, as was the case in 236 instances. For primes less than 400,000 he referred to Ycga (see Vega's ' Tabula,' 1797, Vol. II. T, I., and Hulsse's Jega, 1840, T. V.). Ojily 38 errors were found in Chebnac. On the last page is a small table con- taining the number of figures in the periods of the reciprocals of 794 primes below 9901 (779 of which are below 3000). Burckhardt mentions in the preface that he has nearly completed the manuscript of the fourth, fifth, and sixth millions, which will be published, if the sale of the first three millions is as EEPORT— 1873. sufficiently favourable to induce the bookseller to undertake them. There are three pages on the use of the tables. This work, though containing the first million, was published after the second and third. Five errors are pointed out at the beginning of Case's ' Seventh Million.' Burckhardt (Second Million), 1814. The arrangement is the same as for the first million ; and the table extends from 1,020,000 to 2,028,000. This was the first published of the three millions ; and the method of calculation &c. is explained in the introduction, the least factor alone being given. If the others are required, the process is of course to divide the number by this factor and enter the table again with the quotient. To facilitate the division, on the first page (p. viii) a table is given of the first 9 multiples of all primes to 1423. Burckhardt (Third Million), 1816. The arrangement is the same as in the other millions : the table extends from 2,028,000 to 3,036,000. Rees'sCyclop8edia(vol.xxviii. Art. 'Prime Numbers'), 1819. Attached to the article "Prime Numbers" in liees's ' Cyclopfcdia,' is a table of 23 pp., giving a list of primes up to 217,219 arranged in decades — a very convenient table, as there are 910 primes on each page. It is stated (and truly) that the primes are given to twice the extent that they are to be found in any previous English work. In the course of the article the author says, "And a work lately published in Holland, not only contains the prime numbers up to ] ,000,000, but also the factors of all composite numbers to the same extent — a performance which, it must be allowed, displays the industry of its author to much more advantage than either his genius or judgement." Tliis can only refer to Chbh- NAc's table, which was published at Dcvcnter (Davcntria) in 1811 ; and it is a matter of regret that an English writer on mathematics should have thought only deserving of a sneer a work the performance and extension of which had been consistently urged by Euler and Lambert and afterwards by Gauss. One would expect the article of such a writer on the theoiy of numbers to bo very poor ; and such is the case. He has not thought it worth while to state where the table he gives has been copied from ; it is no doubt taken from Vega (' Tabulje '), 1797, Vol. II. T. I. Dase (Seventh Million), 1862. The least divisor of all numbers from 6,000,001 to 7,002,000 (multiples of 2, 3 and 5 excluded), and therefore also a table of primes between these limits. The arrangement is as in Bueckhaedt, there being 9000 numbers to the page. This work was undertaken by Dase at the suggestion of Gauss ; and the letter of the latter is printed in the preface. In it Gauss adverts to, and expresses his concurrence in, Felkel's desire Ihat the factorial tables should be extended to ten millions ; he states that a manuscript containing the fourth, fifth, and sixth millions (viz. 3,000,000 to 6,000,000) was some years before presented by Crelle to the Berlin Academy, and he expresses a hope that it will soon be jrablished ; he therefore suggests that Dase should complete the portion from 6,000,000 to 10,000,000. Dase accordingly undertook the Avork, and at the time of his death in 1862 had finished the seventh miUion entirely and the eighth million nearly ; while many factors for the ninth and tenth millions had been determined. The seventh million (as also the two follow- ing) were published after Dase's death by a committee of his fellow-towns- men as a memorial of his talent for calculation. Dase (Eighth MiUion), 1863. The arrangement is the same as in the seventh million ; and the table extends from 7,002,001 to 8,010,000 ; the paging runs from 113 to 224. ON MATHEMATICAL TABLES. 89 There is a short preface of 2 pp. by Dr. Eosenberg, who edited the work, which was left nearly complete by Dase. Dase and Rosenberg (Ninth Million), 1865. The arrangement is the same as in the pre\doiis two millions ; and the table extends from 8,010,000 to 9,000,000. The work left incomplete by Dase at his death was finished by Dr. Rosenberg ; the paging runs from 225-334. It is stated in the preface that the tenth million (the last which the tables were intended to include) was nearly completed ; but we believe it has not yet appeared. It will have been seen from the above accounts that CnERNAc's, Btjeck- haedt's, and Dase's tables together contain all the published results with re- gard to factors of numbers ; and by means of them we can find aU the simple divisors of numbers between one million and three millions and between six millions and nine millions easily, and between unity and one million at sight. There is, however, the gap from three millions to six millions ; and it is very much to be regretted that this is not filled up. Gauss states a table of divisors from three millions to six millions exists in manuscript at Berlin ; and Eurckhardt also formed a similar table ; so that this portion has apparently been twice calculated (by Crelle ? and Eurck- hardt). Gauss's letter is dated 1850 ; and it is a calamity that the anticipations con- tained in it have not been realized, as a manuscript unpublished does more liarm than if it were non-existent, by checking others from attempting the task. The completion of Gauss's scheme (viz. the publication of tables to ten millions) is very desirable, as these tables may be regarded as data in regard to investigations in the theory of numbers (see references to memoirs of Euler and Gauss in Cheenac, and Gauss's letter). The tenth million also seems to be still unpublished, though seven years ago we had Dr. Eosenberg's assurance that it was nearly completed. If the whole ten millions were published, we should much like to see a list of all the primes up to this point published separately. Oakes, 1865 (Machine table). The object is to find the prime or least factors of numbers less than 100,000 ; and for this purpose there are three tables, A (1 page large 8vo), E (4 pp. folio), and C (1 page obi. folio), and nine perforated cards, the one to be employed depending on the group of 10,000 that contains the argument. The mode of entry is somewhat compli- cated ; and the table can only be regarded as a matter of curiosity ; for in the method of arrangement of Eurckhasdt or Dase the least factors of aU numbers under 100,000 only occupy a little over 11 pp. or six leaves of small folio or large 8vo size — while the present apparatus consists of six leaves of large and diiferent sizes, and nine cards, besides requiring an involved course of procedure. Col. Oakes does not explain the principle on which his method depends. The following is a list of tables contained in works that are described in §4. TaUes of Divisors.— Dosso:s, 1747, T. XVII. (to 10,000) ; Maseres, 1795 (to 100,000) ; Yega, 1797, Vol. II. T. I. (to 102,000) ; Lambert, 1798, T. I. (to 102,000) ; Earlow, 1814, T. I. (to 10,000) ; Hantschl, 1827, T. VII. (to 18,277): *Salomon, 1827, T. II. (to 102,011); Hulsse's Vega, 1840, T. V. ; KoHLEE, 1848, T. VIII. (to 21,524) ; Houel, 1858, T. VII. (to 10,841); Eanktne, 1866 (to 256). See also Gruson, 1798, § 3, art. 1. List of Prime Numhers.—JiomoTi, 1747, T. XVIII. (10,000 to 15,000) ; Vega, 1797, Vol. II. T. I. (102,000 to 400,000) ; Lambert, 1798, T. II. 40 REPoiiT — 1873. (multiples of primes); T. YI. (to 102,000); Baelow, 1814, T. Y. (to 100,103) ; Hulsse's Yega, 1840, T. Y. (102,000 to 400,313); Minsingeh, 1845 [T. II.] (to 1000) ; Btene, 1849 [T. I.] (to 5000) ; Wackeebaeth, 1867 (to 1063) ; Paekhuest, 1871, T. XXIII. (to 12,239). Art. 9. Sexagesimal and Sexcentenary Tables. Originally all calculations ■were sexagesimal ; and the relics of the sj'stem still exist in the division of the degree into 60 minutes, and the minute into 60 seconds. To facilitate interpolation, therefore, in trigonometrical and other tables, several large sexagesimal tables have been constructed, which are described or referred to below. They are, we believe, scarcely used at all now, for several reasons — first, on account of the somewhat cumbrous size of the complete tables, and secondly because for most purposes logistic logarithms (see § 3, art. 18) are found more expeditious and convenient. A third reason is that both Eernotjlli's and Taylor's tables were published by the Commissioners of Longitude, and, like the other publications of the Board, were advertised so little that their existence never became generally known. Bernoulli, 1779. A sexcentenary table to 600 seconds, to every second, giving at once the fourth term of any proportion of which the first term is •600" and each of the other two are less than 600". The table is, of course, of double entry ; it may perhaps be best described as giving the value of -^~, correct to tenths of a second, x and y each containing numbers of seconds less than 600", .r being expressed in seconds alone, and y in minutes and seconds (though the latter can be turned into seconds at sight, as the number of seconds in the necessary integer number of minutes is given at the top of each page). The .r's run down the left-hand column, and the y's along the top line ; and the arrangement is thus : — The portion of .rfroni 1" to 60" and the whole range of y is given ; this occupies 30 pp. ; then the portion for x from 60" to 120", and for y from 60" to 600"; and so on. The chief use of the table consists in the fact that in astronomical tables the difi'erences are usually given for every 10', so that the interpolation gives rise to a proportion of the kind described above : in some cases the use of the table would be preferable to that of logistic logarithms. Taylor, 1780 [T. I.] (pp. 240). The table exhibits at sight the fourth term of any proportion where the first term is 60 minutes, the second any number of minutes less than 60, and the third any number of minutes and seconds xmder 60 minutes. If the second term consists of minutes and seconds, the table must be entered twice (once for the minutes and once for the seconds). The table can of course also be put to other uses. There is also added a table of the equation of second difference, giving the correction to be applied on this account in certain cases. [T. II.] (pp. 250, 251). Giving the thirds answering to the decimals in every column of [T. I.] where the result is expressed in minutes, seconds, and decimals of a second. [T. III.] (pp. 263-312). A millesimal table of proportional parts adapted to sexagesimal proportions, giving the result of any proportion in which the first term is 60 minutes, the second term any number under 60 miimtes, and the third term any absolute number under 1000. It is in fact the same as the sexagesimal table [T. I.], only that the third term is expressed in seconds, and is given only to 1000 (16' 40"), and the result is also expressed in seconds (in [T. I.] the third terms are given both in minutes and seconds) and ON MATHEMATICAL TABLES. 41 in seconds wholly, so that the expression of the result in seconds wholly is the chief characteristic of [T. III.]. This table is followed by 3 pp. to convert sexagesimals into decimals and vice versa, and numbers into sexagesimals and vice versa. The other tables are weights and measures &c. There are numerous examples given in the introduction. [T. IV.]. Another table occupying one page (p. 252) should be noticed ; it gives squares, cubes, fourth, fifth, and sixth powers of any number of minutes up to GO' : thus the square of 3' is 9" ; the cube, 27'" ; the fourth power 1'" 21'" ; the fifth 4'" 3% &c. The words sursolid and square cube are used for the fifth and sixth powers. On the present work see also Beverley (1833?) (§ 4). It was the author of this table (Taylor) who afterwards calculated the logarithmic trigonometrical canon to every second. The following are references to works in § 4 : — - Sexagesimal tables :— Lynx, 1827, T. Z ; Bagay, 1829, T. XXIV, (lo- garithms with sexagesimal arguments); Beverley (1833 ?),T. VI. (pp. 232 &c.) and T. XV.; Shortkede (Com. log. Tab.), 1844; Gordon, 1849, T. XVII. (half sines, &c., expressed sexagesimally). Tables for the conversion of sexagesimals into decimals, and vice versa: — Douglas, 1809, T. III., Supplement ; Ducom, 1820, T. XX. ; HtJLssE'a Vega, 1840, T. IV. Art. 10. Tables of natural Trigonometrical Functions. A history of trigonometrical tables by Hutton is prefixed to all the editions of his ' Tables of Logarithms ' published during his lifetime * ; and, in his Article on Tables in the ' English Cyclopaedia,' De Morgan has given what is by far the most complete and accurate account of printed tables of this kind that has been published. Information about the earlier tables is also to be found in Montucla and Delambre (see references in De Morgan). For many years, when Mathematics had not passed beyond Trigonometry, the method of construction and calculation of the * Canon Trigonometricus ' formed one of the chief objects of the science, and the works on the subject were comparatively numerous, though now, of course, of purely historic interest only. Prior to the introduction of sines from the Arabians by Albategnius, trigonometrical calculations were always made by chords. The imit-arc was the arc whose chord was equal to the radius (viz. 60°) ; and both arc and radius were divided into 60 equal parts, and these subdivided again into 60 parts, and so on. (It thus appears that it was not the right angle that was divided into 90, 60 and 60 pai-ts, &c., but that the unit-angle was 60°, so that the division was strictly sexagesimal throughout. It is curious that in some modern tables (see Beverley, T. VI. and XV. &c.) the original arrangement has been restored, for convenience of interpolating by Taylor's sexagesimal table). Thus in the earliest existing table, viz. the table of chords in the Syntaxis of Ptolemy (died a.d. 178), the chord of 90° is 84° 51' 10". Purbach (born 1423) and llegiomontanus (born 1436) calcu- lated sines, the former to radius 600,000 and the latter to the same radius and also to radius 1,000,000; but it is not certain whether they were printed. The first known printed table, according to De Morgan, is a table of sines to minutes, without date, but previous to 1500. Peter Apian first published a table with the radius divided decimally (1533). Tangents were first pub- * It also forms Tract XIX. vol. i. pp. 278-306 of his ' Mathematical Tracts,' 1812. 42 EEPORT — 1873. lished by Eegiomontauus (1504) ; and tho first complete canon giving all the six ratios of the sides of a right-angled triangle is due to llheticns (1551), -who also introduced the semiquadrantal arrangement. Eheticus's canon was to every ten minutes to 7 places ; and Vieta first extended it to evenj minute (1579). The first complete canon published in England was by Blundevile (1594), although a table of sines had appeared four years earlier. It may be added that Eegiomontauus (1504) called his table of tangents (or rather cotangents) Tabula foscimda, on account of its great use ; and till the in- troduction of the word tangent by Finck (1583), a table of tangents was called a Tabula fcecunda or Canon foecundus ; Finck also introduced the term secant, the table of secants having previously been called Tabula benefica by Mauro- lycus (1558), and Talula fommdissima by Yieta. The above historical sketch has been compiled from Hutton and De Morgan ; so that most of tho statements contained in it arc not derived from our own inspection of the works mentioned. It is only inteaded to give an idea of the history of the natural canon ; and from the experience we have had of the value of second-hand information in mathematical bibliography, we should not re- commend great reliance to be placed ou any one of the facts. A good deal of information about llheticus, Vieta, &c. is given by De Morgan, whom we have scarcely ever found inaccurate, even in trifling details, Avhen describing works he has examined himself. We have seen several of the works noted, but not BulRcient to make any corrections of importance to the current histories. The next author of importance to Eheticus was Pitiscus (1G13), whose im- portant canon, which stiU remains unsuperseded, is described below. The in- vention of logarithms in the following year changed all the methods of calcula- tion ; and it is worthy of note that Napier's original table of 1614 (see § 3, art. 17) was a logarithmic canon of sines and not a table of the logarithms of numbers. Almost at once the logarithmic superseded the natural canon ; and since Pixiscus's time no really extensive table of pure trigonometrical functions has appeared. Natural canons are now most common in Nautical collections, where the tabular results are generally given to 5 or places only. Traverse tables (multiples of sines and cosines) have not been included (see § 2, art. 12). Massalotjp (described below), however, is really a table of this kind, although constructed for a different purpose. Finck [1583]. Canon of sines, tangents, and secants in separate tables, quadrantally arranged, for every minute of the quadrant, to 7 decimal ])laces. The sines occupy pp. 138-173, the tangents pp. 176-221, and the secants pp. 224-269. De Morgan says that Finck calculated his own secants. There is no date on the titlepage ; but the i^reface and the colophon are both dated 1583. The name tangent is introduced by Finck on p. 73, and that of secant on p. 76. These names were speedily adopted : thus Clavius, at the end of his edition of ' Theodosius ' (Eome, 1586), reprints Finck's tables, and uses his terms both in the headings of the tables and in the trigonometry. He does not mention either Finck or Eheticus by name, but speaks of them as recentiores (p. 188). Pitiscus, in his trigonometry appended to Abraham Shultet's ' Sphsericorum ' (Heidelberg, 1595), uses the names tangent and secant, and refers to Finck or Eheticus for the requisite canons ; and in his larger trigonometry (Augsburg, 1600) he reprints Finck's tables to five deci- mals, placing the sines, tangents, and secants together in one table. Blun- devile, in his ' Exercises ' (London, 1594), reprinted the tables from Clavius. All these woiks are before us ; but a more detailed account would be of only historical or bibliographical interest. ON MATHEMATICAL TABLES. 43 Rheticus, 1596 (' Opus Palatinum '). Complete ten-deeimal trigonome- trical canon for every ten seconds of the quadrant, semiquadrantally arranged, with differences for all the tabular results throughout. Sines, cosines, and secants are given on the versos of the pages in columns headed respectively Perjpendiculum, Basis, Hypotenusa ; and on the rectos appear tangents, cose- cants, and cotangents, in columns headed respectively Perpendiculum, Hypo- tenusa, Basis*. This is the celebrated canon of George Joachim llheticus, the greatest of the table-computers, to whom also is due the canon of sines described below under Pitiscus, 1613. At the time of his death (1576) Rheticus left the canon all but complete ; and the trigonometry was finished and the whole edited by Valentine Otho under the title ' Opus Palatinum,' so- called in honour of the Elector Palatine Frederick IV., who bore the ex- pense of publication. The edition before us is in two volumes, the second containing the ten-decimal canon and occupying 540 pp. (2-541) folio ; then follow 13 pp. of errata numbered 142-153 and 554. At the end of the first volume is a canon of cosecants and cotangents (in columns headed Hypotenusa and Basis respectively) to 7 places for every 10 seconds, in a semiquadrantal arrangement. It occupies 180 pp. (separate pagination, 2-181) ; and there seems no reason why it should have been printed at all, as the great ten-decimal canon completely supersedes it. Besides, it is exceed- ingly incorrect, as comparison with the latter shows at once. On this point De Morgan says that its insertion "was merely the editor's want of judg- ment ; it is clearly nothing but a previous attempt made before the larger plan was resolved on ;" while Hutton writes, " But I cannot discover the reason for adding this less table, even if it were correct, which is far from being the case, the numbers being uniformly erroneous and- different from the former through the greatest part of the table." Mention of it is introduced by Hutton with the words, " After the large canon is printed another smaller table," &c., while in the copy before us it ends the first volume, the second containing the great canon. It is also to be inferred from De Morgan's ac- count that the whole work generally is bound in one (very thick) volume. The tangents and secants in the early part of the great canon were found to be •inaccurate ; and the emendation of them was intrusted to Pitiscus, who "corrected the first eighty-six pages, in which the tangents and secants were sensibly erroneous " (De Morgan) ; and copies of this corrected portion alone were issued separately in 1607, as well as of the whole table with the correc- tions. We have not seen one of these corrected copies ; but vide De Morgan's fuU account, ' English Cyclopaedia,' Article " Tables," and ' Notices of the Roy. Astron. Soc.,' t. vi. p. 213, and ' Phil. Mag.' June, 1845, The pagina- tion of the other parts of the work is ' De Triangulis globi cum angulo recto,' pp. 3-140 ; ' De Fabi'ica Canonis,' pp. 3-85 ; ' De Triquetris rectarum line- arum in planitie,' pp. 86-104 ; ' De Trianguhs globi sine angulo recto,' pp. 1-341 ; ' Meteoroscopium,' pp. 3-121 (the first three by Rheticus and the rest by Otho). In 1551 Rheticus had published a ten-minute seven-place canon in his ' Canon Doctrinae Triangulorum,' Leipzig, with which the present work must not be confounded. And in 1579 Vieta published his ' Canon Mathematicus, seu ad triangula cum Adpendicibus,' for every minute of the quadrant. This * The explanation of these terms is evident. The sines and cosines arc perpendiculars and bases to a hypotenuse 10,000,000,000 ; the secants and tangents are hypotenuses and perpendiculars to a base 10,000,000,000, and the cosecants and cotangents are hypo- tenuses and bases to a perpendicular 10,009,000,000. The object Eheticus had in view was to calculate the ratios of each pair of the sides of a right-angled triangle. 44 REPORT — 1873. and several other works that we have examined will be noticed at length in a future Keport. On Eheticns's other works see Pitiscus, 1613, below. Gcrnerth has given a list of 598 errors that he found in the first seven or eight figures of the ten-decimal canon in the ' Zeitschrift f. d. osterr. Gymn.' YI. Heft, S. 407 (also published separately). He also gives an account of the contents of the ' Opus Palatinum,' from which it appears tliat in his copy the difterent parts of it were bound up in a diflPerent order from that in which they appear in the copy wc have examined (which seems to be anomalous in this respect) ; and he omits the 121 pp. of the ' Meteoroscopium.' The great in- accuracy of the small canon is also noticed by him ; and it is on this account that he gives uo errata list for it. Pitiscus, 1613 [T. I.] (pp. 2-271, calculated by Rheticus). Natural sines for every ten seconds throughout the quadrant, to 15 places, semiqua- drantally arranged, with first, second, and third differences. (On p. 13, Fer- pendkulum and Basis are printed instead of Sinus and Sinus complementi). [T. II.] (pp. 2-61, calculated by Rheticus). Natural sines for every second from 0° to 1°, and from 89° to 90°, to 15 places, with first and second diff'erences. [T. III. and IV.] (pp. 3-15). The lengths of the chords of a few angles, to 25 places, with verifications &e., followed by natural sines and cosines for the tenth, twentieth, and fiftieth second in every minute to 35', to 22 places, with first, second, third, fourth, and sometimes fifth differences. The numbering of the pages thus recommences in each table (except. T. IV.) ; and each has a separate titlepage. On the first two the date is printed do . lo . XIII. instead of cIo . loc . xiii. The rescue of the MS. of this work from destruction by Pitiscus (as told by himself in the preface) forms a striking episode in the history of mathematical tables. The alterations and emendations in the earlier part of the corrected edition of the ' Opus Palatinum ' were made by Pitiscus; and he remarked that a table of sines to more places than ten was requisite to enable the corrections to be conveniently made. He had his suspicions that Rheticus had himself cal- culated a ten-second canon of sines to fifteen decimal places; and on application to Valentine Otho, the original editor of the ' Opus Palatinum,' the latter, who was then an old man, acknowledged that such was the case, but could not remember where the MS. was (" ob memoriaj senilis debilitatem "). He thought that perhaps he had left it at Wittemberg ; and accordingly Pitiscus sent a messenger there to search for it ; but after considerable expense had been in- curred he returned without it. After the death of Otho, when the MSS. of Rheticus, which had been in his possession, passed into the hands of James Christmann, the latter discovered the canon among them, when it had been given up for lost. As soon as Pitiscus knew this he examined the MSS. page by page, although they were in a very bad condition (situ et squalore obsitas ac pa3ne foBtentes), and to his great satisfaction found :^(1) the ten-second canon of sines to 15 places, with first, second, and third differences (printed in the work under notice) ; (2) sines for every second of the first and last degrees of the quadrant, also to 15 places, with first and second difi'erenccs ; (3) the commencement of a canon of tangents and secants, to the same number of decimal places, for every ten seconds, with first and second dif- ferences ; (4) a complete minute-canon of sines, tangents, and secants, also to 15 decimal places. From this account, talcen in connexion with the ' Opus Palatinum ' and the contents of the present work, one is able to form some idea of the enormous computations undertaken by Rheticus ; ON MATHEMATICAL TABLES. 45 his tables not only to this day remain unsuperscded and the ultimate authori- ties, but also formed the data whereby Vlacq calculated his logarithmic canon. Pitiscus says that for twelve years Eheticus constantly had some com- puters at work (duodecim totos annos semper aliquot Logistas aluit) ; and how much labour and expense on his part would have been wasted but for the zeal of Pitisciis is painful to contemplate ; as it was, it is matter of regret that llheticus did not live to see the publication of either of his canons, the first of which appeared twenty years, and the other thirty-seven years after his death. It was Pitiscus's intention to add Eheticus's minute-canon of tangents and secants ; but they laboured under the same defect as those in the (uncorrected) ' Opus Palatinum,' and on this account ho was dissuaded from so doing by Adrianus Romanus. The matter spoken of above as [T. III. and IV.] was due to Pitiscus himself, and was introduced at the advice of the same mathematician. The enormous work undertaken by Eheticus needs no eulogy ; and the earnestness and love of accuracy displayed by Pitiscus, not only rendered apparent by his acts but also evident in the prefaces to his several works, will always render his an honoured name in science. The present work is exceedingly rare ; and the copy wc have examined is in the library of the Greenwich Observatory. It, the ' Opus Palatinum,' and Vlacq's ' Arithmetica Logarithmica,' 1628, and ' Trigonometria Artifici- alis,' 1633, may be said to be the four fundamental tables of the mathemati- cal sciences. Gernerth (in the work cited under Rhexictis, 1596, supra) has given a list of 88 errors that he detected in the first 7 or 8 places of the canon of sines; he detected altogether 110; but 22 he states were given by Vega in his ' Logarithmisch-trigonometrische .... Tafeln und Pormeln,' Vienna, 1783 : this was Vega's first publication of tables ; and we have not seen the work. Grienberger, 1630. Sines, tangents, and secants, to 5 places, for every minute from 0° to 45° (with foot entries also ; but the table is only half a complete canon, as e.g. sin 50° could not be taken out from it). There are five more figures added to the sines, but separated from them by a point (this is not a true decimal point, as is evident from the rest of the work, where no trace of decimals occurs), the object the author had in view in adding them being that when the sines had to be multiplied by large numbers, the re- sults might still be correct to the last unit (radius 100,000). Grienberger stated that more than 35 years before (about 1595) he had calculated a canon of sines to 1 6 places, and made considerable progress with the secants when the ' Opus Palatinum ' appeared and caused him to lay aside his work. This he regretted exceedingly at the time of writing the present work, as he was not able to add the five extra figures to the tangents and secants, which he had transferred from his MS. in the case of the sines. The ' Opus Pala- tinum' contained enough figures; but some of them were doubtful, and he wished no doubt to attach to any part of his table. The book is a duodecimo volume, and would scarcely have been noticed here, but from the f;ict of j)art of it having been the result of an original calculation. Napier's bones are mentioned, but not logarithms. The preface contains Grienberger's 39-figure value of TT (see ' Messenger of Mathematics,' July 1873) ; and it was in con- nexion therewith that we sought the work out, and learnt with some surprise of Grienberger's incomplete and unpublished calculations. The copy we examined is in the British Museum. Massaloup, 1847, T. I. The first five hundred multiples of the sines and 46 REPORT — 1873. cosines of all angles from. 1° to 45° at intervals of 10' to two places. The table occupies 442 closely printed pages. T. II. gives the first 109 multiples of the sine of all angles from 0° to 15° at intervals of 1' to two places. The above is the mathematical description of these tables ; but in the book, which is intended for surveyors &c., the multiples correspond to differ- ent lengths (1.0,1.1,.. ..50.0 Eutheu) of the hypothenuse ; and the sine and cosine columns arc headed Rohe and Grundlinie, and are given in lluthen. As the arguments are at intervals of a Fuss (= jL of a lluthe) the table exhibits the results apparently to 3 places. The arrangement in T. I. is different from that in T. II., as while in the former the lluthen and Fiisse run down the column, and the minutes along the top line (so that aU the multiples of the same sine or cosine are given consecutively), in T. II. the minutes run down the column, and the Fiisse along the top line (so that the same multiples of different angles are given consecutively). In this table also the results are given to 3 places, if the method of statement used in the book be followed. As it has been assumed that a Ruthe = 10 Puss, while fre- quently it = 12 Fuss, T. III. is given to convert decimals into duo- decimals, or, more strictly, lluthen Decimalmaass into Werkmaass and Bergmaass. T. I. and II. are of course simple traverse tables. Junge, 18(34. Natural sines and cosines for every ten seconds of the quadrant to G places. The table is one of the clearest we have seen, the figures being distinct, and plenty of space being left between the columns &c., so as to give a light appearance to the page, though its large size is rather a disadvantage. The tabular results were interpolated for by Thomas's calculating machine from the natural sines in Ht/lsse's tables ; and the last figure may bo in error by rather more than half a unit. The connexion between the tables and Thomas's machine, referred to in the title and in the preface, merely amounts, we suppose, to this — that Avhilo' computers in general use log sines, those who possess Thomas's machine will find it easier to dispense with logarithms and use natural sines and ordinary arithmetic. *Clouth. Natural sines and cosines (to G places) and their first nine multiples (to 4 places) for every centesimcil minute of the quadrant, arranged seraiquadrantaUy, the sines and their multiples occupying the left-hand pages, and the cosines the right ; the arguments axe also expressed in sexagesimal minutes and seconds, the intervals being then 32"-4. We have not seen the work itself, but only a prospectus, containing 2 pp. (108 and 109) as specimens. Judging from this, the book would contain 208 pp. In the copy of the pro- spectus before us, the words " Mayen (chcz I'auteur) " are covered by a piece of paper on which is printed "Halle, Louis Nebert, Libraire-Editeur." There is no date; but we should judge the table to have been only recently published. We have also seen advertised ' Tafcln zur Bercchnung goniometrischer Co-ordinaten,' by F. M. Clouth — no doubt a German edition of the same work. The following is a classified list of trigonometrical tables described in iSines, iangents, secants, and versed sines. — (To 7 places) HAKiscni,, 1827j T. V. ; WiLLicH, 1853, 1.B; IIutton, 1858, T. IX. (To 6 places) Galbraith, 1827, T. VL ^'mes, tanrjents, awl secants. — (To 7 places) Sir J. Mooee, 1G81 [T. III.] ; ON MATHEMATICAL TABLES. 47 Vlacq, 1G81 [T. I.J ; Ozanam, 1G85 ; Sheravin, 1741 [T. IV.] ; Hent- RCHBN (Vlacq), 1757 [T. I.] ; ScncLZE, 1778 [T. V.] ; Lambert, 1793, T. XXVI. ; Douglas, 1S09 [T. III.]. (To 6 places) Oughtred, 1657 [T. I.] (centesimal division of the degree) ; Ursinus, 1827 [T. V.] ; Eeardmore, 1862, T. 38. (To 5 places) Houel, 1858, T. II. ; Peters, 1871 [T. V.]. ^ines and taivjents (only). — (To 7 places) Uates, 1781 [T. II.] ; Vega, 1797, T. III. ; Hobert and Ideleu, 1799 [T. I.J (centesimal) and B (cen- tesimal) ; (?) *Salomon, 1827, T. XII. ; Turkish Logaruhms f 1834J ; HtJLSSE's Vega, 1840, T. III. (To 6 places) Trotter, 1841 [T. IV.]. (To 5 places) Schmidt, 1821 [T. III.] ; Eankine, 1866, T. G ; Wacker- BARTH, 1867, T. VIII. (To less than 5 places) Parkhurst, 1871, T. XXX. and XXXI. TancjenU and secants (only). — Donn, 1789, T. V. (4 j)laees) ; [Sheep- shanks, 1844] [T. IV.] (4 places). Sines (alone). — (To 15 places) Callet, 1853 [T. VII.] (centesimal). (To 7 places) Donn, 1789, T. Ill ; Hassler, 1830 [T. V.]. (To 6 places) Maskelyne (Eequisite Tables, Appendix), 1802, T. I.; Ducom, 1820, T. XIX. ; KERiG.iiT, 1821, T. IX. ; J. Taylor, 1833, T. XX. ; Norie, 1836, T. XXVI.; Griefin, 1843, T. 19; J, Taylor, 1843, T. 32; Domke, 1852, T. XXXVI. (To 5 places) Lambert, 1798, T. XXV. ; Maskelyne (Requisite Tables), 1802, T. XVII. ; BowDiTCH, 1802, T. XIV. ; Moore, 1814, T. XXIV. ; Wallace, 1815 [T. III.] ; Gregory, &c., 1843, T. X. Midtiples o/ suies.— Schulze, 1778 [T. VI.] ; Lambert, 1798, T. XXV. Versed sines (alone).— (To 7 jjlaces) Sir J. Moore, 1681 [T. IV.] ; [Sir J. Moore, 1681, Versed sines'] ; Dodson, 1747, T. XXVI. ; Douglas, 1809, [T. IV.] ; Farley, 1856 [T. I.]. (To 6 places) Maskelyne (Requisite Tables, Appendix), 1802, T. II. ; Mackay, 1810, T. XLI. ; Lax, 1821, T. XVII. (and coversed &c. sines) ; Riddle, 1824, T. XXVIIL ; IS'orie, 1836, T. XXXVI.; Rumker, 1844, T. III. ; Inman, 1871 [T. VIIL] and [T. IX.]. Sines &c. expressed in radicals.- — Lambert, 1798, T. XIX. ; Ursinus, 1827 [T. III.] ; Vega, 1797, Appendix. Miscellaneous. — Sin" -U, Andrew, 1805, T. XIII ; sin" x and tan- x, Pasuuich, 1817, T. II. ; suversed, coversed, sucoversed sines, Lax, 1821, T. XVII. ; I sin .r, Stanseury, 1822, T. Y; sexagesimal cosecants and cotan- gents, Beverley (1833 ?), T. VI. (pp. 232 &c.) ; sexagesimal sines. Id. T, XV.; sin lHt;LSSE'sVEGAT.IV.1840;sin^~, [Sheepshanks, 1844] [T. VL] ; I sin X expressed sexagesimally, Gordon, 1849, T. XVIII. ; see also Schlo- milch [1865 ?]. Note. — A list of tables in which both natural and logai-ithmic functions are given side by side in the same table is added at the end of § 3, art. 15. Art. 11. Lengtlis of Circular Arcs. Tables of the lengths (or longitudes) of circular arcs are very frequently given in collections of logarithmic and other tables ; but we have seen none of sufficient extent to be published separately. Angles are measured either by degrees, minutes, &c., or by the ratio which the corresponding arc bears 48 REPORT— 1873. to the unit arc, or arc equal in length to radius. The latter method is usually described in English text-books under the title " Circular Measure ; " so that in the descriptions in § 4 we have spoken indifferently of the length of the arc of 0;°, the longitude of >t°, or the circular measure of .r°. The tables of circular arcs usually give the circular measure of 1°, 2°,. . up to 90°, 180°, or sometimes 360°, of 1', 2', . . . . GO', of 1", 2", .... GO", and very often of 1'", 2'",. . . .60'" also. By means of such a table any number of degrees, minutes, &c. cau be readily expressed in circular measure. The following is a detailed list of the lengths of circular arcs contained in ■works described in § 4 : — (To 44 places) Hobert and Ideler, 1799, G (centesimal division), (To 27 i^laces) Academie de Prusse, 177G [T. II.] ; Schulze, 1778 [T. VII.]; Lamdert, 1798, T. XXIII. (To 25 places) Callet, 1853 [T. V.] (sexagesimal and centesimal). (To 15 places) Hantschl, 1827, T. X. (To 12 places) Schmidt, 1821 [T. IV.]; Muller, 1844 [T. IV.]. (To 11 places) Vega, 1794, T. II.; Hulsse's Vega, 1840, T. II.; Kohler, 1848 [T. V.]. (To 10 places) Shortrede, 1849, T. III. ; Bruhxs, 1870. (To 8 places) Vega, 1797, T. III. ; Pearson, 1824 [T. III.]. (To 7 places) Dodson, 1747, T. XXV. ; Ursinus, 1827 [T. III.] ; Grtt- soN, 1832, T. VI.; Trotter, 1841 [T. VII.] ; Shortrede (tables), 1844, T. XXXVIII.; Warnstorff's ScntrjiAcuER, 1845 [T. II.]; Willicu, 1853, T. D ; Bremiker's Vega, 1857, T. II. ; Hutton, 1858, T. XI. ; Dupuis, 1868, T. IX. ; Peters, 1871 [T. III.] (To 6 places) Bremiker, 1852, T. II. (To 5 places) Wackerbartu, 1867, T. IV. See also Vega, 1800, T. II.; Byrne, 1819 [T. II.]; *ScnLoMiLCH [1865?]. Art. 12. Tables for the e.vpresslon of hours, minutes, S,-e. as decimals of a day, and for the conversion of time into sjicice, and vice versa. The largest table we have seen to convert hours, minutes, &c. into decimals of a day is Houel, 1866. Tables of this kind are not numerous. Three hundred and sixty degrees of space or arc are equivalent to twenty- four hours of time ; so that 1" corresponds to 15°, 1'" to 15', and 1" to ] 5" ; 1" is therefore 4 thirds of time = 4' ; 36' ==2'" 24^ &c. Small tables to convert space (arc, or longitude) into time arc not unfrequcntly given in collections (generally nautical) of tables. A complete tabic of the kind gives the numbers of hours and minutes corresponding to 1*^, 2°, . . . . 360° ; and the same figures also denote the number of minutes and seconds, and seconds and thirds (of time) corresponding to 1', 2', 360', or 1", 2", 360" respectively. In this Report •■, ", % &c. arc used to denote hours, minutes, seconds, and thirds (of time), and °, ', ", "' for degrees, minutes, &e. of space — a distinction which it is often convenient to adopt. Littrow, 1837. T. I.-IV. (5 pp.) are small tables for the conversion of arc into time &c. All the other tables, which occupy more than nine tenths of the tract, are astronomical. Hoxiel, 1866 (Time Tables), T. II. To convert hours, minutes, and seconds into the decimal of a day (pp. 15). Any number of hours, minutes, and seconds (and fractions of a second, as proportional parts arc added) ox MATHEMATICAL TABLES, '49 can bo readily expressed as a decimal (to seven places) of a day, and vice versa by means of it. Tlic following are tables described in § -1 : — Tables for the conversion of Time into Space, and vice versa. — -Cross- well, 1791, T. XIII.; BowDixcir, 1802, T. XII.; Ilios, 1809, T. XYI.- I'u^iiiGAN, 1821, T. XIII. ; Stansbuky, 1822, T. I. ; Pearson, 1824 [T. I.] ; Galbraith, 1827, T. XII. (Introd.); Warnstorff's Schumacher, 1845 TT II- KiinLER, 1848 [T. I.] ; Gordon, 1849, T. XI. ; Domke, 1852, T. XLVll. and XLVIII. ; Bremiker, 1852, T. II. ; Thomson, 1852, T. I. ; Bremiker's Veg\, 1857, T. III. ; HouEL, 1858, T. I. ; Peters, 1871 [T. II.]. Tables to express Degrees, Minutes, S,-c. as decimals of a rir/M angle, or Hours, Minnies cj-c. as decimals of a day, and vice versa, 6,-c. — IIobert and Ideler, 1799, C. I.-IV., D. I.-III., E. I.-IV., F. ; Galbraith, 1 827, T. XI, (Introd.); Hantschl, 1827, T. XII.; Beverle\- (1833?), T. VI, (p. 127) ; KiiHLER, 1848, T. IX. ; Peters, 1871 [T, I.], Art. 13. Tables of {Briygian) Logarithms of Numbers. The facts relating to the invention of Briggian (or decimal) logarithms arc as follows: — In 1614 Napier published his 'Canon ilirificus ' (see § 3, art. 17), which contained the first announcement of the invention of logarithms, and also a table of logarithmic sines, calculated so as to be very similar to what are now called hyperbolic logarithms. HenrtBriggs, then Professor of Geo- metry at Gresham College, London, and afterwards Savilian Professor of Geo- metry at Oxford, admired this work so mucli that he resolved to visit Napier. " Naper, lord of Markinston, hath set my head and hands at work with his new and admirable logarithms. I hope to see him this summer, if it please God ; for I never saw a book which pleased me better, and made me more wonder," This he says in a letter to Usher (Usher's ' Letters,' p. 3G, accord- ing to Ward). Briggs accordingly visited Napier, and stayed with him a whole month (in 1615), He brought with him some calculations he had made, and suggested to Napier the advantages that would result from the choice of 10 as a base, having publicly explained them previously in his lectures at Gresham College, and written to Napier on tlie subject. Napier said that he had already thought of the change, and pointed out a slight improvement, viz. that the characteristics of numbers greater than unity should be posi- tive,^ and not negative, as Briggs suggested. Briggs visited Napier again in 1016, and shoAved him the work he had accompHshcd, and, as he himself says, would have gladly paid a third visit in 1617, had Napier's life been spared (he died April 4, 1617). The work alluded to is Briggs's ' Logarithmorum Chilias Prima,' which was published (privately, we believe) in 1617, after Napier's death, as in the short preface he states that why his logarithms are dif- ferent from those introduced by Napier " spcrandum, ejus librum posthumum, abunde nobis propediem satisfacturum." The liber postlmmus was Napier's ' Constructio,' which appeared in 1619, edited by his son (see § 3, art. 17). Briggs continued to labour assiduously, and in 1624 published his 'Arith- nietica Logarithmica,' giving the logarithms of the numbers from 1 to 20,000, and from 90,000 to 100,000 (and in some copies to 101,000), to 14 places. To the above facts we must add that Napier made a remark, both in Wriglit's translation of the ' Uescriptio ' (1010) and in the ' llabdologia' (1617), to (lie effect that lie intended in a second edition to make an alteration equivalent to taking the logarithm of 10 equal to unity. We have thought it proper to give the circumstances attending the inveu- 1873. ]5 50 REPORT — 1873. tiou of Briggian logarithms in the ahove detail, as there seems every proba- bility that the relations of Napier and Briggs may become a subject of con- troversy among those who have never taken the trouble to examine the original sources of information, Hutton, in his ' History of Logarithms ' (prefixed to all the early editions of his logarithmic tables, and also printed in vol. i. pp. 306-340 of his ' Tracts,' 1812), has unfortunately interpreted all Briggs's statements with regard to the invention of decimal logarithms in a manner clearly contrary to their true meaning, and unfair to Napier. In reference to the remark in Briggs's preface to the ' Chilias,' that it is to he Jiojjed that the posthumous work will explain why the logarithms are different from Napier's, Hutton proceeds : — " And as Napier, after communication had with Briggs on the subject of altering the scale of logarithms, had given notice, both in Wright's translation and in his own ' Rabdologia,' printed in 161 7, of his intention to alter the scale (though it appears very plainly that he never intended to compute any more), without making any mention of the share Avhich Briggs had in the alteration, this gentleman modestly gave the above hint. But not finding any regard paid to it in the said posthumous work, published by Lord Napier's son in 1619, where the alteration is again adverted to, but still without any mention of Briggs, this gentleman thought he could not do less than state the grounds of that alteration himself. " Thus, upon the whole matter, it seems evident that Briggs, whether he had thought of this improvement in the construction of logarithms, of making 1 the logarithm of the ratio 10 to 1 before Lord Napier or not (which is a secret that could be known only to Napier himself), was the first person who com- municated the idea of such an improvement to tlie world ; and that he did this in his lectures to his auditors at Gresham College in the j'ear 1615, very soon after his perusal of Napier's < Canon Mirificus Logarithmorum ' in the year 1614. He also mentioned it to Napier, both by letter in the same year and on his first visit to him in Scotland in the summer of the year 1616, when Napier ap- proved the idea, and said it had already occurred to himself, and that he had determined to adopt it. It would therefore have been more candid in Lord Napier to have told the world, in the second edition of this book, that Mr. Briggs had mentioned this improvement to him, and that he had thereby been confirmed in the resolution he had already talcen, before Mr. Briggs's com- munication with him, to adopt it in that his second edition, as being better fitted to the decimal notation of arithmetic which was in general use. Such a declaration would have been but an act of justice to Mr. Briggs ; and the not having made it cannot but incline its to suspect that Lord Napier was desirous that the world should ascribe to him alone the merit of this very useful improvement of the logarithms, as well as that of having originally in- vented them ; though, if the having first communicated an invention to the world be sufficient to entitle a man to the honour of having first invented it, Mr. Briggs had the better title to be called the first inventor of this happy improvement of logarithms." The above comments of Hutton's are all the more unfortunate because they occur in a history that is generally accurate and truthful. It is needless to say that, the facts being as above narrated, there is not the smallest ground for imputing iinfairness to Napier ; but Hutton seems to have some- how become possessed of such an idea and read all the facts by the light of it. On the other hand, however, some of the accounts are scarcely fair to Briggs. Mr, Mark Napier, in his ' Memoirs of John Napier,' has successfully refuted Hutton ; but he has fallen into the opposite extreme of extravagantly eulogizing Napier at the expense of Briggs, whom he reduces to the level of a mere ON MATHEMATICAL TABLES. 51 computer ; and in these terms Mr. Sang has also recently spoken of the latter. Mr. Napier attributes Huttou's assertions to national jealousy (!) ; and it will be a matter of regret if any other writers should follow his example in at- tempting to glorify Napier by depreciating Briggs. The words of the latter, in the 1031 translation (and amplification, see below) of his ' Arithmetica ' of 1G24, are : — " These numbers were first invented by the most excellent John Neper, Baron of Marchiston ; and the same were transformed, and the founda- tion and use of them illustrated with his approbation [ex ejusdem sententia] by Henry Briggs." No doubt the invention of decimal logarithms occurred to both Napier and Briggs independently ; but the latter not only first an- nounced the advantage of the change, but actually completed tables of the new logarithms. Thus, as regards the idea of the change, Napier and Briggs divide the honour equally ; while, on the principle that " great points belong to those who make great points of them," nearly all belongs to Briggs. On the subject of Briggs and the invention of logarithms, see the careful and impartial life of Briggs in Ward's ' Lives of the Professors of Gresham College,' London, 1740, pp. 120-129, and also ' Vitfe quorundam eruditis- simorum et illustrium virorum' &c., scriptore Thoma Smitho, Londini, 1707 (Vita Henrici Briggii), as well as ' Memoirs of John Napier of Merchiston,' by Mr. Mark Napier, Edinburgh, 1839, and the same author's ' Naperi libri qui supersunt' (see § 3, art. 17). See also Hutton's account (reference given above) and Phil. Mag., October and December (Supplementary No.) 1872, and May 1873. It is proper to add that the date we have given for Briggs's first visit to Napier, viz. 161.5, is diflferent from that assumed by other writers, viz. 1616; ■\ve have, however, little doubt that the former is correct, as it in all respects {fgrees with the facts. The reason that Ward, Hutton, &c. assign Briggs's first visit to 1616, and the publication of the ' Chilias' to 1618, is, no doubt, due to the fact that they supposed Napier to have died in 1618 ; but Mr. Mark Napier has shown that the true date is 1617 ; and this brings all the facts into agreement (see Phil. Mag. December 1872, Supp.). Like Napier, Briggs was not very particular about the spelling of his name. In Wright's translation it appears as Brigs on the titlepage, Brigges on the first page of the preface, and Briggs in the eulogistic verses. Although we have spoken of logarithms to the base 10 &c., we needscarcely observe that, although exponents and even fractional exponents were in a sort of way introduced by Stevinus, neither Napier nor Briggs, nor any one tiU long after, had any idea of connecting logarithms with exponents. To return to the original calculation of the logarithms of numbers. Briggs, ns has been stated, published the logarithms of the numbers from 1 to 20,000 and from 90,000 to 100,000 to fourteen places, in his ' Arithmetica.' There was thus left a gap from 20,000 to 90,000, which was filled up by Adrian Vlacq (although Briggs had in the mean time nearly completed the necessary calculations ; see Phil. Mag. May 1873), who published at Gouda, in li)28, a table containing the logarithms of the numbers from unity to 100,000 to 10 places of decimals. Having calculated 70,000 logarithms and copied only 30,000, Vlacq would have been quite entitled to have called his a new work. He designates it, however, only a second edition of Briggs, the title running, "Arithmetica logarithmica sive logarithmorum chiliadca centum, pro numeris naturali serie crescentibus ab TJnitate ad 100000 Editio socuuda aucta per Adrianum Vlacq, Goudanum Gouda), excudebat Petrus Eammasenius. 1628." This table of Vlacq's was published, with an English explanation prefixed, in London in 1631, under the title, " Logarith- micall Arithmctike, or Tables of Logaritbmes for absolute numbers, from au £ 2 5,2 UEPORT — 1873. unite to 100000 London, printed by George Miller, 1G31" (full titles arc given in § 5). Speaking of Briggs's ' Arithmetiea Logarithmiea' of 1G24, De Morgan, in his article on Tables in the ' English Cyclopedia,' says : — " After his [Briggs's] death, in 1(331, a reprint was, it is said, made by one George Miller ; the Latin title and explanatory parts were replaced by English ones — •' Logarith- micall Arithmctike ' &c. We much doubt the reprint of the tables, and think that they were Briggs's own tables, with an English explanation prefixed in place of the Latin one. Wilson (in his ' History of Navigation,' prefixed to the third edition of llobertson) says that some copies of Vlacq, of 1628, were purchased by our booksellers, and published at London with an English ex- planation lu'cmised, dated 1031 . Mr. Babbage (to whose large and rare col- lection of tables we were much indebted in the original article) has one of these copies ; and the English explanation and title is the same as that which was in the same 3'ear attached to the asserted reprint of Briggs. We have no doubt that Briggs and Vlacq were served exactly in tlie same manner." On referring to Robertson (fourth edition, p. xvi), there is found to be no further information than that contained in the above extract. That De Morgan's suggestion is quite correct, and that Miller's and Vlacq 's tables are both printed from the same types, we have assured ourselves by a most careful comparison, which leaves no doubt whatever that the two works are printed from the same type throughout. We are thus enabled to state that the same errata-list suffices for both ; and this is important, as Vlacq (1G28, or 1031) is still the most convenient and most used ten-figure table in ex- istence. Briggs's friends were annoyed at Vlacq's publication ; but it must be borne in mind that their objections have reference, not so much to the table (which is the only thing of practical importance now) as to the prefixed tri- gonometry, which Vlacq curtailed in his '•' second edition." George Miller also published some copies of the original 'Arithmetica' of 1624, with the same title- page and introduction as were prefixed to the copies of Vlacq of 1628 ; and this was distinctly wrong, as the titlepage does not in this case describe the con- tents correctly. It thus appears that Briggs's table was published in 1624, and Vlacu's in 1628 — that copies of the tabular portions of both these works were obtained by George Miller, and published by him in 1631, with the same (English) title- page and introduction, Avhich, though correctly describing the contents of Vlacq, is quite inaj^propriate for Briggs. This has led to a verj^ great amoinit of confusion, which has been greatly increased by the fact that on the title- pages Briggs's and Neper's names occur, and that Vlacq only called his work a second edition. It is in consequence exceedingly common to see Vlacq's work assigned to Briggs or Neper ; and it is almost invariably ascribed to one or other of the latter in the catalogues of libraries. Vlacq's 'Aritlimetica' of 1628 was also published with the same date, with a French title (" Arithmetique Logarithmctique " &c.) and introduction. Vlacq modestly describes his share of the calculation in the words : — " La description est traduit du Latin en Francois, la premiere Table augmentee, et la secoude composee par Adriaen Vlacq." Miller's (1631) copies of Vlacq are not so rare as the extract from De Morgan might imply. We have seen five of them, and only three or four of the original (1628) works (including both Latin and French). In 1031 Vlacq published his ' Trigonometria Artificialis' (§4). This work contains, among other tables, the logarithms of the numbers from unity to 20,000, printed also (with the exception of the last sheet, referred to fur- ther on) from the same type. ON MATHJiMATlCAL TABLES. 53 No further calculation of logarithms of numhers took pluco till the end of the last century, when the great French manuscript tables (the ' Tables Du Cadastre' — see description of them below) were comiKited under the direction of Prony. These, as is well known, hare never been published. In 1794 Vega published his ' Thesaurus Logarithmorum Completus/ which contains a complete ten-figure table from 1000 to 101,000. It was not, how- ever, the result of a fresh calculation, but was cojued from Ylacq, after ex- amination and correction of many errors (see Vega's ' Thesaurus,' § 4). In 1871 Mr. Sang published his seven-figure table of logarithms of numbers to 200,000, the second half of which was obtained by a new calculation. It is thus seen that, with the exception of the Tables du Cadastre, and the second half of Mr. Sang's table, every one of the hundreds of the tables that have appeared has been copied from Eriggs or Vlacq ; and considering the enor- mous number of calculations in which logarithms have been employed, and the vast saving they have eftected of labour, it must be admitted that (apart from the fact that the great tables of Ertggs and Ylacq remain still unsuperseded) great historical interest attaches to the original com- putation. Ylacq's ten-figure table contains about 300 errors (leaving out of consi- deration errors atfecting only the last figure by a unit). The greater number of these were found cither by Vega, or by Lefort from comparison with the Tables dtt Cadastre : complete references and a small subsidiary list are given in the ' Monthly Notices of the Royal Astronomical Society ' for May and June 1872. While speaking of ten-figure logarithms, we may men- tion PiNETo's table described below ; but Vlacq (1628 or 1631) and Vega (1794) are far preferable : they are unfortunately so rare, however, that not many besides those Avho have access to a good library can make use of them, and, except to a few, the employment of ten-figure logarithms in their most convenient form is denied : we much prefer Vlacq to Vega for use, as the arrangement is more convenient. To return to the history of logarithmic tables to a less number of figures. In 1625 Wingatc published at Paris his ' Arithmetique Logarithmetique,' con- taining seven-figure logarithms to 1000, and logarithmic sines and tangents from GuNTER (see De Morgan ; the full title of the Gouda edition of Wingate (1628) is given by Hogg, p. 408), thus introducing Briggian logarithms into France ; and in 1626 appeared both Henrion's 'Traicte' (§ 4) at Paris, con- taining 20,000 logarithms from Briggs and Gunters logarithmic sines and tangents, and De Decker's ' Nieuwe Telkonst ' (§ 4) at Gouda, giving also logarithms from Briggs and Gunter; then Vlacq began to calculate logarithms, and brought them in 1628 to the state in which they now are. There is a table' of logarithms in Norwood's 'Trigonometric' (1031) ; and in 1633 appeared Eoe's table (§ 4), in which the first four figures of the logarithm are printed at the top of the column. This was an advance halfway to the modern arrange- ment, which was introduced in its present form in John Newton's eight-figure table (1658). On Fatohaber, 1631, and Oughtred, 1657, see § 4. Tables of seven- and five-figure logarithms are too numerovis to notice here separately. The cliief line of descent is Briggs, Vlacq, IIoe, perhaps Newton, the editions of Sherwin, Gardiner; and then both Hutton and Callet bring down the succession to the present day. A very fair account of several logarithmic tables is given by Hogg in section iv. " Elementar- Gcometrie " (B) of his ' Ilandbuch,' who added to the books described in this part of his bibliography a description of the contents. But the reader must be warned against trusting his accounts, except where he is clearly describing 54 BEPOiiT — 1873. works he has seen. Of seven-figure tables we have found Babbage as con- venient as any, and it is nearly free from error ; Callet and Htjtton are also much used ; Shortrede and Sang are both conspicuous for giving the multii^les of the differences instead of proportional parts ; the latter work also extends to 200,000 instead of 100,000 as usual. Of five-figure tables De Morgan's (Useful-Knowledge Society) tables are considered the best, and arc practically free from error. We cannot, however, here particularize the advantages of the different tables, which must be gathered from their full descriptions. Some of them have, of course, been merely included on account of their his- torical value. We may here mention that the subject of errors in these tables will be considered in a subsequent Report. Vega (p. iii of the Introduction to the ' Thesaurus,' 1794) says that Ylacq's ' Arithmetica ' (1628) and ' Trigonometria ' (1633) were printed at Pekin in 1721, under the title " Magnus Canon Logarithmorum, tum pro sinibus ac tangeutibus ad singula dena secunda, tum pro numeris absolutis ab unitate ad 100,000. Typis sinensibus in Aula Pckinonsi, jussu Impcratoris cxcusus, 1721 " (three volumes folio, on Chinese paper), and that a copy had been offered him for sale two years previously (1792). Montucla (' Histoire,' vol. iii. p. 358) says, the name of the Emperor in question was Kang-hi. Rogg also (p. 408) confirms Yega, extracting the title from Brunet's * Manuel du Libraire.' In the preface to his tables (1849) Mr. Filipowski concludes by a sneering remark on the Chinese, saying that Mr. Babbage proved, " as had long been suspected, from what source those original inventors had derived their logai'ithms ; " and we have noticed this tendency to ridicule the Chinese in this matter as detected plagiarists in others. In point of fact there is no more plagiarism than when Babbage or Callet publishes a table of logaritlims with- out the name of Ylacq on the titlepage. The first publication in China, we infer from Rogg, merely professed to be a reprint of Vlacq ; and if logarithms came into general use, it is natural that they would be published, as with us, without the original calculator's name. The fault is with those who form preconceived opinions on subjects they have not investigated. A Turkish table of logarithms is described in § 4. A small table of logarithms to base 2 is noticed below, imder Montferrier, 1840. We may mention a little book, ' Instruction clcmentaire et pratique sur I'usage des Tables de Logarithmes,' by Prony (Paris, 1834, 12mo), which explains the manner of using of tables of logarithms &c., adapted to Callet, In many seven-figure tables of logarithms of numbers the values of S and T are given at the top of each page, with Y, the variation of each, for the purpose of deducing log sines and tangents. S and T are the values of log — —, and loo- - — L for the number of seconds denoted by certain numbers (sometimes ° X only the first, sometimes every tenth) in the number-column on each page. Thus, in Callet, 1853, on the page of Avhich the first number is 67200, ^ , sin 6720" _, ^ , tan 6720" ... ,, ,„ ,, ... . S=log ,,„ .,-, and T=log — ^.-o,-, , while the Ys are the variations of ° b/20 5720 each for 10". To find then, say, log sin 1° 52' 12"-7, or log sin 6732"-7, we have 8=4-6854980, and log 6732-7 = 3-8281893, whence, by addition, we have 8-5136873; but Y for^lO" is -2-29 ; whence the variation for 12"-7 is —3, and the log sine required is 8-5136870. Tables of S and T are fre- quently called, after their inventor, Delambre's tables. It is only since the completion of this Report, and therefore too late to ON .AlAXHJiMATlCAL TVBLISS. 55 make any use of it, that we have received from Professor Bicrcns do Haan a copy of a very valuable tract, ' Jets over Logarithmentafels,' extracted from, the ' Yerslagcii en Mededeelingen der Kouinklijke Akademie van Weten- schappen, Afdeeling Natuurkuudc,' Deel xiv. Amsterdam, 1862, 8vo (pp. SO), which contains by far the most complete list of authors or editors of loga- rithmic tables of all kinds, with the dates and places of publication (from 1014 to 1862), that we have seen, and must be nearly perfect. Some remarks arc made on those of them that de Haan has examined himself ; and there is ap- pended a valuable index of reference to papers on logarithms that have ap- peared in any Journal or Society's Proceedings. We may also refer to the paper of Gernerth's noticed under Rheticus, 1596 (§ 3, art. 10), Avhich contains a number of last-figure errors in logarith- mic ancl other tables. Gernerth was desirous of ascertaining the care bestowed on the editing of mathematical tables, and considering that it was best measured by the accuracy of the last figure, he confined himself to the exa- mination of this point alone (except in the cases of Rheticus and Pitiscus, where tlie first seven or eight figures were included) , and detected very many errors. He altogetlicr examined tables by eighteen authors ; but generally, where the errors were numerous, he has given only five per cent, of those that he found. Also, as this sheet is passing through the press, we add references to two papers in the ' Monthly Notices of the Royal Astronomical Society ' for April and May, 1873, " On the Progress to accuracy of Logarithmic Tables," and " On Logarithmic Tables ;" in the former of which the number of Vlacq's original errors that were reproduced in succeeding works is discussed, while the latter contains remarks on logarithmic tables both of numbers and trigo- nometrical fanctions. An abstract of the first appears also in the ' Journal of the Institute of Actuaries,' vol. xvii. pp. 352-354. Briggs, 1617. Logarithms of numbers from unity to 1000 to 14 places of decimals. This was the first table of Rriggian logarithms calculated or published. Neither author's name nor date nor place appears on the title- page of the work, which is a mere tract of 16 pp. (at all events in the Brit. Mus. copy) ; but that it was published by Briggs in 1617 is beyond doubt (see ' Phil. Mag.' he. cit. below). The preface concludes with the motto "In tenui; sed non tenuis fructusve laborve." On the work see the introductory remarks to tliis Article, aud also 'Phil. Mag.' December (Supplementary No.) 1872. It is stated by Hutton and all the other writers to be an 8-place table ; but it really is as described above. One reason for the universal error is that copies are so extremely rare that we have only been able to see one *, viz. that in the British Museum, in the catalogue of which it is entered under Logarithms, and marked as of [1695 ?]. The book is not in the printed Bodleian Catalogue. It is peculiarly interesting as being the first publication of decimal logarithms. Nearly all the descriptions and bibliographies will be found very erroneous, several confounding it with "Wright's translation of Napier's ' Canon ' (see § 3, art. 17). Briggs, 1624. Logarithms of numbers from 1 to 20,000, and from 90,000 to 100,000, to 14 places, with interscript differences. The characteristics to the logarithms are given ; aud this has led to the table being sometimes erro- neously described as being to 15 places. The table occupies 300 pages. * We think we remember to have met with another among the Birch lilSS. in tho British Miisewm. 56 Hi; i' OKI— 1873. Several lists of errata in this work have been given — viz. by Vlacq in his ' Arithinctica,' by Shekwin in his tables, by Vuga (folio, 1794), by Ly.FOET (' Annalcs de rObscrvatoire dc Paris'). The introduction occujncs 88 pages, and is in Latin. In some copies there is an additional chiliad added, so that the range of the second portion of the table is from 90,000 to 101,000 ; and there is a table of square roots of numbers up to 200, to 10 places, occupying the last two pages : these copies are very rare. There is one in the Library of Trinity College, Cambridge, with the following note in it by Dr. Brinkley : — " This is a very scarce copy, having an addition very rarely to be met with. Vide Ilutton's preface to his ' Logarithms,' p. 33, Avho could never find a copy with the addition." Mr. Merrifield has also one of the.se copies. On this v.'ork sec the introductory remarks to this Article. Tables du Cadastre. On the proposition of Carnot, Prieur, and Brunct, the French Government decided in 1784 that new tables of sines, tangents, &c., and their logarithms, should be calculated in relation to the centesimal division of the quadrant. Prony was charged with the direction of the work, and was expressly required " non sculement a composer des Tables qui ne lais- sasscnt ricn a dcsirer quant a I'exactitudc, maisa en faire le monument de calcul le plus vaste et le plus imposant qui eiit jamais etc execute ou meme congu," — - an order faithfully carried out. Prony divided the calculators &c. into three sections : the first consisted of five or six mathematicians (including Lcgendre), who were engaged in the purely analytical work, or the calculation of the fundamental numbers ; the second section consisted of seven or eight calcu- lators possessing some mathematical knowledge ; and the third comprised the ordinary computers, 70 or 80 in number. The woi'k, which was done wholly in duplicate, and independently by the two divisions of computers, occupied two years. As a consequence of the double calculation, there arc two manuscripts in existence, one of which has been long deposited in the Archives of the Obser- A'atory ; the other, though supposed to be in the Archives of the Bureau des Longitudes, was in reality in the possession of Prony's heirs, by whom it was presented to the Library of the Institute in 1858. Each of the two manuscripts consists essentially of 17 large folio volumes, the contents being as follows : — Logarithms of numbers to 200,000 8 vols. Natural sines 1 vol. Logarithms of the ratios of arcs to sines from O^-OOOOO to 1 , . , 0''-05000, and log sines throughout the quadrant . . J ' Logarithms of the ratios of arcs to tangents from 0«-00000 1 to O''0-5000, and log tangents throughout the 14 „ quadrant J It would take too much space to state the intervals &c. in detail. Speaking generally, the trigonometrical functions are given for every hundred-thousandth of the quadrant (10" centesimal or 3"-24 sexagesimal). The tables were all calculated to 14 places, with the intention of publishing only 12 ; but M. Le- fort, who has recently examined them, states that the twelfth figure may be in error by as much as 0-8 of a unit in this place, though a little additional care would have rendered it more accurate. The Institute copy has also a table of the first 500 multiples of certain sines and cosines ; and the Observatory copies have an introduction containing, among several other subsidiary tables, the first OS MAXilEMATiCAL TAULES. 57 20 powers of r^ to 28 figures. It may be mentioned that the logarithms of 10,000 primes were calcuhitod to 19 places, and tlie natural sines for every minute (centesimal) to 22 places. This account of the ' Tables du Cadastre ' has been abridged from a memoir by M. Lefort, in t. iv. (pp. [123]-[150]) of tlie ' Annales de I'Observatoire do Paris ' (1858), where an explanation of the methods of calculation, with the formulas &c., is given. The printing of the table of natural sines was once begun. M. Lefort says that he has seen six copies, all incomplete, although including the last page. De Morgan also men- tions that he had seen some of the proofs. Babbage compared his table with the ' Tables du Cadastre ;' and M. Lefort has given, by means of them, most important lists of errors in Vlacq and Beiggs ; but these are almost the only uses that have been made of tables the calculation of which required so great an expenditure of time and money. " In 1820," says De Morgan, " a dis- tinguished member of the Board of Longitude, London, was instructed by our Uovernment to propose to the Board of Longitude of Paris to print an abridg- ment of these tables, at the joint expense of the two countries. £5000 was named as the sum which our Government was willing to advance for this purpose ; but the proposal was declined " (Peuny Cyclopaedia, Article " Prony "). The value of the logarithms of numbers is now materially lessened by Mr. Sang's seven-figure table from 20,000 to 200,000 (see Saxg, 1871, in this Article). Hogg (p. 241) gives the title " Notice sur Ics grandes tables logarithm, et trigonom. calculees au Bureau du Cadastre," Paris, an IX. (=1801\ and on the subject gives a reference to Bcnzeuberg's ' Angewandte Geora.' iii. p. 557. Hill, 1799. Pive-figure logaritlims from 1 to 100 and from 1000 to 10,000, printed at full length, and with characteristics- — no difiercnces (pp. 23-^8). The author was an accountant; and the table was intended for commercial purposes, its use in which is explained in the book, Reishammer, 1800. These are commercial logarithms, intended for merchants &c. When the number is less than unity, the logarithm of its reciprocal (which the author calls the Jor/aritJime nef/afif)is tabulated; if greater than unit}', its own logarithm (lor/ariihme 2^ositif). The first table (which only occupies 2 pages) gives the locjarWimes neijatifs of the frac- tions from ^i-y to 1, at intervals of -j-J^y ^o ^ places (the characteristics are given, and not separated from the other figures). This is followed by the principal table, which occupies 117 pages. On the first page are given the lo(jarilhmcs nc</atlfs of 128 fractions, viz. of all proper fractions whose deno- minators arc 60, 48, 40, or 32, arranged in order thus : — ^'^, -L, J^, ^L, J-y , • • • -as' ou' ■§&• '^^^ r^st of ^^c logarithms are j^ositifs ; and the argu- ments proceed from 1 to 111, with the 128 fractions just described inter- mediate to each integer. Thus we have l^^j, l-J^, &c., 2-j,\, 2-Jyr, &c., as arguments. The arguments then proceed from 111 to 207 at intervals of •.,15^, from 207 to 327 at intervals of ^, thence to 807 at intervals of §, and from 808 to 10,400 at intervals of unity, — all to 5 places. The characteristics are given throughout. A page of proportional parts is added. There arc besides several small tables, to facilitate the calculations, only one of which requires notice. It gives on a folding sheet the 128 fractions previously described, expressed as fractious with denominators 100 and 10, and also (when the numerator is integral) expressed as fractions with de- nominators GO, 48, 40, 32, 30, 24, 20, IG, 15, 12, 8, 6, 5, 4, 3, 2. Thus -^ = 10y''^-^100, and=l77'.j-rl0; as it cannot be expressed in lower terms 58 BJiroiiT — 1873. (or higher terms with any of the above denominators); it only appears as 5 ia the 48 column. In reference to a work by Girtanuer (179-i) which we have not seen, but which appears to be very similar to the present, De Morgan justl}'^ remarks, " But it will not do : Mohammed must go to the mountain. When coin- age, weights, and measures are decimalized, the use of logarithms will foUow as a matter of course. It is useless trying to bring logarithms to ordinary fractions." Rees's Cyclopaedia (Art. "Logarithms," vol. xxi.), 1819. Seven-ligure logarithms of numbers from 1000 to 10,000, with differences ; arranged in groups of five. Schron, 1838. Three-figure logarithms to 1400, and five-figure logarithms to 14,000, with corresponding degrees, minutes, &c., and proportional parts. Of the 20 pages 4 are occupied with explanations &c. The arrangement is as in seven-figure tables. Steinberger, 1840. The titlepage is misleading ; the logarithms do not extend from 1 to 1,000,000, but only from 1 to 10,000. The only pretext for giving 1,000,000 as the limit is that, of course, two additional figures may be obtained by interpolation ; but on this principle ordinary seven-figure tables should be described as extending, not to 100,000, but to 10,000,000. The first five figures of the logarithms are printed in larger type than, and separated by an interval from, the last two, so that the table may be more conveniently used either as a five- or seven-figure table ; the change of figure is denoted by an asterisk prefixed to all the logarithms affected. The figures, though large, are not clear, the appearance of the page being dazzling ; the 6's and 9's also seem too large for the other figures, and after all are not very readily distinguishable from the O's. No differences or proportional parts are given. Montferrier's Mathematical Dictionary, 1840. Under the Article "Logarithmes," in t. iii. (the supplementary volume) is given a table of four-- figure logarithms of numbers from 1000 to 10,000 (pp. 271-279). In the same volume (p. 2.52, facing letter L) is given a table of logarithms of the numbers from 1 to 420 to base 2 to five places, the only table of th& kind wo have met with. Babbage, 1841. Seven-figure logarithms of numbers from 1 to 1200 and from 10,000 to 108,000, with diflferences and proportional parts (the last 8000 are given to 8 places). Degrees, minutes, and seconds are also added, but they arc divided from the numbers by a thick black line, and are printed in somewhat smaller type, so that they are not so obtrusive as in Callet and others. On the last page there are a few constants. Great pains were taken Avith the preparation of this table (which is stereo- type), with the view of ensuring the maximum of clearness &c., and -wdth success. The change of figure in the middle of the block is marked by a change in type in the fourth figure in all the logarithms affected. This is, we think, with the exception of the asterisk, the best method that has been used. The chief defect, or rather point capable of improvement, is that tho three leading figures in the logarithms are not separated, or in any way dis- tinguished, from tho rest of the figures in the block, as is the case in Callet and others. The table was read (wholly or partially) altogether nine times with different tables of logarithms (four of those readings were made after the stereotyping), and is no doubt all but perfectly correct. One feature of this table is that every last figure that has been increased is marked with a dot subscript. ON MATIIliMATlCAL TAULJiS. 59 We know of only two errors : viz., in log 52943 the last figure should bo 5 instead of 6 ; and in log 102467 the last two figures should be 02 instead of 92. The occurrence of the former of these errors is very remarkable, as the logarithm is correct in Vega (folio, 1794), with which the table was read twice (sec Sang, 'Athenaeum,' June 8, 1872, and Glaisher, 'AthensBum,' June 15, 1872, or ' Journal of the Institute of Actuaries,' July 1872 and January 1873). The latter is given in Gould's (American) 'Astronomical Journal,' vol. iv. p. 48. Copies of the book were printed on papers of different colours — yellow, brown, green, ifec, as it was considered (no doubt justly) that black on a white ground fatigues the eye more than any other combination *. Yellow or light brown seem the colours most preferred by computers, green not being very satisfactory. In the preface to his tables (1849), Mr. Filipowski writes : — " Babbage's ' Tables of Logarithms,' which probably are the most accurate of all ; for, by the aid of his ingenious calculating machine, he was enabled to detect a variety of errors in former tables." This is untrue. [Scheutz, 1857.] Five-figure logarithms, from 1000 to 10,000, calcu- lated and printed by Scheutz's calculating machine : specimens of a few other tables are added. A history and description of the machine &c. is given. Sang, 1859. Pive-figure logarithms, from 1000 to 10,000, arranged as in a seven-figure table ; no differences. Gray, 1865. The table in this tract is rather an auxiliary table to facilitate the calculation of logarithms to twelve places, than a table itself. The tables at the end of the work (see p. 2 of the Introduction) give log(l + -001n), log (l + -00rn), log (1 + -O0r70> from «==0 to k=999, at intervals of unity, to twelve places. The use of the sequantities in the cal- culation of logarithms is well-known (see, e.g.. Introduction to Shokteede's Tables, vol. i. 1849). Pages 43-55 are occupied with the history of the metliod, and will be found valuable and interesting. The rest of the book is devoted to explanations &c. Weddle's method of calculating the logarithms of numbers by resolving them into the reciprocals of series of factors of the form 1 — •!";•, r being a digit, and then using a subsidiary table of the logarithms of these factors, is fidly explained, as also are some improved methods of Mr. Gray's own, depending substantially on the same principle ; and aU arc illustrated with full numerical examples. The whole constitutes the most complete account of the simplest and best of the known methods for the calculation of isolated logarithms that we have met with ; and any one engaged on work of this kind would do well to consult it. Of course for calculating a table, the method of differences, as Mr. Gray remai'ks, is the best. A portion of this tract appeared in the * Mechanics' Magazine ' for 1848 ; and the whole is reprinted from the ' Assurance Magazine and Journal of the Institute of Actuaries.' Pineto, 1871. This work consists of three tables ; the first (Table auxiliaire) contains a series of factors by which the numbers whose logarithms are required are to bo multiplied to bring them within the range of Table 2, and occupies three pages. It also gives the logarithms of the reciprocals of the factors to twelve places. Table 1 merely gives logarithms to 1000, to ten places. Table 2 gives logarithms from 1,000,000 to 1,011,000 * " Of all the things that are meant to bo read, a blaek monumental inscription on white marble in a bright light is about the most difficult." — Do Morgan. 60 REl'ORT— 1873. to ten places ; the left-hand pages contain the logarithms, and the right- liand Images the proportional parts, which are given for every hundredth of the differences. The change in the line is denoted by an asterisk ; and the last figure is underlined -when it has been increased. The mode of using the tables is as follows : — If the first figures of the number lie between 1000 and 1011, the logarithm can be taken out directly from table 2 ; if not, a factor M is found from the auxiliary table, by which the number must be multiidicd in order to make its initial figures lie between these limits, and so bring it within the range of table 2. After performing this multiplication the logarithm can be taken out ; and to neutralize the effect of the multiplication, as far as the result is concerned, log ( — J must be added ; this quantity is therefore given in an adjoining column to M in the auxiliary table. A similar procedure gives the number answering to any logarithm, only that another factor (approximately the reciprocal of M) is given, so that in both cases multiplication is used. The laborious part of the work is the multiplication by the factor M ; but this is compensated to a great extent by the ease with which, by the proportional parts, the logarithm is taken out. Great pains have been taken to choose the factors M (which are 300 in number) so as to minimize this labour ; and of the 300 only 25 consist of three figures all diflPcrent and not involving or 1. Whenever it was possible, factors containing two figures alike or containing a 0, or of only one or two figures, have been found. The process of taking out a logarithm is rather longer than if Ylacq or Yega were used ; but, on the other hand, the size of this book (only about 80 pp. 8vo) is a great advantage, both of the former works being large folios. Also both Vlacq and Vega are so scarce as to be very difficult to procure ; so that Pineto's table will be often the only ten-figure table available for any one who has not access to a good library ; and on this account it is unique. Though the principle of multiplying by a factor, which is subsequently cancelled by subtracting its logarithm, is frcquentl}'^ employed in the construction of tables, this is, we believe, the first instance in which it forms part of the process of iisvi{j the table. By taking the numbers to 12 instead of 10 places, in a manner explained in the introduction, greater accuracy in the last place is ensured than results from the use of Vlacq or Vega. It is not stated whether the table is stereotyped ; so we pi'esume it is not. On the last page (p. 56) are given the first hundred multiples of the modulus and its reciprocal to 10 places. (Notices and examples taken from Pineto's tables will be found in the ' Quarterly Journal of Mathematics ' for October 1871, and the ' Messenger of Mathematics ' for July 1872.) Sang, 1871. Ten-figure logarithms, from 1 to 1000, and seven-figure logaritlims, from 20,000 to 200,000, with differences and multiples (not pro- portional parts) of the differences throughout. The advantages arising from the table extending from 20,000 to 200,000, instead of from 10,000 to 100,000, are, that whereas in the latter the dif- ferences near the beginning of the table are so numerous that the propor- tional parts must either be very crowded or some of them omitted, and even if they are aU given the interpolation is inconvenient, in a table extending from 20,000 to 200,000 the differences are halved in magnitude, while the number of them in a page is quartered ; the space gained enables multiples instead of proportional parts to be given. The table is printed without rules (except one dividing the logarithms from the numbers) ; and the numbers are separated from the logarithms by ON MATHEMATICAL TABLES. Gl reversed commas. The absence of rules does not appear to us by any means an unqualified advantage ; and a farther drawback is that numbers and logarithms are printed in the same type. The change of figure in the line is denoted by an Arabic nokta (a sign like the diamond in a pack of cards) ; and this, tliough very clear for O's, leaves the other figures unchanged, and is greatly inferior in all points of vievr to the simple asterisk prefixed, or the small figure as used by Babbage. In spite of these drawbacks the table is very convenient, and has advantages possessed by no other, as, in addition to the greater ease with which the interpolations can be performed, greater accuracy is obtained — the last figure being often inaccurate by one or tv/o units in logarithms inter- l^olated from the usual seven-figure tables. We find, however, that computers prefer Babbaoe, except for numbers beginning with 1. The logarithms of the numbers between 100,000 and 200,000 were calcu- lated de novo by Mr. Sang, as if logarithms had never been computed before ; and a very full account of the method and manner in whicli the calcula- tions were performed is given by him in the * Edinbm-gh Transactions,' vol. xxvi. pt. iii. (1871). Tliis is the only calculation of common logarithms of numbers since the days of Vlacq, 1628 (except the French manuscript tables). Two errors in the book (which is stereotyped) were pointed out in the * Athenajum' for Juno 8 and 15, 1872, viz. the last figures of log 389G2 and 52943 should be 2 and 5 instead of 3 and 6 respectively. Mr. Peter Gray has kindly communicated to us the following six im- portant eiTors which have been discovered and communicated to Mr. Sang (or found on revision) and circulated by him in certain later copies of his tables : — Page 203, log 118530, /or 9503 read 8503 „ log 118537, „ 9539 „ 8539 „ log 118538, „ 9576 „ 8576 220, log 127340, „ 9348 „ 9648 312, log 173339, „ 9863 ., 8963 „ 354, for number 19540 read 19440. The following is a classified list of the tables of logarithms contained in works that are described in § 4 : — ■ Tables of Lor/arithns of Numbers (to more than 20 places). — Sharp, 1717 [T. IV.] (61 places) ; SmmwiN, 1741 [T. I.] and [T. II.] (61 places) ; HoBEitT and Ideler, 1799 [T. III.] (36 places); Byrne, 1849 [T. IV.] (50 places) ; Callet, 1853 [T. III.], I. and II. (61 places) ; Huttox, 1858, T. 5 and 6 (61 places, early editions onlv) ; Parkhurst, 1871, T. II., III., and IX. (102 places), and T. XVIII. (Ol'places). (To 20 places) Gardiner, 1742, and (Avignon) 1770 [T. IV.] and [T. V.]; Paekhttrst, 1871, T. XIII. and XIV. (To 15 places) Douglas, 18i)9, T. IV., Supplement. (To 11 places) Boeda and Delambrk, 1800 or 1801 [T. II.] ; Kohlee, 1848 [T. III.] ; Callet, 1853 [T. II.], I. and II ; Hovel, 1858, T. V. (table to calculate logarithms) ; IIxiTXOjr, 1858, T. II. and III. (To 10 places) De Decker, 1626 [T.I.]; Henrion, 1626 LT.I.]; Vlacq, 1628 and 1631 [T. I.] ; Vlacq, 1633 [T. II.] ; Vega, 1794 [T. I.] ; Hantschl, 1827, T. IV. ; *-Saloiion, 1827, T. VIII. ; Parkhurst, 1871, T. XII. (To 8 places) John Newton, 1658 [T. I.] ; Houel, 1858, T, IV. (table to calculate logarithms) ; PAEKnuRST, 1871, T. XXXVII. (To 7 places) Faulhaber (Logarithmi), 1631 ; Norwood, 1631 ; Roe, 1633, C2 REPOiiT— 1873. T. I.; OuGiiTRED, 10.37 [T. II.]; Sir J. Moore, IGSl [T. I.]; Vl\cq, 1681 [T. II.] ; OzANAM, 1685 ; Gardiker, 1742, and (Avigiioii) 1770 [T. I.]; Sherwin, 1741 [T. III.]; Dodson, 1747, T. XXXII.; Hentschen (Vlacq), 1757 [T. II.]; Schtjlze, 1778 [T. I.] ; Donn, 1789, T. I. ; Taylor, 1792 [T. I.] and [T. II.] ; Vega, 1797, T. I. ; Vega, 1800, T. I. ; Borda and Delambre, 1800 or 1801 [T. I.] ; Douglas, 1809 [T. I.], and Supple- ments ; Lalande, 1829 [T. I.] ; Hassler, 1830 [T. I.] ; Grttsoi^, 1832, T. 1. ; Turkish Logarithms (1834) ; [De Morgan] 1839 [T. II.] ; Farley, 1840, T. II. ; IIulsse's Vega, 1840, T. I. ; Trotter, 1841 TT. IX.] ; Shortrede (Tables), 1844, T. I. ; Minsinger, 1845 [T. I.] ; Kohler, 1848 [T. I.] ; Shortrede, 1849, T. I. ; Willich, 1853, T. XX. ; Callet, 1853, T. I.; Bremiker's Vega, 1857, T. I.; Hution, 1858, T. I.; Schron, 1860, T. I.; Waceerbarth, 1867, T. I.; Dupuis, 1868, T. I. and II.); Bruhns, 1870, T. I. (To 6 places) Dunn, 1784 [T. I.] ; Adams, 1796 [T. I.] ; Maskelyne (Re- quisite Tables, Appendix), 1802, T. III. ; Mackay, 1810, T. XLV. ; Wallace, 1815 [T. I.] ; DucoM, 1820, T. XXI. ; Lax, 1821, T. XVIIL ; Kerigan, 1821, T. X.; lliDDLE, 1824, T. V.; Ursinus, 1827 [T. I.]; Galbraith, 1827, T. II.; ^Salomon, 1827, T. VII.; J. Taylor, 1833, T. XVIII. ; NoEiE, 1836, T. XXIV. ; Jahn, 1837, Vol. I. : Farley, 1840 [T. I.] ; Trotter, 1841 [T. L] ; Griffin, 1843, T. 17 ; J. Taylor, 1843, T. 4 ; Eumicer, 1844, T. I. ; Coleman, 1846, T. XX. ; Barer, 1846, T. I.; Domke, 1852, T. XXXII. ; Bremiker, 1852, T. I. ; Thomson, 1852, T. XXIV. ; Eaper, 1857, T. 04 ; Beardmore, 1862, T. 30 ; Inman, 1871 [T. VII.]. (To 5 places) Bates, 1781 | T. I.] ; Maskelyne (Requisite Tables), 1802, T. XVIII. ; Bowditch, 1802, T. XVI. ; Lalande, 1805 [T. I.] ; Rios, 1809, T. XV. ; Moore, 1814, T. IV. ; De Peasse, 1814 [T. I.] ; Pasquich, 1817, T. I. ; Reynaud, 1818 [T. I.] ; Schmidt, 1S21 [T. I.] ; Stansbury, 1822, T. X.; [Schumacher, 1822?]. T. V. (arguments in degrees &c.); Hantschl, 1827, T. I. ; Bagay, 1829, T. XXIII. ; K5hler, 1832 [T. I.] ; [De Morgan], 1839 [T. I.] ; Gregory &c., 1843, T. XI. ; Muller, 1844 [T. I.] ; Stegmann, 1855, T. I. ; HoiJEL, 1858, T. I. ; Galbraith and Haughton, 1860 [T. I.], and [T. II.]; ^Schlomilch [1865?] ; Rankine, 1866, T. I. ; "Wackerbarth, 1867, T. I. (To 4 places) [Encke, 1828] [T. I.]; [Sheepshanks 1844] [T. L] ; Waenstorff's Schumacher, 1845 [T. III.]; HoIjel, 1858, T. VI.; Anony- mous [1860 ?] (on a card) ; Oppolzeh, 1806. See also Shortrede (Comp. Log. Tab.), 1844 ; Parkhuest, 1871, T. XXVII. and XXVIII. Art. 14. Tables of AntllogaritJims. In the ordinary tables of logarithms the natural numbers are all integers, while the logarithms tabulated are only approximate, most of them being incommensurable. Thus interpolation is in general necessary in order to find the number answering to a given logarithm, even to five figures. It was natural therefore to form a table in which the logarithms were exact quantities, -00001, -00002, -00003 to -99999, &c., and the numbers in- commensurable. Few of such tables have been constructed, as for most purposes the ordinary tables are sufficiently convenient, and computers much prefer to have only oue work to refer to. The earliest antilogarithmic table is DoDsoN, 1742 ; and the only others of any extent are Shortrede (1844 and 1849) and Filipowski (1849), described in § 4. Mr, Peter Gray has a large tAvclve-figurc antilogarithmic table far advanced towards completion ; but whether it will be published is uncertain. ON MATHEMATICAL TABLES. (13 Dodson, 1742 (Antilogarithmic Canon). Numbers to elcvcu places corresponding to logarithms from -00000 to 1-00000, at intervals of -00001, arranged like a seven-figure logarithmic table, -with interscript differences, and proportional parts at the bottom of the page. The changes in the fourth figure in the middle of the column, both in the numbers and the differences, are marked by points and commas, but not very clearly. Tliere is an intro- duction of 84 pages ; and the tables occupy about 250 pages. In page ix of the Introduction an extract is given from Wallis, who states that Harriot began, and Warner completed, a table of antilogarithms, which was ready for press fifty years before. This was told Wallis by Dr. Pell, who had assisted Warner in the calculation ; and Wallis mentions that he had himself seen the calculation thirty years before, among Harriot's or Warner's papers. Dr. Pell subsequently informed WaUis that the papers were in the hands of Dr. Busby, and that he (Dr. Pell) hoped to publish them shortly. Dr. Pell died in 1685 ; and at the time Wallis wrote Dr. Busby was also dead, and the printing had not been begun. Speaking of this manu- script De Morgan remarks : — " All our efforts to trace it, by help of published letters <fcc., lead to the conclusion that, if existing, it must be among Lord Macclesfield's unexamined manuscripts at Sbireburn Castle : this is by no means improbable." See, however, some additional information and im- portant remarks by De Morgan, ' Budget of Paradoxes ' (1872), pp. 457, 458. A list of thirty-six errors affecting the first eight figures in Dodson's canon is given by Filipoavski in the preface to his 'Antilogarithms' (1849). Mr. Peter Gray (' Insurance Record,' June 9, 1871) says that in 1847 he had collected a list of 125 errors in Dodsou ; these he communicated to Shorteede, and they were corrected in the plates of his tables (1849). Dodson's work is unique of its kind, and it remained the only antilogarithmic canon for more than a century after its completion, till in 1844 Shorteede published the first edition of his tables ; in 1849 he published his second edition ; and in the same year Filipowski's tables appeared. For hyperbolic antilogarithms (viz. e^ and e~*) see under miscellaneous tables (§ 3, art. 25). The following are antilogarithmic tables described In § 4 : — AntUogarithmic Tables. — Gaediner, 1742, and (Avignon) 1770 [T. VI.] (20 places); Dodson, 1747, T. XXXIII.; [Sheepshanks, 1844] [T. VII.]; Shorteede (Comp. Log. Tab.), 1844 ; Shoetrede (tables), 1844, T. II., and 1849, T. II. ; FiLiPowsKi, 1849, T. I. ; Caelet, 1853 [T. II.], III. ; Stegmank, 1855, T. IL; HoIjel, 1858, T. VL ; Hxitton, 1858, T. IV.; Anontmohs [1860 ?] (on a card) ; Paekhtjest, 1871, T. XXVIL, XXVIIL, and XXXV. Art. 15. Talks of (Briggian) Loganthmic Trigonometrical Functions, A general account of the introduction of Briggian logarithms is given in § 3, art. 13 ; and Napier's ' Canon Mirificus' (1614), containing a Napierian logarithmic canon, is described in § 3, art. 17. The first table of decimal logarithms of numbers was published by Brtggs in 1617, and the first (decimal) logarithmic canon by Gunter in 1620 (see below), giving the results to 7 places. The next calculation was by Vlacq, who appended to his ' Centum Chiliades ' in the ' Arithmetica ' of 1628 a minute logarithmic canon to 10 places, obtained by calculating the logarithms of the sines &c. of Ehetictjs. After the publication of his ' Ai-ithmetica ' in 1624, Beiggs devoted himself to the calculation of logarithmic sines &c., and at his death in 1631 had all but completed a ten-decimal canon to every hundredth of a 6i REPORT — 1873. flegree. This was published by Tlacq at liis own expense at Gouda in 1633, under the title •' Trigonometria Britannica ' (see below) : the intro- duction was written by Gellibraud, by whose name the book is sometimes cited. In the same year Vlacq published his ' Trigonometria Artiiicialis,' containing a ten-aecond canon to ten decimals. Guntee's original table contains a good many errors in the last figures ; and a very slight comparison shows whether any particular table was copied from Gunter or Vlacq ; IIenrion, 1626, and de Decker, 162G (§ 4), are from the former, Faulhaber (§ 4), 1631, from the latter. Briggs appreciated clearly the advantages of a centesimal division of the quadrant, and, by taking a hundredth of a degree instead of a minute, made a step towards a reformation in this respect ; and Hutton has truly remarked that, but for the appearance of Vlacq's work, the decimal division of the degree might have become recognized, as is the case with the corresponding division of the second*. The next great advance on the ' Artificialis' was more than a century and a half afterwards, when Michael Taylor (1792) published his seven-decimal canon to every second (§ 4). On account of its great size, and for other reasons, it never came into very general use, Bagay's 1829 (§ 4) being preferred ; the latter is now, however, very difficult to procure. The only other canon to eveiy second we have seen or heard of is Shorirede's, 1844 and 1849 (§ 4), which is the most complete as regards proportional parts &c. that we know of. The canon is in modern editions issued separately. Lalande (' Encyclopedic Methodiquc. if atheraatiques,' Ast. Tables) states that in April 1784 he received from M. Robert, cure of St. Genevieve at Toul, a volume of sines for every second of the quadrant, and soon after the tangents ; but he had heard that Taylor, in England, was engaged in publishing log sines and cosines to every second, and that the Board of Longitude had contributed £300 to the expense. These volumes were pur- chased by Babbage at the sale of Delambre's library, and they appear in the Babbage Catalogue (only the title of the table of sines is given ; but it is to be presumed that the library contains both, as two volumes arc spoken of). Babbage lent them in 1828 to the Board of Longitude ; and some errata in Taylor, 1792, were found by means of them. [They are now (1873) in the possession of Lord Lindsay, who has purchased the whole of Mr. Babbage 's mathematical library.] No ten-decimal canon to every second has been calculated. The French manuscript tables are described in § 3, art. 13. Of logarithmic trigonometrical canons that have appeared the number is very great. We may especially mention Callet, 1853; Bremiker's Vega, 1857; Hutton, 1858; Schron, 1860; Dupuis, 1868; and Bruhns, 1870. _ * The centesimal division of the degree is of paramount imporiance, wliereas the cente- simal division of the right angle is of next to none at all ; and had the French mathemati- cians at the end of the last century been content with the former, it is not unlikely that their tables woidd have superseded the sexagesimal ones still in use, instead of liaving been almost totally ignored by computers. Thehundredlh part of a right angle is almost as arbitrary a unit as the ninetieth ; and no advantage (but on the contrary great inconvenience) would re- sult from the change ; but to divide the nonagcsimaldegree into centesimal minutes, and these into centesimal seconds, &c., in other words to measure angles by degrees and decimals of a degree, wo\ikl ensure all the advantages of a decimal system (a saving of work in interpo- lations, multiphcations, &e.). This Briggs and his followers. Roe, Oughtred, John Newton, &c., perceived and acted upon two hundred and fifty years ago ; and they seem to liave shown a truer appreciation of the matter than did the French mathematicians. It may be taken for granted that the magnitude of the degree will never be altered; but there is no reason why sexagesimal minutes and seconds should not be replaced by decimals of a degree ; and this is a change which might, and we hope will hereafter be made. ON MATHEMATICAL TABLES. 65 The cliief tables in which the angle is divided completely centesimaUy are Callet 1853, BoRDA and Delambre, and Hobert and Ideler. For the meaning of S and T (Delambre's tables), see § 3, art. 13, near the end of the introductory remarks. Gunter, 1620. Log sines and tangents for every minute of the quadrant (semiquadrantally arranged) to 7 places. This is the first (Briggian) loga- rithmic trigonometrical canon calculated or published. The book is ex- tremely scarce ; and we have only seen one copy of it, viz. that in the British Museum, where it is bound up with Briggs's ' Logarithmorum Chilias Prima.' There is engraved on the titlepage a diagram of a spherical triangle, S P Z. De Morgan (who had never seen a copy) says that it also contains logarithms of numbers as far as 1000 ; but this is not correct. The British-Museum copy has written in ink on the titlepage, " Eadius autem verus est 10,000,000,000." This has reference to the fact that the logarithm of the radius is taken to bo 10, and is true in one sense, but not in the usual one, which is that, this being the radius, the sines &c. are true to the nearest unit. Custom has veiy properly decided to consider the radius of a logarithmic canon the same as what would be the radius of the resulting natural canon if the logarithms were replaced by their numbers. We have not seen the second edition, in which no doubt the logarithms of numbers mentioned by De Morgan were added ; or it is just possible that some copies of Briggs's ' Chilias ' (1617) were issued with the ' Canon,' both being bound together in the copy we have seen, and that this has given rise to the assertion. Gunter's ' Canon ' was also issued under an English title, ' A Canon of Triangles,' &c. (Bodleian Catalogue) : see Phil. Mag. (Suppl. No.) Dec. 1872. For a life of Gunter, see Ward's ' Lives of the Professors of Gresham College,' pp. 71-81. Briggs, 1633 (' Trigonometria Britannica '). Natural sines (to 15 places) and tangents and secants (to 10 places), also log sines (to 14 places) and tangents (to 10 places), at intervals of a hundredth of a degree from 0° to 45°, with interscript differences for aU the functions. The division of the degree is thus centesimal; but the corresponding argu- ments in minutes and seconds are also given, the intervals so expressed being 36". This table was calculated by Briggs ; but he did not live to publish it. The trigonometry is by Gellibrand. Gunter, 1673. At the end of the work is given a table of log sines and tangents for every minute of the quadrant to 7 places, followed by seven- figure logarithms of numbers to 10,000. The table of log sines &c. is printed as it appeared in Gxjnter's ' Canon Triangulorum,' 1620, as the last figures in very many instances differ from the correct values, which were first given by Vlacq in the ' Arithmetica ' &c. (1628). This is the fifth edition of Gunter's works; but we remember to have seen it stated somewhere that the works themselves (separate) were regarded as the first edition in this enumeration. Berthoud, 1775. At the end of the ' Recueil des Tables n^cessaires pour trouver la longitude en mer,' is a table of log sines to every minute of the quadrant to 6 places (pp. 25-34). Callet, 1827 (* Log Sines &c.'). Log sines and tangents for every second to 5°, and log sines, cosines, tangents, and cotangents from 0° to 45°, at intervals of ten seconds, with differences, all to seven places. 1873. s 66 REPORT— 1873. These are the same as Callet 1853 [T. IX. and X.] (§ 4), and were pub- lished separately, De Morgan states, to accompany Babbage's logarithms of numbers ; they are in consequence printed on yellow paper ; but it is, both in colour and texture, very inferior to that used by Babbage. Airy, 1838. Log sines and cosines from O*" to 24'', at intervals of 10^ to 5 places. The proper sign is prefixed to each quantity : no dififer- ences. The sines are on the left-hand pages, the cosines on the right-hand. As was remarked by De Morgan, this is an eightfold repetition of one table : it occupies 48 pp. The table is improperly described as having been " computed under the direction " &c. : it is, of course, only a simple re- arrangement. The following is a complete classified list of tables on the subject of this article contained in the works that are described in § 4, with several other lists appended. Log sines, tangents, secants, and versed sines, — (To 7 places) Wimcir, 1853, T. B ; Hunoif, 1858, T. IX. (To 5 places) Eios, 1809, T. XYI. (also log coversed &c.). Log sines, tangents, and secants. — (To 10 places) Vlacq, 1628 and 1631 [T. II.]; Faxjlhaber (Canon), 1631. (To 7 places) Sir J. Moore, 1681 [T. III.] ; Sherwin, 1741 [T. IV.] ; BoEDA and Delambre, 1800 or 1801, T. VI. (centesimal) ; Douglas, 1809 [T. II.]. (To 6 places) Dunn, 1784 [T. II.] ; Adams, 1796 [T. II.] ; Wallace, 1815 [T. II.] ; J. Taylor, 1833, T. XIX. ; Noeie, 1836, T. XXV. ; Trotter, 1841 [T. in.]; Griffin, 1843, T. 18; J. Taylor, 1843, T. 5; RuJiker, 1844, T. II. ; Coleman, 1846, T. XXIH. ; Eaper, 1846, T. IV. ; Domke, 1852, T. XXXV. ; Eaper, 1857, T. 68 ; Inman, 1871 [T. IV.]. (To 5 places) Maskelyne (Requisite Tables), 1802, T. XIX.; Bow.. DITCH, 1802, T. XVII. ; Moore, 1814, T. V.'; Galbraith, 1827, T. V. ; Geegort &c., 1843, T. IX. ; Hotjel, 1858, T. II. (To 4 places) Gordon, 1849, T. IX. (cosecants). Log sines and tangents (onh/). — (To 11 places) BoRDAand Delambre, 1800 or 1801 [T. III.] (centesimal), and [T. V.] (logarithmic diiferences of sines and tangents). (To 10 places) Vlaco, 1633 [T. I.]; Roe, 1633, T. I. (centesimal division of the degree) ; Vega, 1794, T. II. (To 8 places) John Newton, 1658 [T. II.] and [T. III.] (arguments partly centesimal). (To 7 places) de Decker, 1626 [T. II.] ; Henrion, 1626 [T. II.] ; Norwood, 1631 ; Vlacq, 1681 [T. I.] ; Ozanam, 1685 ; Gardiner, 1742, and (Avignon), 1770 [T. II.]; DoDsoN, 1747, T. XXXIV.; Hentschen ( Vlacq), 1757 [T. I.]; ScnuLZE, 1778 [T. III.] and [T. V.]; Donn, 1789, T. III.; Taylok, 1792 [T. III.] ; Vega, 1797, T. II. ; Lambert, 1798, T. XXVI. ; HoBEETandlDELEE, 1799 [T. I.] (centesimal) ; Vega, 1800, T. II. ; (?) *Salo- MON, 1827, T. IX.; Bagay, 1829, Appendix; Lalande, 1829 [T. U.j; Hasslee, 1830 [T. IL-IV.]; Getjson, 1832, T. VII.; Turkish logarithms [1834] ; Hulsse's Vega, 1840, T. II. ; Shortrede (Tables), 1844, T. III., and 1849, Vol. II. ; Kohler, 1848 [T. IV.'\ ; Callet, 1853 [T. VI.] (cente- simal), [T. IX.] and [T. X.] ; Beemiker's Vega, 1857, T. II. and III. ; HuTTON, 1858, T. VIII. ; Scheon, 1860, T. H. ; Dupuis, 1868, T. VI., VII., and VIII. ; Beuhns, 1870, T. II. and III. _ (To 6 places) Oughtbed, 1657 [T. I.] (centesimal division of degree) j DucoM, 1820, T. IX. ; ITRsiNtrs, 1827 [T. II.] and [T. V.]; J. Taylor, 1833, ON MATHEMATICAL TABLES. 67 T. XIX. ; NoRiE, 1836, T. XXV. ; Jahn, 1837, Vol. II. ; Parley, 1840 [T. II.] ; J. Taylor, 1843, T. 5 ; Kumker, 1844 ; Domke, 1852, T. XXXIV. ; Bremiker, 1852, T. II. (To 5 places) Bates, 1781 [T. II.] ; Lalande, 1805, T. II, ; De Prasse, 1814 [T. II.] ; Pasquich, 1817, T. II. ; Reynaud, 1818 [T. II.] ; Schmidt, 1821 [T. II.] ; KoHLER, 1832 [T. II.] ; [Db Morgan], 1839 [T. III.] ; Galbraith and Haughton, 1860 [T. III.] ; Wackerbarth, 1867, T. III. (To 4 places) [Encke, 1828] T. II.; Beverley (1833?), T. XVII. ; MuLLER, 1844 [T. IV.]; [Sheepshanks, 1844] [T. III.]; Warnstorff's Schumacher, 1845 [T. IV.J ; Thomson, 1852, T. XVI.; Oppolzer, 1866; Parkhurst, 1871, T. XXX. and XXXI. (Miscell.) Shortrede (Comp. Log. Tab.) 1844. Log sines and secants (only). — (To 5 places) Stansbury, 1822, T. H. Lo(j sines (alone*) (for small arcs, sines = tangents). — (To 7 places) Gardiner, 1742 [T. II.], and (Avignon) 1770 [T. II.] ; Hulsse's Vega, 1840, T. II. ; KiiHLER, 1848 [T. IV.]. (To 6 places) Mackay, 1810, T. XLVI. ; Kerigan, 1821, T. VIII. ; Hanxschl, 1827, T. II. ; Farley, 1840 [T. III.] ; Rapbr, 1846, T. III. ; Raper, 1857, T. 66 and 67 ; Beardmore,'1862, T. 37 ; Inman, 1871 [T. III.]. (To 5 places) [Schumacher, 1822?] T. VI.; [De Morgan] 1839 [T. IV.] ; Raper, 1846, T. II. ; Thomson, 1852, T. XII. (To 4 places) [Sheepshanks, 1844] [T.II.]; Parkhurst, 1871, T. XXXVIII. (Expressed otherwise) Acad:6mie de Prusse, 1776 [T. I.] ; Callet, 1853 [T. VII.] (centesimal) (15 places). Lot/ tangents (alone*) (for small arcs, sines = tangents). — (To 7 places) Gardiner (Avignon), 1770 [T. II.]. (To 6 places) Mackay, 1810, T. XLVII. ; Hantschl, 1827, T. III. Log versed sines (alone). — (To 7 places) Sir J. Moore, 1681 [T. IV.] ; [Sir J. Moore, 1681, versed sines] ; Douglas, 1809 [T. IV.] ; Parley, 1856 [T. 11.]. (To 6 places) Rumkeb, 1844, T. IV. (To 5 places) Kerigan, 1821, T. XI.; J. Taylor, 1833, T. XXI., and 1843, T. 30. (To 4 places) Donn, 1789, T. V. JS^ote. — Log rising (in nautical tables) =log versed sine. See next page. Log secants (alone). — (To 5 places) Thomson, 1852, T. XI. Miscellaneous.— IjOg sec x, \ log sec x, and | log sin x, Croswell, 1791, T. I. ; log difF. sin., Borda and Delambre, 1800 or 1801 [T. V.] (centesimal) ; log I (1 + cos x), log I (1 ± sin .r) &c., Rios, 1809, T. XVI. ; log tan *|, Stansbury, 1822, T. Z»; log i (1 - cos x) &e., Stansbury, 1822, T. R. ; log 1 (1— cos x), NoRiE, 1836, T. XXXI. ; log h (1— cos x), Coleman, 1846, T. XXL; log I (1-cos x\ Gordon, 1849, T.'XVIII. ; log! (1-cos x), Thomson, 1852, T. XIII. ; log cosec .r-r54000, Thomson, 1852"; T. XV. ; log sin "^^ Thomson, 1852, T. XXIII.; log \ (1-cos x), Raper, 1857, T. 69; \ log \ (1 — cos x) and log \ (1— cos x), Inman, 1871, T. V. and VI. The following arc tables generally met Avith in nautical collections : — Log sines, tangents, and secants to everg quarter-point. — (To 7 places) * Tables of sines and tangents are not unfreqiiently printed with the sines on the versos and the tangents on the rectos of tlie leaves, or vice versa, so that practically they arc sepa- rated ; hxit in such cases the table has usually been regarded merely as one of sines and tangents. f2 68 JtEPORt — 1873. NoRiE, 1836, T. XXIII. ; Shoetrede (Tables), 184^ T. V. ; Doxif, 1789, T. II. (sines and cosecants only). (To 6 places) Eiddle, 1824, T. lY. ; Galbraith, 1827, T. lY. ; J. Taylor, 1833, T. XYII. ; Trotter, 1841 [T. II.] ; Griffin, 1843, T. 16 ; J. Taylor, 1843, T. 3; Coleman, 1846, T. XIX.; Domke, 1852, T. XXXTL; IIaper, 18.57, T. 65. (To 5 places) Adams, 1796 [T. III.] ; Bowditch, 1802, T. XYI. ; Moore, 1814, T. III. Log. I elapsed time, mid time, and rising. — (To 5 places) ,Donn, 1789, T. lY. ; Maskelyne (Requisite Tables), 1802, T. XYI. ; Bowbitch, 1802, T. XIII. The tbree Tables are separated in the following : — (To 5 places) Mackay, T. XLYIII.-L. ; Moore, 1814, T. XXIII. ; Norie, 1836, T. XXYII.- XXIX. ; DoMEE, 1852, T. XXXYII.-XXXIX. We have thought it worth while to collect into one list below all the tables, giving log sines &c. to every second. It must be particularly noticed, how- ever, that in the great majority of cases only the functions for the first few degrees of the quadrant are given to every second in the tables z'eferred to, which should in all cases be sought in § 4. Tables of logarithmic trigonometrical functions to seconds. — Gardiner, 1742 [T. il. I, and (Avignon) 1770 [T. II.] ; Schulze, 1778 [T. III.] ; Taylor, 1792, T. III. (for the whole quadrant) ; Yega, 1794, T. II. ; Yega, 1797, T. II. ; Yega, 1800, T. II.; Ducom, 1820, T. IX. ; Xerigan, 1821, T. Yllt. ; [Schumacher, 1822?] T. VI.; *Salomon, 1827, T. IX.; Bagay, 1829, Appendix (for the whole quadrant) ; Hassler, 1830 [T. II.] ; Jahn, 1837, Yol. II. ; [De Morgan] 1839 [T. lY.] ; HUlsse's Yega, 1840, T. U. ; Muller, 1844 [T. lY.] ; Shortrede (Tables), 1844, T. III. and 1849, Yol. II. (for the whole quadrant); Rarer, 1846, T. II.; Kohler, 1848 [T. lY.] ; DoMKE, 1852, T. XXXIY. ; Bremiker, 1852, T. II. ; Callet, 1853 [T. IX.]; Bremiker's Yega, 1857, T. II. ; Raper, 1857, T. 66; Hutton, 1858, T. YIII. ; Wackerbarth, 1867, T. III. ; Dupuis, 1868, T. YI. and YII. ; Bruhns, 1870, T. II. ; Inman, 1871 [T. III.] and [T. YIII]. We have formed the following lists of tables in § 4, which (not only in tho same work, but side by side in the same table) give both natural and logarithmic functions : — Tables containing both natural and logarithmic functions (^in the same table). —(To 15 places) Callet, 1853 [T. YII.] (centesimal). (To 7 places) Sir J. Moore, 1681 [T. III.] ; Ylacq, 1681 [T. I.] ; OzANAM, 1685 ; SherwixN, 1741 [T. lY.] and [T. Y.] ; Hentschen (YLAca), 1757 [T. I.] ; Schulze, 1778 [T. Y.] ; Donn, 1789, T. III. ; Lambert, 1798, T. XXYI. ; HoBERT and Ideler, 1799 [T. I.] (centesimal) ; Willich, 1853, T. B ; Hutton, 1858, T. IX. (To 6 places) Oughtred, 1657 [T. I.]; Ursinus, 1827 [T. Y.]. (To 5 places) HoiJEL, 1858, T. II. (To 4 places) Donn, 1789, T. Y. (Mixed) Bates, 1781 [T. II.]. Natural and log versed sines (in the same table). — (To 7 places) Sir J. Moore, 1681 [T. lY.] ; [Sir J. Moore, 1681, versed sines] ; Sherwin, 1741 [T. Y.] ; Douglas, 1809, T. lY. Art. 16. Tables of Hyperbolic Logarithms (viz. logarithms to base 2-71828. . .), The logarithms invented by Napier, and explained in the ' Descriptio ' (1614) and ' Coustruetio ' (1619) (see § 3, art. 17), were uot the same as ON MATHEMATICAL TABLES. 69 those now called h/jjei-holic (viz. to base e) and very frequently also Naj^ierian logarithms. It is also to be noticed that JSTapier calculated no logarithms of numbers. Jonx Speidell, 1019 (see below), first published logarithms to base e bothof numbers and sines. The most complete table of hj'perbolic logarithms is Base's, described below, which could be used, though not so convenieutly, as an ordinary seven-figure Briggian table extending from 1000 to 105,000. It would sometimes be useful to have also a complete seven- place table of hyperbolic logarithms of numbers from 1000 to 100,000, ex- actly similar to the corresponding' Briggian tables, as in some cases it is con- venient to perform calculations in duplicate, first by Briggian, and then by hyperbolic logarithms ; and such a table would be of use in multiplying- five figures by five figures : but hyperbolic logarithms cannot be rendered conve- nient for general purposes. The most elaborate hyperbolic logarithmic table is "Wolfkam's, which prac- tically gives the hyperbolic logarithms of all numbers from unity to 10,000 ioforty-eiglit decimal places. It first appeared, we believe, in Schulze (§ 4), and was reprinted in Vega, folio, 1794 (§ 4). Wolfram was a Dutch Ueutenant of artillery ; and his table represents six years of very laborious work. Just before its completion he was attacked by a serious illness ; and a few logarithms were in consequence omitted in Schuize (see Introduction, last page but two, to vol. i. of Schulze). The omissions were supplied in Vega's ' Thesaurus,' 1794. De Morgan speaks of Wolfram's table as one of the most striking additions that have been made in the sub- ject of logarithms in modern times. Montucla (' Histoire,' vol. iii. p. 360) states that in 1781 Alexander Jom- bert proposed to publish by subscription new tables of hyperbolic logarithms to 21 places for all prime numbers to 100,000, with a table of all odd numbers of two factors to the same limit. The author was Dom Vallej're, advised by Dom Robe, benedictine of St. Maur. Only two hundred subscribers were re- quired before the commencement of the printing, and nothing was asked in advance; but the project fell through, no doubt for want of subscribers. We infer from this account that the table was calculated. The Catalogue of the Royal Society's Library contains, under the name of Prony, the title, " Formules pour calculer I'efl'et d'une machine a vapeur a detente et a un seul cylindre Tables de logarithmes hyperboliques calcu- lees de 100^ en 100" d'unite, fol. lithog.," but without any reference to the place where the book is to be found in the library, so that we have not seen it. Speidell, 1619. Logarithmic sines, tangents, and secants, semiquadi-antally arranged, to every minute, to five places. The logarithms are hyperbolic (viz. to base e), and the first of the kind ever published. When the characteristic is negative SpeideU adds 10 to it, and does not separate the characteristic so increased from the rest of the figures by any space or mark. Thus he prints the logarithm of the sine of 21° 30' as 899625, its true value being 2-99625 ; but the logarithm of the cotangent is given as 93163 ; it would now bo written -93163. The Royal Society has " the 5-impression, 1623," with the " Breefe Treatise of Sphaericall Triangles " prefixed, and also some ordinary hj-perbolic logarithms of numbers (the first published) &c. On this see De Morgan's long account of Speidell's works, who, however, had never seen the edition of 1619, in which the canon occurs by itself without the logarithms of numbers. We cannot enter into the question of Speidell's fairness here. The 1619 copy we have seen (Cambridge Univ. Lib.) has an obUteration where, in the 1623 copy, there occur the words " the S-impression." 70 REPORT— 1873. Rees's Cyclopaedia, 1819 (Art. " Hyperbolic Logarithms," vol. xviii.). Hyperbolic logarithms (to 8 places) of all numbers from 1 to 10,000, arranged in groups of five. The table was calculated by Bablow, and appears also in hia mathema- tical tables (1814). Dase, 1850 (Hyperbolic Logarithms). Hyperbolic logarithms, from 1 to 1000, at intervals of unity, and from 1000-0 to 10.500-0 at intervals of 0-1 to seven places, with differences and proportional parts, arranged as in an ordinary seven-figure table. The change of figure in the line is de- noted by an asterisk prefixed to all the niimbers affected. The table is a complete seven-place table, as by adding log 10 to the results the range -is from 10,000 to 105,000 at intervals of unity. The table appeared in the 34th part (new series, t, xiv.) of the ' Annals of the Vienna Observatory' (1851); but separate copies were printed, in the preface to which Dasegavc six errata. This portion of the preface is reproduced in the introduction by Littrow to the above volume of ' Annals.' The table was calciilated to ten places, and three rejected. It was the author of this table who also com- puted the factorial tables (§ 3, art. 8), and calculated the value of n cor- rectly to 200 decimal places (Crelle's Journal, t. xxvii. p. 198). Filipowski, 1857. Hyperbolic logarithms, from 1 to 1201, to 7 places, are appended to Mr. Filipowski's English edition of Napier's ' Canon Mirificus.' The folloAving is a list of references to § 4 : — Hyperholic logarithms of numhers. — (To 48 places) Schulze, 1778 [T. II.] ; Vega, 1 794 [T. III.l ; Callet, 1853 [T. III.], I., and II. (To 25 places) Lambert, 1798, T. XVI. (To 20 places) Callet, 1853 [T. II.], I. and II. (To 11 places) Borda and Delambre, 1800 or 1801 [T. IV.]. (To 10 places) *Salomon, 1827, T. VIII. (To 8 places) Vega, 1797, Vol. II. T. II. ; Barlow, 1814, T. VI. ; Hant- echl, 1827, T. VI. ; HtLssE's Vega, 1840, T. VI. ; Trotter, 1841 [T. XI.] ; KoHLER, 1848, T. I. (To 7 places) Gardiner (Avignon), 1770 [T. VII.]; L.uibert, 1798, T. XIII.-XVI. ; WiLLicH, 1853, T. A ; Hutton, 1858, T. V. and VI. ; Duruis, 1868, T. III. (To 5 places) Uankine, 1866, T. 3 ; Wackerbarth, 1867, T. V. • See also *Schlomilch [1865 ?]. Art. 17. Napiei-ian Loganthms (not to base 2-71828. . . . ). The invention of logarithms has been accorded to Napier of Merchiston with a unanimity not often met with in reference to scientific discoveries. The only possible rival is Justus Byrgius, who seems to have constructed a rude kind of logarithmic table ; but there is every reason to believe that Napier's system was conceived and perfected before Byrge's in point of time ; and in date of publication Napier has the advantage by six years. Further, Byrge's system is greatly inferior to Napier's ; and to the latter alone is the whole world indebted for the knowledge of logarithms, as (with the exception of Kepler, one of the most enthusiastic of the contemporary admirers of Napier and his system, who does allude to Byrge) no one ever suggested any one else as having been the author whence they had drawn their information, or as having anticipated Napier at all, tiU the end of the last century, when Byrge's claim was first raised, though his warmest advocates always assigned far the greater part of the credit of the invention to Napier. ON MATHEMATICAL TABLES. .^J On Byrge's claim see De Morgan's careful resume (article " Tables," under Justus Byrgius, 1620, in the 'Eng. Cyclop.,' where references are given), and Mr, Mark Napier's ' Memoirs of John Napier of Merchiston,' Edin- burgh, 1834 (where the question how far Napier received any assistance from his predecessors in the discovery is fully discussed). We have also seen ' Justus Byrg als Mathematiker und dessen Eiuleitung in seine Logarith- men,' by Dr. Gieswald, Dantzig, 1856, 4to (pp. 36). Napier's ' Canonis Logarithmorum Mirifici Descriptio ' (which contained the first announcement and the first table of logarithms) was published in 1614 ; and in 1619 (two years after his death, which occurred on April 4, 1617) appeared the ' Mirifici logarithmoiTim Canonis Constructio,' edited by his son Robert, in which the method of constructing the canon is explained. The various reprints and translations of the ' Descriptio ' and ' Constructio ' are described under Napier, 1614 and 1619 ; and the relations between Napier and Briggs with regard to the invention of decimal logarithms are noticed in § 3, art. 13. The most elaborate canon of Napierian logarithms is Ursinus (1624-1625), described below. The diflference between the logarithms introduced Napier and hyperbolic logarithms is explained under Napier (1614). We have paid considerable attention to the early logarithmic tables, and have examined all of them that were accessible to us ; and it is with some regret that we omit to notice them in detail here : the accounts of the smaller tables that immediately suc- ceeded Napier would be of only bibliographical or historical interest ; and to describe them with sufiicient detaU to render the accounts of value would occupy too much space. However, as the works of this period are very rare, it is worth while remarking that there is a copy of Napier's * Constructio ' in the Cambridge University Library (there is none in the British Museum or Royal Society's Library), where also are to be found Ursinus's ' Cursus ' of 1618, Speidell 1619, and Kepler 1624: we have generally, in describing works of this date, mentioned the library containing the copy we have seen. We have found De Morgan to be very accurate (except where he has had to form his opinions from secondhand or imperfect evidence) ; and he has 'devoted much care to the early logarithmic tables, so that we feel the less reluctance in omitting to notice them further here. Napier, 1614. The book consists of 57 pp. explaining the nature of logarithms &c., and 90 pp. of tabular matter, giving natural sines and their Napierian logarithms to every minute of the quadrant (seraiquadrantaUy arranged) to seven or eight figures (seven decimals). Logarithmic tangents are also given under the heading differenUce (they are the differences between the sine and cosine, which, though the latter name is not used, are both on the same line, as a consequence of the semiquadrantal arrangement of the table). The logarithms introduced by Napier were not hyperbolic or Napierian logarithms as we now understand these terms, viz. logarithms to the base e (2-71828 . . ), but somewhat difierent ; the relation between the two being L^ e' = lO^.e 10^ or L = 10^ log, 10^ - 10^ I, I being the logarithm to base e, and L the Napierian logarithm j the relation between N (a sine) and L, its Napierian logarithm is therefore L N = 10,000,000 <5 10,000,000; 72 KEPORT — 1873. the logarithms therefore decrease as the sines increase. A hrief explanation of the principle of Napier's own method is given by Professor Wackerbarth in vol. xxxi. p. 263 (1871) of the 'Monthly Notices of the Royal Astro- nomical Society.' The anthor of that communication there points out that the description in most elementary books of Napierian logarithms, as loga- rithms to the base e, is incorrect ; but this criticism appears to us irrelevant, as by calling certain logarithms Napierian it is not asserted that they arc used at present in the exact form in whicli they were presented by Napier. A glance at the formula written above shows that all the essential features of logarithms to the base e arc contained in Napier's system, and that there is no impropriety in calling the former by his name. De Morgan says that " Delambre proposed to call them [Napier's logarithmsj Napierian logarithms, and to restrict the term hj-pcrbolic to the modern or e logarithms ; but custom has refused," — and no doubt very properly, as, except in mathematical histories &e., there is no occasion to distinguish the two systems from one another. For our own part, we should much prefer to see natural or ' hyperbolic and common logarithms universally called Napierian uni Brigr/ian, after the two great founders of logarithmic tables. A translation of Napier's ' Canon Mirificus ' was made by Edward "Wright (well known in connexion with the history of navigation), and, after his death, published by his son at London in 1616, under the title " A Description of the admirable Table of Logarithmes, &c." (12mo). Ou account of the rarity of this work and the ' Constructio,' the full titles of both are given in § 5. There is a short " Preface to the Reader " by Briggs, and a description of a triangular diagram invented by Wright for finding the proportional parts. Napier's table, however, is printed to one figure less than in the ' Canon Mirificus ' throughout. The edition was revised by Napier himself. On Wright, see Introduction to Button's ' Mathematical Tables.' The ' Canon Mirificus ' was also rej)rinted by Maseres in the sixth volume of the ' Scrip- tores Logarithmici ' (1791-1807); and in 1857 Mr. Filipowski published at Edinburgh a translation of the same work (full title given in § 5 ; the tone of the Introduction renders any comment on it unnecessary). Both the ' Deseriptio ' (the ' Canon Mirificus ') and the ' Constructio ' were reprinted by Bartholomew Vincent at Lyons in 1620 (who thus first published logarithms on the Continent), the title of the former appearing on the titlepage as " Logarithmorum Canonis Deseriptio, sen Arithmeticarum supputationura mirabilis abbreviatio. Ejusquc ususin utraque Trigonometria ut etiam in omni Logistica Mathematica, amplissimi, facillimi &, expeditissimi cxplicatio. Authore ac Inventore Joanne Nepero, Barone Merchistonii, &c., Scoto. [Printer's device with word Vincenti.'] Lugduni. Apud Barth. Vin- centium, M.DC.XX. Cum privilegio Cffisar. Majest. & Christ. Galliarum Regis." The full title of Napier's original edition of 1614 is given in § 5 ; and it will be seen that it is very difi'erent from that written above. Yery many writers (including Montucla) give the title of Vincent's reprint as that of the original work. There is an imperfect copy of Vincent's reprint, containiug only the * Deseriptio ' (the * Constructio ' having been torn out), in the British Museum ; but the Royal Society has a perfect copy. Wright's translation of 1616 is in the British Museum. On the accuracy of Napier's Canon see Delambre, * Astron. Mod.,' t. i. p. .501. Mr. Mark Napier's 'Memoirs of John Napier' gives nearly all that is known with regard to Napier's life, MSS., &c. ; but it is told in a verbose and diff'use manner, and written in a partisan spirit as regards Briggs. A manuscript on arithmetic and algebra, written by Napier and left by ON MATHEMATICAL TABLES. 73 him to Briggs, was privately printed in 1839, under the title " De Arte Logistica Joannis Naperi Merchistonii Baronis libri qui'supcrsuut," edited by Mr. Mark Napier. An historical sketch, mainly derived from the same author's ' Memoirs,' is prefixed. In 1787 was also published ' An account of the Life, Writings, and Inventions of John Napier of Merchiston,' by David Stewart, Earl of Buchan, and Walter Minto, LL.D. Perth, 4to. See also Phil. Mag. Suppl. No., December, 1872, " On some early Logarithmic Tables." Leslie (' Philosophy of Arithmetic,' 2nd edit., 1820, p. 24(3) describes Napier's work as " a very small duodecimo ;" the last word should be " quarto." The page is 7*7 inches by 5*7 inches. We may remark that Napier's name is spelt in a variety of Avays ; we have seen Neper, Naper, Nepair, and Nepper. He always Latinized his name into Neperus or Naperus, but spelt it in the vernacular several ways. ' The family now write the name Napier; and this spelling is generally adopted, and with good reason. Napier, 1619 (' Constructio '). This work contains no table, and is there- fore not properly included in this Report. We have, however, noticed it on account of its being a sequel to the ' Descriptio,' and also on account of its rarity (the fuU title is given in § 5). The only copy we have seen (in the Cambridge University Library), which belonged to Oughtred, contains two titlepages, the first running " Mirifici logarithmorum canonis descriptio. . . . accesserunt opera posthuma ; primo, Mirifici ipsius canonis constructio .... Edinburgi. . . . 1619," and the second being as given in § 5. From this we infer that a reprint of the ' Descriptio ' (1619) was prefixed to the ' Constructio,' but that it was torn out from the copy we have examined. On the reprints, &c. of the ' Constructio,' see under Napier, 1614. Ursinus, 1624—1625. A canon exactly similar to Napier's in the ' Canon Mirificus,' 1614, only much enlarged. The intervals of the argu- ments are 10" ; and the results are given to eight places : in Napier's canon the intervals are 1', and the number of places is 7. The logarithms are strictly Najplerian, and the arrangement is identical with that in the canon of 1614. This is the largest Napierian canon that has been calculated. The copy we have seen is in the British Museum. In 1618 Ursinus published his * Cursus Mathematicus,' of which there is a copy in the Cambridge Uni- versity Library. The only table of Napierian logarithms described in § 4 is Schulze, 1778 [T. v.] (sines and tangents). Art. 18. Logistic and Proportional Logarithms. What arc now called fractions or ratios used to be styled logistic numbers ; and logistic logarithms are logarithms of ratios : thus a table of log -, x [ being the argument and o a constant, would be called a table of logistic or proportional logarithms ; and since log - = log o — log x, it is clear that the tabular results only differ from those of an ordinary table of logarithms by the subtraction of a constant and a change of sign. It appears that Kepler, in his ' Chilias ' described below, originated tables of this kind ; but the step that separates logistic from common logarithms is so small that no great interest attaches to their first appearance. The use of the tabulation of log - in the working of proportions in which the third term is a fixed quantity a is evident. 74 REPORT 1873. There seems a tendency to keep the name logistic logarithms for those tables in which a = 3600" = 1° (so that the table gives log 3600 — log .r, x being expressed in minutes and seconds), and to use the ndirae proportional logarithms when a has any other value. We have not met with any modern table of this kind forming a separate work ; and such tables are usually of no great extent. They are to be found, however, in many collections of tables ; and the logistic logarithms from Callet were published separately at Nuremberg in a tract of 9 pp. in 1843 (see title in § 5). It may be remarked that tables of log - often extend to values of x X greater than a ; and then, in the portion of the table for which this is the case, the mantissae are rendered positive (by the supposed addition of the characteristic — 1, which is omitted) before tabulation, Kepler, 1624. We cannot do better than foUow De Morgan's example, and give a specimen of the table, which contains five columns : — 53- 36-36 5-48 80500-00 19- 19-12 21691-30 124-15 48-18 The sinus or numerus dhsolutus is 805, which (to a radius 1000) is the gine of 53° 36' 36", and the Napierian logarithm is 2169130. The third and fifth columns are explained as follows : — if 1000 represent 24'', then 805 represents 19*" 19"" 12' ; and if 1000 represents 60°, then 805 represents 48° 18' ; there are intorscript differences for the first and fourth columns. Thus, as De Morgan remarks, Kepler originated logistic logarithms. Kepler's tract is reprinted by Maseres in vol. i. of his ' Scriptores Logarithmici ' (1791); and there is also reprinted there " Joannis Keplcri .... supple - mentum chiliadis logarithmorum . . . .Marpurgi, 1625," the original of which we have not seen, but it contains no table. The copy of the 1624 work Ave have described is iu the Cambridge University Library, For an account of Kepler's ' Tabulae Rudolphinae,' see De Morgan. Proportional logarithms for every second, a being 3°, are given almost invariably in collections of nautical tables, usually to four places, but some- times to five. T. 74 of Raper, so frequently referred to in § 4, is a four- place table of this kind, and was, as we have seen stated in several places, first computed by Maskelyne. The reference was made to Raper rather than to any other of the numerous places where it occurs, as his work on Navigation is one of the best-known, and has been through numerous editions. Prof. Everett (Phil. Mag, Nov. 1866) says, quoting Raper, that proportional logarithms as at present used are a source of perpetual mis- takes even to expert computers ; but this must be intended to apply rather to practical men, as for the mathematical calculator they are very convenient. The following is a list of tables on the subject of this article, which are described more fiiUy in § 4. Logistic logarithms for every second to 1°, viz. log 3600 — log x. — (To 4 places) Gakmnek, 1742 and (Avignon) 1770, T. III. (to 4800") ; Dodson, 1747, T. XXXVI. (to 4800") ; Schtjlze, 1778 [T. IV.] (to 3600") ; Vega, 1797, Vol. II. T. IV, (to 3600") ; Gordon, 1849, T. XXI. (to 3600") ; Callet, 1853 [T. XI.] (to 5280") ; Htttton, 1858, T. VII. (to 5280") ; Inman, 1871 [T. I.] (to 3600", intervals of 2"). Proportional logarithms for every second to 3°, viz. log 10,800 — log x. — (To 5 places) Rios, 1809, T. XIV. ; Lax, 1821, T. XIV. ; -Galbeaith, ON MATHEMATICAL TABLES. 75 * 1827, T. X. ; Bagay, 1829, T. XXII. ; Colemak, 1846, T. XXIV. ; Ixma^ 1871 [T. II.] (To 4 places) (viz. T. 74 of Eapeh) Croswell, 1791, T. V. ; Maskeltne (Kequisite Tables), 1802, T. XV. ; Bowditch, 1802, T. XV. ; Andrew, 1805, T. XIV.; Mackat, 1810, T. LI. ; Moore, 1814, T. XXV. ; Ditcom, 1820, T. VII. ; Kerigan, 1821, T. XII. ; Stansburt, 1822 [T. II.] ; Kiddle, 1824, T. XXIX.; J. Taylor, 1833, T. XXXVL ; Beverley (1833?), T. XVIII. ; Norte, 1836, T. XXXIV.; Gregory &c., 1843, T. VIII. ; Griffin, 1843, T. 41 ; J. Taylor, 1843, T. 35 ; Eumker, 1844, T. XXIV. ; Gordon, 1849, T. X. ; DoMKE, 1852, T. XL. ; Thomson, 1852, T. XIX. ; Eaper. 1857, T. 74. Proportional logarithms for every minute to 24'*, viz. log 1440 — log x. — ■ (To 5 places) Galbraith, 1827, T. IX. (To 4 places) Stansbury, 1822, T. G ; Lynn, 1827, T. E; Gregory &c. 1843, T. XII. ; Gordon, 1849, T. XIX. ; Thomson, 1852, T. X. ; Kaper, 1857, T, 21A. Art. 19. Tables of Gaussian Logarithms. Gaussian logarithms have for their object to facilitate the finding of the logarithms of the sum and difference of two numbers whose logarithms are known, the numbers being themselves unknown ; on this account they are often called Addition and Subtraction logarithms. The problem is therefore • given log a and log 6, find log (a ± b) by the taking out of only one logarithm. The utility of such logarithms was first pointed out by Leonelli, in a very scarce book printed at Bordeaux in the year XI. (1802 or 1803), under the title " Supplement logarithmique ;" but it met with no success. Leonelli's idea was to construct a table to 14 places — an extravagant extent, as Gauss has re- marked. The first table constructed was calculated by Gauss, and published by him in vol. xxvi. (p. 498 et seq.) of Zach's ' Monatliche Correspondenz ' (1812) : it gives B and C for argument A, where A = log a?, B = log 1 1 + - ) C = log (1 + x), so that C = A + B ; and the use is as follows. We have identically — log (a + b) = log a + log (l + ^ = log rt -f- B I for argument log - j. The rule therefore is, to take log a, the larger of the two logarithms, and to enter the table with log a — logb as argument, we then have log (a + b) = log n + B, or, if we please, = log 6 + C. For the difference, the formula is log (a — b) =log b + A (argument sought in column C) if log rt — log 6 is greater than -30103, and = log 6 — A (argument sought in column B) if log a — log 6 is less than -30103 ; there are also other forms. Gauss remarks that a complete seven-figure table of this kind would be very useful. Such a table was formed by Matthiessen ; but the arrangement is such that very little is gained by the use of it. This Gauss has pointed out in No. 474 of the ' Astronomische Nachrichten,' 1843, and in a letter (1846) to Schumacher, quoted by De Morgan. Gauss's papers on logarithms and reviews of logarithmic tables from the ' Gottingische gelehrte Anzeigen,' * Astronomische Nachrichten,' &e., are reprinted together on pp. 241-264 of t. iii. of his ' Werke,' 1866. Of these pp. 244-252 have reference to Gaussian logarithms and contain reviews of Pasquich, 1817 (§. 4), and Matihiessen, 76 . REPORT — 1873. « 1818 (below). The largest tables are Zech (reprinted from Hulsse's edition of Vega) and Wittstein, which answers the purpose Gauss had in view the best of all : there is also a good introduction to the latter (in French and German), explaining the use and objects of the tables. Whenever in this Eeport the letters A, B, C are used in the description of Gaussian logarithms, they are always supposed to have the meanings assigned to them by Gauss (which are explained above), unless the con- trary is expressly stated. Of course all Gaussian tables have reference to Briggian (not hyperbolic) logarithms. Leonelli, 1806. This is the German translation of Leonelli's work, and suggested to Gauss the construction of his table in Zach's ' Correspondenz.' The book consists of two parts : in the first there are 9 pages of tables &c. wanted in the construction of logarithms, viz. log cc, log 1-x, log (1-Oa-), . . . . log (l-OOOOOOOOOO.r), for x = 1, 2, 9, to 20 places, and the same for hyperbolic logarithms; also log -1, -2 (9-9), and log l-Oo:, log l-OOOo--, log 1-OOOOO.r, and log 1-OOOOOOO.r, for x = 01, 02, ... . 99. The second part is headed " Theorie der Ergiinzungs- und Verminderungs- Logarithmen zur Berechnung der Logarithmen der Summen und Differenzen yon Zahlen aus ihren Logarithmen," and on pp. 52-54 the specimen table is given ; log x being the argument, it gives log j 1 + - J and log (1 + x) as tabular results to 14 places, for arguments from -00000 to '00104 at intervals of -00001. [It wiU be noticed that the above are the same as Gauss's A, B, and C] The middle page of this table (p. 53) is nearly an inch longer than any of the other pages of the book. The original work, according to Houel, 1858, ^ Avertissement,' p. vi, was published at Bordeaux, An XI., under the title " Supplement logarithmique," (fee. Gauss, 1812. b/^ = log fl + -\\, and C (= log (1 + x)) are given for argument A(= log x) from A = -000 to 2-000 at intervals of -001, thenco to 3-40 at intervals of -01, and to 5-0 at intervals of -1, all to 5 places, with differences. The table occupies 27 small octavo pages. Gauss's paper is re- printed from the ' Correspondenz ' in t. iii. pp. 244—246 of his ' Werke,' 1866 ; but the table is not reproduced there. Matthiessen, 1818. B and C are given to 7 places for argument A, from A = -0000 to 2-0000 at intervals of -0001, thence to 3-000 at intervals of -001, to 4-00 at intervals of -01 and to 5-0 at intervals of -1 ; also for A = 6 and 7, with proportional parts. As C = A 4- B, the last three figures are the same for B and C, so that the arrangement is, column of A, column of first four figures of B, column of first four figures of C, column of last three figures of B and C, proportional parts ; the eye has therefore to look in two different columns to take out a logarithm. There is also another disadvantage, viz. that as there are only four figures of argument, if it is to be used as a seven-figure table three more must be interpolated for. The introduction is both in German and Latin. Mr. Gray, who recalculated a considerable portion of this table, found that it contained numerous errors (see Gray, 1849, below). See also the intro- ductoi-y remarks to this article. Weidenbach, 1829. Modified Gaussian logarithms. Log x (= A) is the argument, and log '^' ^ (= B) is the tabular result. A and B are thus X -~ 1 ^ ON MATHEMATICAL TABLES. 77 " reciprocal," the relation between them being, infact, 10 * + " = 10^ + 10" + 1, so that either A or B may be regarded as the argument. The table gives E to five places with differences, from A = -382 to A = 2-002 at intervals of -001, from A = 2-00 to A = 3-60 at intervals of -01, and then to 5-5 at intervals of -1. The corrimencement of the table being at A = '382 does not render it incomplete, by reason of the reciprocitj' referred to above, since for arguments less than '382 we can take B as the argument. Thus, at the beginning of the table A and B are very nearly equal, viz. A = -382, B = 0-38355 ; A = -383, B = '38255. There is an introduction of 2 pp. by Gauss. The use of the table in the solution of triangles is very apparent, e. g. in the formula cot - = -^^, tan — , in Napier's analogies, &c. 2 a — 2 Gray, 1849. Modified Gaussian logarithms. T. I. Log (1 + x) is the tabular result for log x as argument ; and the range is from log .^• = -0000 to 2-0000 at intervals of -0001, to 6 places, with proportional parts to hundredths (viz. 100 proportional parts of each difference). T. ir. Log (1 — x) is the tabular result for log x as argument; and the range is from log x = 3-000 to 1-000 at intervals of -001, and from 1-0000 to i-9000 at intervals of -0001, to 6 places, with complete proportional parts. The first table might have been copied from ITatthiessen by contracting the 7 places of the latter to 6 ; but it was recalculated by Mr. Gray, and many errors were thereby found in Matthiessen's table (Introduction, p. vi) ; the second t<able was also tke result of an original calculation. Some remarks and references on the subject of Gaussian logarithms &c. will be found in the Introduction to the work. Since writing the above account, Mr. Gray has sent us a copy of his * Addendum to Tables and Formulae for the computation of Life Contin- gencies .... Second Issue, comprising a large extension of the principal table . . . . ' London, 1870, 8vo (26 pp. of tables and an introduction), which is a continuation of the work under notice, and is intended to be bound up with it, a new title having reference to the whole work when so augmented being added. The ' Addendum ' contains a table of log (1 + x) to 6 places for argument log X, from log x = 3-000 to I-OOO at intervals of -001, and from 1-0000 to 0-0500 at intervals of -0001, the latter portion having proportional parts for every hundredth of the differences added : the whole of course the result of an original calculation. Mr. Peter Gray was the first to perceive the utility of Gaussian logarithms in the calculation of life contingencies, and to him is due their introduction as well as the calculation of the necessary tables, which it is evident are valuable mathematically, apart from the particular subject for which they were undertaken. Zech, 1849. Table of seven-figure Gaussian logarithms. Denoting, done by Gauss, log x, log [ 1 + - j, and log (1 -j- x), by A, B, C respectively, then the table gives B to seven places, from A = -0000 to A = 2-0000 at intervals of -0001, from A = 2-000 to A = 4-000 at intervals of •001, and thence to 6-00 at intervals of -01, with proportional parts through- out ; the whole arranged as an ordinary seven-figure logarithm table, and headed Addition table. The Subtraction table gives C to 7 places, from B = -0000000 to -0003000 at intervals of -0000001, thence to -050000 at intervals of -000001, and thence to -30300 at intervals of -00001 to seven places, with proportional parts. as was 78 REPORT— 1873. The addition table occupies 45 pp., the subtraction table 156 pp. The whole is a reprint from Hulsse's Yega of 1849, the paging being unaltered, and running from 636 to 836. The second edition is identical with the first, except that the 3 pp. of introduction are omitted. ■Wittstein, 1866. A fine table of Gaussian logarithms in a modified form. H (=log (1 +.v)) is given to seven places for the argument A ( =log a) for values of the argument from 3'0 to 4'0 at intervals of -l, from 4-00 to 600 at intervals of -01, from 6-000 to 8-000 at intervals of -001, from 8-0000 to 10-0000 at intervals of -0001, and also from -0000 to 4-0000 at the same intervals. Differences and proportional parts (or rather multiples) are given, except on one page (p. 5), where they are given for alternate differences as there is not sufficient space. The arrangement is similar to that of a seven-figure logarithmic table. The figures have heads and tails, and are very clear. On p. 127 there is given a recapitulation to three places, and to hundredths, of part of the table and the formulaj. A complete explanation is given in the introduction to the work'. Gaussian logarithms are very useful in the solution of triangles in such formulae as cot „ — _ t tan (A — B), in which Weidenbach's table would also be useful. The following is a list of tables of Gaussian logarithms contained in works noticed in § 4. Tables of Gaussian hgaritJims. — Pasquicit, 1817, T. III. (5 places) ; [Encke, 1828] [T. III.] (4 places) ; Xohlee, 1832 [T. III.] ; Hulsse's Vega, 1840, T. XII. ; Mt-LLER, 1844 [T. II.] ; [Sheepshanks," 1844] [T. V.] ; KoHLEH, 1848 [T. II.] ; Shortrede, 1849, T. VII. ; Filipowski, 1849, T. II. ; HouEL, 1858, T. III. ; Galbraith and Haughton, 1860 [T. IV.] ; Oppolzer, 1866. Art. 20. Tallies to convert Bri(/c/ian into Hyperholic Logarithms, and vice versa. Tables for the conversion of Briggian into hyperbolic logarithms, and vice versa, are given in nearly all collections of logarithmic tables. Such a table merely consists of the first hundred (sometimes only the first ten) multiples of the modulus -43429 44819 03251 82765 11289. . . ., and its reciprocal 2-30258 50929 94045 68401 79914 , to five, six, eight, and ten or even more places. A list of such tables, contained in works described in § 4, is given below ; tables of this kind, however, rarely exceed a page in extent, and are very easy to construct. It is not unlikely that the list is far from perfect, for in some cases it was not thought worth Avbilc noticing such tables when of small extent and to few places. "We mention Degen (§ 4) as containing one of the largest. The following is a list of tables contained in works noticed in § 4. To convert Briggian into hiiperholic logarithms and vice versa. — (To more than 10 places) Schtjlze, 1778 [T. I.] ; "Degen, 1824, T. II. ; Sjiorxrede, 1849, T. VII. ; Callet. 1853 [T. IV.] ; Paskhxjrst, 1871, T. V. (To 10 places) Schron, 1860, T. I. ; Bruhns, 1870. (To 8 places) Shortrede (Tables), 1844, T. XXXIX. ; Kohler, 1848, [T. I.] ; HotJEL, 1858, T. III. (To 7 places) Bremiker, 1852, T. I. ; Beejiiker's Vega, 1857, T. I. ; Dupuis, 1868, T, V. (To 6 places) Dodson, 1747, T. XXXVII. ON MATHEMATICAL TABLES. 79 (To 5 places) De Phasse, 1814 [T. II.] (?) ; Gaxbeaith and Haughton, 1800 [T. I.] ; Wackerbakth, 1867, T. V. See also Teotter, 1841 [T. I.]; Schlomilch [1865?]; Raxkine, 1860, T. 3 ; and Pineio, 1871 (§ 3, art. 13). Art, 21. InteriJolation Tables. AH the tables of proportional parts (described in § 3, art. 2) are interpolation tables in one, and that the most usual, sense ; and similarly^ multiplication and product tables may be so regarded (see § 3, art. 2). We may, however, especially refer to Scheon, 1860, as its printed title describes it as an interpolation table — a designation not common. The only separate table we have seen for facilitating interpolations, when the second, third, &c. differences are included, is "Woolhottse, noticed below. We may also refer to Godwaed's tables (title in § 5), but they seem of such special application that we have not thought it necessary to describe their contents. Woolhouse, 1865. Papers extracted from vols, xi, and xii. of thd * Assurance Magazine.' There are 9 pp. of interpolation tables (viz. pp. 14-22) . The work contains a clear explanation of methods of interpolation, with developments. The following are references to tables described in § 4, Binomial-theorem coefficients. — Schulze, 1778 [T. XIII.] ; Yega, 1797, Vol. II. [T. VI.]; Barlow, 1814, T. VII.; Hantschl, 1827, T. IX.; Hulsse's Vega, 1840, T. XIII. ; Kohlee, 1848, T. X. ; Parkhuest, 187i; T. XXXII. See also Rouse (§ 3, art. 25). Other interpolation coefficients. — Petees, 1871 [T. IV.], I. and II. Coefficients of series terms. — Vega, 1797, Vol. II. [T. VI.] ; Lambert, 1798, T. XLIV. ; Hulsse's Vega, 1840, T. VIII. ; Kohler, 1848, T. XI. Art. 22. Mensuration Tables. We have made no special search for tables on mensuration (such as areas of circles of given radius, volumes of cones of given base and altitude, &c.), and have only included those that have fallen in our way in the course of seeking for more strictly mathematical tables during the preparation of this Report. As, however, for several reasons it seems desirable that a complete list of such tables should be formed, we shall endeavour to render this Article as nearly perfect as we can in the supplement. One reason, how- ever, wh)^ such tables are not of very high mathematical value is that the measures are generally expressed in more or less arbitrary units, such as yards, feet, inches, &c., or metres &c. We may especially refer to the large table of circular segments in Sharp, 1717 (§ 4). Sir Jonas Moore (1660?). The table is a very small one, and scarcely occupies a third of a folio page. It gives the periphery of an ellipse for one axis as argument (the other axis being supposed equal to, unity) to 4 places, with differences ; the range of the argument is from -00, to 1-00 at intervals of -01. Thus, to find the perimeter of an ellipse, axes 1 and -78, we enter the table at 78 and find 2-8038. If oue axis is not equal to iinity, a simple proportion of course gives the perimeter. After working^ out four examples, the author proceeds : " I have made above 45,000 arith*- metieal operations for this table, and am now well pleased it is finished.. 80 KEPORT— 1873. Some perhaps may find shorter waj's, as I believed I had myself, till advised otherwise by the truly Honourable the Lord Bruncker, &c." This is perhaps the first tabulation of an elliptic integral. Bounycastle, 1831. A table of the areas of segments (pp. 295-300) : the same as T. XIII. of Hantschi. Todd, 1853. T. I. Areas (to 6 places) and circumferences (to 5 places) of circles for the diameter as argument, the range being from diameter ^ to diameter 24 at intervals of Jg^; the decimal fractions (to 4 places) equivalent to J^^, ^2_.^ <^c., are printed at the top of each page. T. II. The same from diameter 24 to 100 at intervals of ^ (4 places only for the circumferences). T. III. The same from diameter 12 to 600 at intervals of unity. Both areas and circumferences are only given to 4 places. T. IV. The same from diameter "1 to 100 at intervals of 4. Areas to 6 places, circumferences to 5. T. V. to VII. stand in exactly the same relation to spheres that T. I. to IV. do to circles, except that T. V. is equivalent to T. I. and II., the intervals being ^ from 1 to 100 ; and T. VI. commences at 1 (not 12), The volumes and superficies are given to 4 places. T, VIII. Areas (exact) and diagonals (to 5 places) of squares for side as argument, from g to 100 at intervals of ^. In all cases the arguments are given in inches, and the results in square and cubic inches ; but in T. III. and VI. the corresponding numbers of linear, square, and cubic feet are also given. The original work, of which this is the second and greatly augmented edition, was published in 1826 ; and the tables were the result of original calculations. There are besides some specific gravities, &c. The following tables are more fully described in § 4. Mensuration tables. — Sharp, 1717 [T. II.], areas of segments of circles ; [T. III.], table for computing the solidity of the upright hyperbolic section of a cone ; Dodson, 1747, T. XXVI., XXVIII., and XXIX. ; Galbeaith, 1827, T. XV. and XVI. (Introd.) ; Hantschl, 1827, T. XIII. ; Troxter, 1841 [T. v.] and [T. XII.]; Willich, 1853, T. C (circumferences and areas of circles) ; Beaedmore, 1862, T. 34 (circumferences and areas of circles) ; Raneine, 1866, T. 4 and 5. Art. 23. Dual Logarithms. Dual logarithms were invented, and the tables of them calculated, by Mr. Oliver Byrne, who, besides the work described below, has published ' Dual Arithmetic ' and the ' Young Dual Arithmetician ' on the subject. A dual number of the ascending scale is a continued product of powers of 1*1, l-Ol, I'OOl, &c., taken in order, the powers only being expressed. To distinguish these numbers from ordinary numbers, they are preceded by the sign \|/ : thus, \i/ 6, 9, 7, 6 = (M)»(l-01)' (1-001)^ (l-OOOl)" ; n,]/ 0, 0, 2 = (1-1)° (1-01)° (1-001)^, the numbers following the \j/ being called dual digits. "When all but the last digit of a dual number are zeros, the dual number is called a dual logarithm ; but the dual logarithms used by Mr. Byrne are " of the eighth position," viz. there are 7 ciphers between the \|/ and the logarithm. A dual number of the descending branch is a continued product of powers of -9, -99, -999, &c., and the dual number is followed by the symbol /|\ ; thus, (-9)3 (-99)2 = '3 '2 /|\; (-999)= (-999999)2 = '0' 0' 3' 0' 0' 2 /|\. In the descending branch also a dual number reduced to the eighth position is ON MATHEMATICAL TABLES. 81 called a dual logarithm, and is to be considered negative if the ascending dual logarithm is taken positive, and vice versa. Byrne, 1867. T. I. contains aU the dual numbers of the ascending branch of dual arithmetic from \|/ 0, 0, 0, 1 to \|/ 7, 3, 1, 9, and their corresjjonding dual numbers and natural numbers. The range of the dual logarithms is from 00000 to 69892175, and of the natural numbers from 1-00000000 to 2-01167234. Marginal tables are added, by means of which all dual numbers of 8 digits, and their corresponding dual logarithms and natural numbers, may be derived : the table occupies 74 pp. T. II. Dual logarithms and dual numbers of the descending branch of dual arithmetic from '0 '0 '0 '1 '0 '0 '0 '0 /|\ to '3 '6 '9 '9 '0 '0 '0 '0 /|\ with corresponding natural numbers. The range of the dual logarithms is from '10001 to '39633845, and of the natural numbers from -99990000 to •67277805. Marginal tables are added, by means of which all intermediate dual numbers of 8 digits and their corresponding dual logarithms and natural numbers may be derived. This table is printed in red, T. I. and III. being in black. It occupies 38 pp. T. III. Natural sines and arcs to 7 places for every minute of the quadrant. The length of the arc is, of course, the circular measure of the angle, so that we have a table of circular measures to minutes : the arrange- ment is quadrantal. Proportional parts are given for 10", 20". . . .90" for each difference ; and these occupy two thirds of the page. There are small proportional-part tables for the arc : the table occupies 90 pp. The author claims that his tables are incomparably superior to those of common logarithms, and asserts that " these tables are equal in power to Babbage's and Callet's, and take up less than one eighth of the space " ('Dual Arithmetic,' part ii. p. ix). Bahhage and Callet seems an error (unless the Callex of 1827 (§ 3, art. 15) is meant), as the latter work con- tains the table of the logarithms of numbers which is the sole contents of the former. Mr. Byrne's works on the subject are : — ' Dual Arithmetic : a new Art,' London, 1863, 8vo (pp. 244) ; ' Dual Arithmetic : a new Art. New Issue, with a complete analysis,' 1864 (pp. 83) [this work contains a table of 3 pp., " to facilitate the conversion of dual numbers into common ones, or the converse "] ; ' Dual Arithmetic: a new Art. Part the Second ' (pp. 218), and the work above described. Mr. Byrne has also published ' The Dual Doctrine of Angular Magnitude and Functions, &c.,' and the ' Young Dual Arithmetician,' neither of which we have seen : the latter contains an abridgment to 3 dual digits of the tables in the work described above. In spite of the somewhat extravagant claims advanced by the author for his system, dual logarithms have found but little favour as yet either from mathematicians or computers. Art. 24. Matliematical Cotistants. In nearly all tables of logarithms there is a page devoted to certa^ frequently used constants and their logarithms, such as n, -, tt-, i/n, aYt:, &c., the radius of the circle in degrees, minutes, &c., the modulus &c. There are not generally more than four or five logarithms involving tt given ; and usually half the page is devoted to constants relatiug to the conversion of weights and measures. It is only necessary, therefore, here to refer to works in which tliere is a better collection than usual of constants. 1873. s 83 REPORT — 1873. A very good collection is given by Matnaed (described below), and also by Byrne, 1849. This portion of the present Eeport is very far from complete, as the values of mathematical constants have, as a rule, appeared in periodical publications, while those only that are most used by the general computer are to be found in collections of mathematical tables. We refrain, therefore, from giving references to several periodicals we have met with containing constants, as they belong properly to a subsequent portion of the Report ; and it is hoped that, after the completion of the examination of the memoirs, a pretty complete list, either of the constants themselves, or at all events of the places where they are to be found, will be given. We may, however, notice a paper of Paucker's in the first volume of ' Grunert's Archiv der Mathematik und Physik,' in which a number of constants involving tt are given to a great many places, and Gauss's memoirs on the lemniscate-functions (' Werke,' t. iii. pp. 426 &c.), where e~^, e~^^, e~^'^, &c. are calculated to about fifty places. On Euler's con- stant, see ' Proceedings of the Eoyal Society,' t. xv. p. 429 ; t. xvi. pp. 154, 299 ; t. xviii. p. 49 (Shanks) ; t. xix. p. 514 (Glaisher) ; t. xx. pp. 27, 29 (Shanks). On e, the base of the Napierian logarithms, log^2, log^S &c., see, besides the places just referred to, ' Poy. Soc. Proc' t. ■vi. p. 397, and ' Brit. Assoc. Report' (Sections) 1871, p. 16, and also Shanks 1853 (§ 4). Several constants are to be found in the different works of Maseres. Mr. Maynard and Mr. Merrifield have independently calculated log^M and log^??i (M and m being the modulus and its reciprocal) to 30 places (' Assurance Magazine,' t. vi. p. 298). The value of it has been calculated to 500 places of decimals by Shanks and Richter independently, and to 707 places by the former alone : see references, ' Messenger of Mathematics,' December 1872 and July 1873. Mr. Shanks's latest value appears in the ' Roy. Soc. Proc' t. xxi. p. 319. It is proper here to remark that Rutherford's 208-decimal value of tt, given in the ' Phil. Trans.' 1841, p. 283, is erroneous after the 152nd place : this value is reproduced in BniNE, 1849 (§ 4), and in Matnakd; so that it is erroneous also in both of these works. [Maynard.] A good table of constants involving tt, such as tt V2, tt "^ Vtt, &c., and some few involving e &c., to a great many (generally 30) places. There are also other constants not included in the subjects of this Report. The copy of these constants that we examined consisted of six leaves without a cover, and which were evidently extracted from some work. Mr^ C. W. Merrifield, E.R.S., subsequently called our attention to the particil- larly good collection of constants in ' The Millwright and Engineers' Pocket Companion ; . . . . By William Templeton .... Corrected by Samuel May- nard. . . . Eifteenth edition, carefully revised,' London, 1871, 8vo, and lent us a copy ; and on examination it appeared that it was to this work that Maynard's collection belonged, where it occupies pp. 169-180. There are, altogether, 58 constants involving tt, and their logarithms, given generally to 30 places, and 13 others that may also be properly styled mathematical. It is mentioned that part of the table had previously appeared in Keith's ' Measurer ' (twenty-fourth edition, 1846). Templeton's work contains several other tables (areas of circles, &c.), and square roots which would have been included in this Report had we seen the book earlier ; as it is they will be noticed in the Supplement. On Rutherford's value of tt, quoted by May- nard, see introductory remarks to this article. ON MATHEMATICAL TABL13S. 83 The following is a list of references to § 4. Lists of Constants.— Bomos, 1747, T. XXVIL ; Galbeaith, 1827, T LXIII. ; Hanischl, 1827, T. XI. ; [De Morgan], 1839 [T. V.] ; Fasley, 1840 [T. III.] ; MuLLER, 1844 [T. IV.] ; Shortrede (Tables), 1844, T. II. MtLLER, 1844 [T. IV.] ; Eaper, 184G, T. V. ; Kohler, 1848 [T. III.] Byrne, 1849 [T. III.] ; Bremiker, 1852, T. II. ; Willich, 1853, T. XX., &c. Shanks, 1853 (constants to a great many places) ; Beemikee's Vega, 1857 HoiJEL, 1858, T. VIII. ; Hutton, 1858, T. XII. ; Galbeaith and Hatohton, 1860 [T. IV.] ; Wackerbaeth, 1867, T. IV., V., and XXI. ; Bruhns, 1870. Note. — Binomial-theorem coefHcients and coefficients of series-terms are noticed under Interpolation Tables in § 3, ai't. 21. Art. 25. Miscellaneous Tables, figurate Numbers, Sfc. We have placed in this article tables which could not properly be described under any one of the previous twenty-four heads. The list is not, however, a long one, as we have frequently placed doubtful tables in the article which most nearly applied to them. We may refer especially to Jonoourt's table of triangular numbers (de- scribed below), which is perhaps unique. Eetshammer's commercial loga- rithms and Montferriee's binary logarithms are described in § 3, art. 13, Picarte's table to facilitate the performance of divisions is described in § 3, art. 7. We may also particularly notice Degen's large table (§ 4) of "log 1-2. . . .x). There is a table of binomial-theorem coefficients in Eoijse (see below) ; and other tables of the same kind are referred to under Interpolation Tables in § 3, art. 21. Tables of endings of squares are noticed in § 3, art. 4 ; and tables for the solution of cubic equations, viz, + {x — x^), in § 3, art, 5. Browne, 1731, Pp. 6 and 7 are occupied by a table headed " Area of the circle in degrees and to the 10,000th part of a degree." CaUing ^ . a, > if B o 360 ' it gives a, 2a, 3a 100a, 200a, 300a, and 360a to 7 figures. There are also three other columns in which the results only differ by a change of decimal point. Through a mistake in the printing in the copy before us, all the odd pages are upside downi. Heilbronner, 1742. On pp. 922-924, the numbers from unity to 140, 72, and 100 are expressed in the scales whose radices are 3, 2, and 12 respectively. Joncourt, 1762 [T. I.]. A table of triangular numbers up to that of 20,000, viz, ^''^"r^ ^' for all numbers from n = 1 to 20,000 (the table occupies 224 pp.). [T. II.] Cubes of numbers from 1 to 600. There are 36 pages of explanation &c., in which it is shown how [T. I,] may be used in the extraction of square roots, &c, De Morgan refers to this book as "De la Nature. . . .de Nombres trigonaux," 1762, so we suppose some copies with the introduction &c. in French were published. The Eoyal Society's copy has " Dec. 23, 1762," written in ink underj>.eath the printed date. The book is handsomely printed. The Babbage Catalogue also gives the same work with an English title. ' The Nature and Notable Use of the most simple trigonal numbers, with e2 84 REPORT — 1873. two additional tables, &c., translated from the Latin of E. de Joncourt by the author's self.' Phillips, 1829. This is not properly a table at all. Names and an abbreviated way of writing them are suggested for all numbers up to 9 followed by 4000 figures, the chief peculiarity of the system being that 1000 is called ten hundred, and 10,000 a thousand, and so on. The only explanation of the object of the table is contained in the curiously untrue remark that, by adopting the author's names, " we obtain a clearer view of calculations which are generally called inconceivable only because we have hitherto adopted no terms to express and hmit them." On Sir R. Phillips, and the value of his works, see De Morgan's 'Budget of Paradoxes' (1872), pp. 143-145. D. Galbraith, 1838. A piece contains 4, 5. . . .56 squares, and the table is to show the number of dozens in any number of pieces up to 100, &c. It contains ^ for x = 4, 5 56, and y = 1, 2, 3 100, 200, 300, 400, and 500, the value of x being constant over any one page : thus X = 15, 1/ = 65, we have given 81-3 for jL (15 x 65) = 81j'^y . The table was calculated to give the number of handkerchiefs in any number of pieces, «S:c. De Morgan, 1843. Degen's table (§ 4) of log (1, 2. . . .w) is reprinted to six places by De Morgan at the end of his article on " Probabilities " in the ' Encyclopaedia Metropolitana.' The last figure is not corrected : the table occupies pp. 486-490. Rouse (no date). The tables, which are neither elaborate nor very nume- rous, are not of sufficient mathematical value to render it necessary to do more than give a general idea of their contents. In the body of the work are a num- ber of small tables of this kind : — A and B (of equal skill) play 21 games ; and the odds in favour of A's winning 1,2.... 20, 21 are given as tabular results. Similar tables are given for 20, 19 .... 2 games played. Then we have the same when the odds in favour of A are 6 to 5, 5 to 4, 5 to 3, &c., — the maximum number of games, however, being six. On a folding sheet at the end is given the number of ways in which 1, 2, 3. . . .60 points can be thrown with 1, 2. . . .10 dice, and also the number of ways iu which 52 cards can be combined into 4 hands in any given manner (thus, 5 diamonds, 4 hearts, 3 spades, and 1 club can be obtained in 3421322190 ways); the factor and the result when the suits are not specified are also given. The mode of formation of the table is obvious. On a folding sheet at the beginning of the book is given (a + 6)" at full length hrn = l,2.... 30. The following is a list of miscellaneous tables contained in works that are described in § 4. For greater convenience a brief description of the contents of each table is appended to the reference to it. Figurate Numbers. — Lambert, 1798, T. XXXVII. Hyperbolic Antilogarithms {viz. powers of e) and their Briggian logarithms, — ScHULZE, 1778 [T. I.] ; Vega, 1797, Vol. II. T. III. ; Lambert, 1798, T, XI. ; Hulsse's Vega, T. VII. ; Kohler, 1848, T. III. ; Shoetrede, 1844 [T. XL], III. ; Hutton, 1858, T. XII. ; Callet, 1853 [T. II.], III. Miscellaneous. — Sharp, 1717 [T. I.] | multiples of j); Dodson, 1747, T. XX. (combinations), T. XXIII. (permutations), T. XXXV. (seconds in any number of minutes less than 2°) ; Schulze, 1778 (Pythagorean triangles) ; Maseres, 1795 (miiltiples of primes); Vega, 1797, Vol. II. [T. VII.] and [T. VIII.] (piling of shot) ; Lambert, 1798, T. II. (multiples of primes), T. ON MATHEMATICAL TABLES, 85 III. (products of consecutive primes), T. XVII. (numbers of the form 2''3"'5''7«), T. XXIV. (0, ^^...for ^ = 10,000" m, &c.), T. XXXII. (Functiones hyperbolicae circularibus analogse) ; Borda and Delambke, 1800 or 1801 [T. V.] (log sin {x + 2)— log sin x, &c. centesimal) ; Peaeson, 1824: [T. II.] (1°, 2° as decimals of the circumference) ; Degen, 1824, T. I. (large table of log (1.2 a-)), T. III. (multiples of log 2, log 3, &c.) ; XJESiNtrs, 1827 [T. IV.] (length of chords subtending given angles) ; Hantschl, 1827, T. XI. (multiples of constants) ; Hartig, 1829 (contents of solids ex- pressed in Puss and Zoll) ; [De Morgan], 1839 [T. VI.], (log (1.2.3 x)); Hulsse's Vega, 1840, T. IV. (chord table), T. IX. F and G {x }f^, &c.) ; Shortrede (tables), 1849, T. IV. and V. (for calculating logarithms and anti- logarithms), and T. VIII. (log (1.2.3 x)); Domke, 1852, T. XXX. (I •«'' + 1^ I I ; Shanks, 1853 [T. I.] (terms of tan -'i and tan -'^) ; ScHRON, 18G0, T. III. /hyp. log 10" and 1 + -^V *Schl6milch [1865?] (elliptic quadrants); Everett, 1866; "Wackerbaeth, 1867, T. II. (log {1.2.... x), log (1.3.... .r), log (2.4.... a;)); Parkhuest, 1871, T. IV., VI.-VIII., X., XI., XV.-XVII., XIX., XXIV., XXIX., XXXVI. See also KuLiK, 1848, T. 2-10 and 11 (Theory-of-number tables and multiples of and i\ (§ 3, art. 4). § 4. Worlcs containinri Collections of Tables, arranged in aljjhahetical order. [The titles of the works can be found by reference to § 5.] Academie de Prusse (1776). This collection of tables only contains two that come within the scope of this Report. [T. I.] (vol. iii. pp. 172-207). Table of sines, expressed as arcs whose length is equal to that of the sine ; viz. for x (expressed in degrees and mi- nutes) as argument there is given the angle (expressed in degrees, minutes, seconds, and tenths of a second) whose circular measure is sin x, the argu- ment X being given to every minute of the quadrant. There are no differ- ences ; and the arrangement of the table is quadrantal (not semiquadrantal). The table is due to Schulze. [T. II.] (Vol. iii. pp. 258-271). Lengths of circular arcs, viz. the circular measures of 1°, 2°, 3°, 360°, of 1', 2', . • . . 60', and of 1", 2", 60" to 27 places. This table is by Schulze, in whose collection it also appears : see Schulze [T. VII.]. Both these tables are included under the head " Tables auxiliaires " in the third volume. The whole work is attributed in the Royal Society's Catalogue to Schulze, and, from internal evidence we have little doubt, correctly. Adams, 1796 [T. I.]. Six-figure logarithms to 10,860, written at length, with characteristics. Differences arc added. [T. II.] Log sines, tangents, and secants for everj^ minute of the qua- di-ant, to 6 places ; with tables at the bottom of the page to facilitate inter- polations. [T. III.] Log sines, cosines, tangents, cotangents, secants, and cosecants for every quarter point, to 5 places. 86 REPORT — 1873. X Andrew, 1805. T. XIII. Squares of natural semichords, viz. sin^ -^ from a.' = 0° to .r=120°, at intervals of 10", to seven places, -with differences and proportional parts for seconds. This valuable table occupies pp. 29-148 of the work. T. XIV. Proportional logarithms to 3°, at intervals of a second, to four places ; same as T. 74 of Eapee. The other tables are nautical. Anonymous [18G0 ?]. Four-figure logarithms of numbers from 100 to 1000, with proportional parts, on a card (about 12 in. by 10 in.). On the back, numbers (to four figures) to logarithms from -000 to 1-000, at intervals of "001, with proportional parts. Printed by J. Sittenfeld, published by Veit and Co., Berlin. No date. The Brit.-Mus. copy received April 2, 1860. Bagay, 1829. T. XXII. Proportional or logistic logarithms for every second to 3° (or 3") to five places ; same as T. 74 of Eaper, except to five instead of four places. T. XXIII. Seven-figure logarithms, from unity to 21,600 (with the cor- responding degrees, minutes, and seconds), to seven places, with differences, but not proportional parts. T. XXIV. Logarithms of sexagesimal numbers, viz. logarithms of num- bers of seconds in aU angles from 6° 10' 0" to 12°, at intervals of 1", to five places. Appendix. — Table of log sines and tangents for every second of the qua- drant to seven places (without differences). The change in the middle of the column is beautifully clearly marked by a large black nucleus, surrounded by a circle, printed instead of zero. Only the first logarithm affected is so de- noted ; but the mark is so striking that it readily attracts the eye. The table was formed by interpolation from Callet, corrected by Taylor (see p. ii of the ' Avertissement ') ; 76 errors were thus found in Taylor. Some errata are given at the end of the work. All the other tables are astronomical. This work, which has now become rare, is much esteemed. Barlow, 1814. T. I. Squares, cubes, square and cube roots (to 7 places), reciprocals (to 9 places as far as 1000, afterwards to 10), and all factors of numbers from 1 to 10,000, Thus, for the factors of 4932 we have given 2^. 3^ 137. T. II. The first ten powers of numbers from 1 to 100. This table was taken from Hutton [T. IV.] and Vega (Tabulte), vol. ii. T. IV. The errors given in this Report in Hutton are not reproduced in this table. T. III. Fourth and fifth powers of numbers from 100 to 1000. T. IV. For the soJittion of the irreducible case in cubic equations ;- viz. y^-y is tabulated from 2/=l-0000 to 1-1549, at intervals of -0001, to 8 places. T. V. Prime numbers from 1 to 100,103 (this table is incorrectly described on the titlepage to it as extending to 10,000 only). T. VI. Hyperbolic logarithms, to 8 places, of numbers from unity to 10,000 (this table is incorrectly described on the titlepage to it as only extending from 1000 to 10,000) T. VII. Differential coefficients, viz. the first six binomial- theorem coeffi- . , n(n~l) n(n—l)....(n — 5) „ cients, ^-^ ^ \ . . .-^ — —i^ ^g ', from n=-01 to 1-00, at intervals of •01, to 7 places. ON MATHEMATICAL TABLES. 87 These tables occupy 256 pp., and are followed by 78 pp. of formulae, weights, and measures, &c. There is a full introduction, stating whence the tables were derived, or, if computed, from what formulae, ifec. The hyperbolic logarithms were taken from "Wolfeam's table in Schulze ; and the reciprocals, factors, square and cube roots, and several other ^tables were the result of independent cal- culations. The squares, cubes, square and cube roots, and reciprocals from this table were reprinted and stereotyped, at the suggestion of De Morgan, in 1840 (see Barloav's tables, 1840, in § 3, art. 4). The reprint thus gives T. I., the column of factors being omitted. A list of 90 errors in T. I. of the original work is given in the reprint ; and 25 errors in T. YI. are given by Prof. Wackerbarth in the ' Monthly Notices of the Eoyal Astronomical Society ' for April 1867. Bates, 1781. [T. I.] Five-figure logarithms to 10,000, without dif- ferences. [T. n.] Log sines and tangents (to 5 places), and natural sines and tan- gents (to 7 places), for every minute of the quadrant, semiquadrantally arranged: no differences. The tables (which have a separate titlepage, bearing the date 1779) are preceded by 211 pp. of trigonometry, and followed by an Appendix on the motion of projectiles in a non-resisting medium. The work was intended for use in the Military Academy, Belmont, near Dublin. Beardmore, 1862. Only 23 pages (pp. 84-106) of this work contain tables that come within the scope of this Beport. T. 34. Areas and circumferences of circles, to 3 places, for diameters •1, -2, -9, and from 1-00 to 100, at intervals of -25. T. 35. Squares, cubes, fifth powers, square and cube roots (to 3 places), and reciprocals (to 9 places) for numbers from 1 to 100, the squares and square and cube roots being given as far as 1100. T. 36. Six-figure logarithms of numbers from 100 to 1000. T. 37. Log sines from 0° to 4-5° 50', at intervals of 10', to 6 places. T. 38. Natural sines, tangents, and secants for 1°, 2°, . . . . 90"^, to 6 places. The other tables relate to hydraulics, rainfall, &c. The work was first published in 1850 ; and a second edition, in an extended form, was issued in 1851. Beverley [1833?] T. VI. (p. 127). Any number of minutes less than 12'' expressed as a decimal of 12'', to 4 places. T. VI. (pp. 232-243). Sexagesimal cosecants and cotangents for every minute from 20° to 90°. A sexagesimal cotangent is the cotangent when the radius is taken = 60' (or 1°); viz. it bears to 60' the same ratio that the ordinary cotangent does to unity, and is usually expressed in minutes, seconds, and decimals of a second. The same, of course, holds for sines, cosines, &c. Thus the sexagesimal sine of 30° is 30', cosecant 30°= 120', &c. In this table the quantities tabulated are not sexagesimal functions, but sexagesimal functions divided by 3 (and are therefore to radius 20') : we thus have cosec 30°=40'. The table is given to two decimal places of a second. T. XV. Sexagesimal sines, tangents, secants, and versed sines (viz. to rad. 60') to every degree to 90°, to one decimal place of a second, with differences. T. XVII. Log sines and tangents, from 18° to 90°, at intervals of 1', to 4 places. T. XVIII. Proportional logarithms for every second to 3°, to 4 places ; same as T. 74 of Rapek. 83 REPORT — 1873. Mr. Beverley made some improvements in Tatior's Sexagesimal Table (§ 3, art. 9), and devised a plan to introduce them into Taylor's table without reprinting it. He accordingly made application to the Board of Admiralty to be allovred to do so in the copies that remained unsold ; but this was refused. He then offered to purchase all the unsold copies of Htjtton's ' Products ' and Taylor's tables, in order to introduce his improvements ; but his applica- tion was refused after the terms had been agreed upon, because he asked for six months' credit. In the Appendix he complains that " the immense labour that the calculation of his tables required him to exert had then ruined his constitution, and brought him to the verge of a prematui-e grave." It is to be presumed tliat the Admiralty had some grounds for their refusal ; but it is certain that no use has been made of Hutton or Taylor since the time of Mr. Beverley's application. No pains at any time seem to have been taken to circulate or make known any of the books published by the Board of Longitude, so that none of them have ever come into general use. Mr. Beverley died in 1834, at the age of 39 ; and the present work was published after his death, as it contains a notice of his life by " J. B.", and evident traces of revision. He often refers to his Taylor's Sexagesimal Table, but no doubt it was never published. We have seen 'The Book of Formulae &c., Cirencester, 1838,' by the same author ; but it contains no tables. Borda and Delambre, An IX. (1800 or 1801). [T. I.] Seven-figure logarithms of numbers from 10,000, to 100,000, with differences and pro- portional parts for aU. The line is broken when a change takes place in the middle of it. It may be remarked that while in all modern tables of logaritlims of numbers three figures are common to the block, and four only are given in the columns, in this table there are but two leading fig-ures, and five are found in the columns, so that the lines are broken in very few instances. [T. II.] Eleven-figure logarithms of numbers to 1000, and from 100,000 to 102,000 (the latter vsdth differences). [T. III.] Log sines, cosines, tangents, and cotangents for centesimal argu- ments, viz. from 0' to 10", at intervals of 10", and from 0' to 50", at in- tervals of 10' to 11 places, without differences (", ', " being used to denote .centesimal degrees (or grades as they are sometimes called), minutes, and seconds). [T. IV.] Hyperbolic logarithms of numbers from 1 to 1000 to 11 places. [T. v.] Log differences of sines for every 1", 2", . . . 10" throughout the quadrant, and the same for tangents for 1" and 2", to 7 places, viz. log sin 2" -log sin 1", log sin 3" -log sin 2" throughout the quadrant of 100", logsin 4"-log sin 2", log sin 6"— log sin 4" throughout the quadrant, &c. It is to be noticed, however, that in this mode of description of the table log sin 0" must be treated throughout as instead of — oo ; for facing 1" we have given log sin 1" (not log sin 1"— log sin 0") in the first column; jand facing 2" in the second we have log sin 2" &c. [T. VL] A great centesimal table, gi\'ing log sines, cosines, tangents, co- tangents, secants, and cosecants from 0" to 3", at intervals of 10" (with full proportional parts for every second), thence to 50" at intervals of 1\ with full proportional parts for every 10"). A page of tables for converting sexagesimals into centesimals &c., com- pletes the work, which is a thick small-sized quarto, with clearly printed and not too heavy pages. After the printing of the work Prony asked Delambre to examine the Tables du Cadastre (Avhich are to every 10" throughout. the quadrant to 12 places ; but see § 3, art. 13) ; and this gave Delambre the opportunity of reading them with Borda's table of sines and ON MATHEMATICAL TABLES. 89 tangents in this work : the result was the detection of a great number of last-place errors, which are given on pp. 117-119 (see p. 114, Preface de I'cditeur). There are other errata given on p. 116. De Morgan remarks that Delambre is wrong in saying that Hobeet and Idelee's tables, 1799 (§ 4), subdivided the quadrant as minutely as those which he and Borda had published ; but this is not the case, as the latter are as stated above. The mistake is one into which any one accustomed to describing tables would naturally fall, as the mode of arrangement gives the impression that the portion of [T. VI.] to 3" is to every second, and that that from 3" to 40^ is to every ten seconds : at first sight it is not easy to see why this was not the form of table adopted ; but the reason for the arrange- ment being as it is was no doubt that the sine and cosecant, tangent and co- tangent might be placed exactly on the same footing, as the proportional parts are the same for each pair. [Mr. Lewis, of Mount Vernon, Ohio, men- tions that Bremiker has fallen into the same mistake as De Morgan did, thus giving additional proof of how misleading is the arrangement of the table to those who have not had occasion to use it : see ' Monthly Notices of the Eoyal Astronomical Society,' May 1873, vol. xxxiii, pp. 455-458.] Bowditch, 1802. T. XII. For the conversion of arc into time. T. XIII. Log I elapsed time, mid time, and rising ; same as T. XVI. of Maskelyxe, 1802. It is stated in the preface that this table was first jmblished by Mr. Douwes, of Amsterdam, about 1740, and that he re- ceived £50 for it from the Commissioners of Longitude in England. 1024 (small) errors contained in this table in the second edition of Eequisite Tables are said to be here corrected. T. XIV. Natui'al sines for every minute to 5 places. T. XV. Proportional logarithms for every minute to 3° ; same as T. 74 of RirER. T. XVI. Log sines, tangents, and secants for every quarter point to 5 places, and five-figure logarithms to 10,000. T. XVII. Log sines, tangents, and secants for every minute of the qua- drant to 5 places : arguments also in time (90°=twelve hours), and the com- plement to 1 2"^ given also. The other tables are nautical. On the titlepage it is stated that the tables are "corrected from many thousand errors of former publications ; " most of them doubtless only affect- ing the last figure by a unit. Bremiker, 1852. T. I. Six-figure logarithms to 1000, and from 10,000 to 100,010, with proportional parts ; with degrees, minutes, and seconds corresponding to every tenth number of seconds, and ten times each such number; the change in the line is denoted by a bar over the 3rd figure in all the logarithms affected. The table is followed by the first hundred multiples of the modulus •434 . . . and its reciprocal to 7 places. T. II. Log sines (left-hand pages) and tangents (right-hand pages) for every second to 5° to 6 places, and log sines and tangents for every ten seconds of the quadrant to 6 places, with differences, and proportional parts beyond 5°. This is followed by small tables giving the circular measure of 1°, 2° . . . 180°, 1', 2', . . . , 60', 1", 2". . . 60" to 6 places; and for the conversion of arc into time &c. The last page contains a few constants. There is an introduction of 82 pp., containing, among other things, an in- vestigation " De erroribus, quibus computationes logarithmicae afficiuntur." Nine errors in this work are pointed out by Prof. Wackerbarth in the ' Monthly Notices of the lloyal Astronomical Society " for April 1867. Bremiker's Vega, 1857. T. I. Seven-figure logarithms to 1000, and 90 KEl'OKT— 1873. from 10,000 to 100,000, with differences and all the proportional parts on the page. The change of figure in the line is denoted by a bar placed over the fourth figures of all the logarithms affected. S and T (see § 3, art. 13) are given at the bottom of the page, as also are the numbers of degrees, minutes, and seconds corresponding to every tenth number in the number-column of the table. At the end of this table is a table containing the first hundred multi])les of the modulus -434 . . . and its reciprocal 2-302 ... to 7 places. T. II. Log sines and tangents from 0° to 5° to every second, to seven places : no differences. At the end of this table is given a page of circular arcs, containing the circular measure of 1°, 2°, . . . 180°; 1', 2', . . . 60'; 1", 2 ', . . . 60" to seven places. T. III. Log sines and tangents for every ten seconds of the quadrant, to seven places, with differences : proportional parts are added after 5°. T. III. is followed by a page containing tables for the conversion of arc into time : the other tables are astronomical. On p. 547 are a few con- stants. The tables are stereotyped. An edition with an English Introduction, edited by Prof. W. L. F. Fischer, was published in 1857 (title in § 5) ; the contents are the same as in the above work, the tables being printed from the same plates. Bruhns, 1870. T. I. Seven-figure logarithms of numbers to 1000, and from 10,000 to 100,000, with differences, and all the proportional parts. The all is printed in italics, because in Eabbage, Callet, &c. only every other table of proportional parts near the beginning of the table is given, for want of space. In this work there is no inconvenient crowding, as even where the side-tables are very numerous, the type, though small, is still very clear. The constants y and T, for the calculation of sines and tangents (§ 3, art. 13), are added, and placed at the bottom of the page, as also are the numbers of degrees, minutes, and seconds in every tenth number of the number-column (regarded as that number of seconds), and the same for each of these numbers multi- pHed by 10. T. II. Log sines, cosines, tangents, and cotangents to every second from 0° to 6°, to seven places, with differences throughout, and. proportional parity, except in the portion of the table from 10' to 1° 20', where the size of the page would not admit of their insertion. T. III. Log sines, cosines, tangents, and cotangents from 6° to 45° to every ten seconds, to seven places, with differences and proportional parts. Of course room could not be found for the proportional parts of all the dif- ferences ; but throughout all the table on no page are there less than six proportional-part tables. On p. 186 tlie first hundred multiples of the modulus and its reciprocal are given, to ten places ; and at the end of the book are tables of circular arcs, viz. the circular measure of 1°, 2°, . . . 180°, 1', 2', . . . 60', 1", 2", . . . 60", to ten places, a page for the conversion of arc into time, and some constants. In T. I. the change in the line is denoted by a bar placed over the fourth figure of all the logarithms affected, the similar change when the third figure ie decreased being denoted in the other tables by an asterisk; a final 5 in- creased has a bar superscript. It is incorrectly stated in the preface that the practice of marking all the last figures that have been increased was intro- duced by ScHEON ; for this innovation was due to Babbage (see his preface, p. x). Dr. Bruhns may, however, merely mean that the mark (viz. a bar sub- script) introduced by Schron (1860) fatigues the eye and is of next to no use ; and if so, we entirely agree with him. In Babbage the increase is ON MATHEMATICAL TABLES. 91 denoted by a point subscript, which the reader scarcely notices ; but in Schron the bar catches the eye at once and is confusing. The cases also in which it is necessary to know whether the last figure (unless a 5) has been increased are excessively rare ; and in fact any one who wants such accuracy should use a ten-figure table. On the whole, this is one of the most convenient and complete (considering the number of proportional-part tables) logarithmic tables for the general com- puter that we have met with ; the figures have heads and tails ; and the pages are light and clear. Purther, we believe it is published at a low price. Byrne, 1849 (Practical . . . method of calculating &c.). [T. I.] Primes to 5000, pp. xiii and xiv. [T. II.] A very small table to convert degrees &c. into circular measure, p. XV. [T. III.] List of constants (69 in number), chiefly relating to tt (which Mr. Byrne denotes by jj), such as 2 n-, 36 tt, y^g- tt, tt^S, V"", &c. (pp. xviii to xxiii) : the value of n is inaccurate ; see § 3, art. 24. [T. IV.] Logarithms of numbers from unity to 222, to 50 places (pp. 77-82). Callet, 1853. [T. I.] Seven-figure logarithms to 1200, and from 10,200 to 108,000 (the last 8000 being to 8 places). Differences and proportional parts are added ; but near the beginniug of the table, where the differences change very rapidly, only the proportional parts of alternate differences are given, through want of room on the page (this is also done by Babe age and others). The constants S and T (see § 3, art. 13) for calculating the log sines and tangents of angles less than 3°, as also Y the variation for 10", are given in a line at the top of the page (see p. 113 of the Introduction). To the left of each number in the number-column are placed not only the degrees, minutes, &c. corresponding to that number of seconds, but also, in another column, those corresponding to ten times that number. When the change of figure occurs in the middle of the block of figures the line is broken — the best theoretical way of overcoming the difficulty. De Morgan and others, however, have expressed a strong dislike to it ; and we agree with them. [T. II.] I. Common and hyperbolic logarithms of numbers from 1 to 1200 to 20 places, the former being on the left and the latter on the right-hand pages. II. Common and hyperbolic logarithms of numbers from 101,000 to 101,179 to 20 places, with first, second, and third differences, the hyper- bolic logarithms being on the right-hand pages. (Note. AU the common ■logarithms from 101,143 to 101,179, with one exception, contain errors.) III. Common and hyperbolic antilogarithms from -OOOOl to -00179 at intervals of -00001, and from -000001 to -000179 at intervals of -000001, respectively, to 20 places, with first, second, and third differences. [T. III.] I. Common logarithms (to 61 places) and hyperbolic logarithms (to 48 places) of all numbers to 100, and of primes from 100 to 1097; and (II.) from 999,980 to 1,000,021 : the hj'-perbolic logarithms occupy the right- hand pages as before. [T. IV.] The first hundred multiples to 24 places, and the first ten mul- tiples to 70 places, of the modulus -434 . . . and its reciprocal 2-302 . . . [T. v.] Ratios of the lengths of degree &c. (ancient and modern) to the radius as unit, viz. the circular measure of 1°, 2°, . . . 100°, 1', 2', . . . 60', 1", 2", . . . 60", and of the corresponding quantities in the centesimal divi- sion of the right angle (1» . , . 100" ; 1^ . . 100' ; 1". . .100") to 25 places. [T. VI.] Log sines and tangents for minutes (centesimal) throughout the quadrant (to seven places), viz. from 0" to oO", at intervals of V, with differences. 9.2 REPORT— 1873. The order of the columns is sine, tangent, difference for sine, difference for tangent, cosine ; but this arrangement only holds up to 5", when differences are added for the cosine also. A change in the figiu-e at the toj) of the column is denoted in the column by a line subscript under all the figures of the firsf^^ logarithm affected, which arrests the eye at once. [T. VII.J Natural and log sines (to 15 places) for every 10' (ten minutes centesimal) of the quadrant. It is as well here to note that in the log sine and cosine columns only nine figures are given, as the preceding figures are olitainable from [T. VI. J ; two, however, are common to both : thus from [T. VI.] we find log sin 10'=7-1961197, and in [T. VII.] we have given, corresponding to log sin 10^ 969843372; so that log sin 10'=7-19611969 843372. It will therefore be noticed that the log sines are in strictness given to 14 (and not 15) places. Further, it appears that the last figure has not been, or at all events not been always, corrected; for log sin 50"= log -^ = -34948500216800940...., and the logarithm in [T. VII.] ends with the figures 6800. This is the only one we have examined. At the end of [T. VII.] is given a page of tables to connect decimals of a right angle with degrees, minutes, and seconds, ikc. [T. VIII.] consists of proportional-part tables, and occupies 10 pp. : by means of them any number less than 10,000 can be multiijlied by a single digit with great ease ; the use of this in interpolation is evident. A full explanation is given on pp. 32-36 of the Introduction to the work. [T. IX.] Log sines and tangents for every second of the first five degrees, to seven places, without differences (sexagesimal). [T. X.] Log sines and tangents for every ten seconds of the quadrant, to seven places, with differences (sexagesimal). [T. XL] Logistic logarithms, viz. log 3600" — log x" from x = 0" to a- = 5280" = 1° 28'; 3600" = 1°. The other tables have reference to Boi'da's method for the determination of the longitude at sea. On the whole, this is the most complete and practically useful collection of logarithms for the general computer that has been published. In one not very thick octavo volume, 11 important tables are given ; the type is very clear and distinct, though rather small. In the logarithms of numbers an attempt has been made to give rather too much on the page ; but for general usefulness this collection of tables is almost unique. The introduction, of 118 pp., is the worst portion of the work ; it is badlj' arranged, confused, and, worst of all, has no index ; so that it is very hard to find the explanation of any table required, if it is explained at all. On p. 112 the value of e is given; but the figures after the 8th group of five are erroneous, and should be 47093 69995 95749 66967 6 (see Erit. Assoc. Report, 1871, Transactions of Sections, p. 16). On pp. 12 and 13 of the introduction are two tables that deserve notice ; the first gives the square, 4th, IGth .... 2''°th roots of 10 to about 28 significant figures (leaving out of consideration the ciphers that follow the 1 in the higher powers). The second gives powers of '5 as far as the 60th. With regard to errors, an important list is given by Lefort in the ' Comptes Uendus,' vol. xliv. p. 1100 (1857) ; and these of course apply to the later iirages. Manj^ errors of importance, as also some information as to the sources whence CaUet derived his tables, are given. See also Gauss in Zach's ' Monatliche Correspondonz,' November 1802 (or 'Werke,' t. iii. p. 241), for four errata, and Gernerth's paper (referred to at the end of the introductory ON MATHJiMATIOAL TABLES. 93 remarks in §3, art. 13), and also Hutxon's tables (editions 1783-1822), Gernerth remarks (p. 25) that errors pointed out by Hutton in 1822 still re- mained uncorrected in the tirage of 1846. We may also refer to a paper by Herrmann, entitled " Verbesserung der II. Callet'schen Tafel der gemeinen Logarithmen mit 20 Decimalen, nebst Vorschliigen fiir die weitere Porde- rung dieses Zweckes," printed in the ' Sitzungsberichte der Kaiserlichen Akademie der Wissensdhaftcn,' Vienna, 1848, part ii. pp. 175-190. On p. liii of their work, Hobert and Ideler (1799) remark that they found that in general the natural sines of Callet were calculated accurately, but that in the log sines the last two figures were generally doubtful ; they mention also that they found many other faults in the work, but, being un- certain how far these are corrected in the stereotype edition, they only give one : viz., on p. 117 of the introduction, in the eighth place in the value of f there is a 2 for a 3 ; and this fault renders erroneous the multiples of /. A list of 380 errors is given on pp. 348 and 349 of the book, in all of which the error is + 1 in the last place, and also an error in a natural sine is given. The above error in /is corrected in the tirage of 1853. On p. 120 of EoRDA and Delambre there are given six eiTors in the ste- reotyped tables of Callet. A good many errors are also given at the end of Vega's Manual (1800). Many other errata are noted in other books ; but it seems useless to give references without at the same time examining whether the errors have been subsequently corrected, and, if so, in what tirages. Hobert and Ideler consider that Callet obtained his log sines most pro- bably by interpolation from the ' Trigonometria Artificialis ' of Vlacq. The number of tirages of this work has been very great : it was first published in 1783, we believe; but the type from which the earlier tirages were printed was subsequently reset, as the size of the page in the editions published in this century is larger than that of the first, which had tlierefore more right to the title " Tables portatives." The tirage we have described above is that of 1853 ; and we have seen one of 1862, " revue par J. Dupuis " (Dupuis was himself subsequently the editor of a set of logarithmic tables, described in this section). There is also a stiU more recent edition, edited by M. ISaigey. We have an impression that the Callet of 1793 was the first logarithmic table stereotyped ; but we have not investigated the matter. Coleman, 1846. T. XIX. Log sines, tangents, and secants to every quarter point, to 6 places. T. XX. Six-figure logarithms to 10,000, arranged in decades, with pro- portional parts above 1000. T. XXI. Logarithms for finding the apparent time or horary angle, viz. 1 • / , 1 — cos 07 \ „ log semi- versed sines f = log ^ ) ^1'°°^ ^ to 9^ at intervals of 5% to 5 places, with proportional parts. T. XXIII. Log sines, tangents, and secants for every minute of the quadrant, to 6 places. T. XXIV. Proportional logarithms for every second to 3° ; same as T. 74 of IIaper, only to 5 instead of 4 places. It must be observed that on the first page (extending to 10') the logarithms are not given completely, the last figure, two figures, or three figures being printed as ciphers. This is done, we presume, because in the cases to M'hich the table is intended to be applied accuracy in these places is not required. The same is done in several other copies of this table occurring in other nautical collections. Opposite is given 4 . 88 . . instead of — oo. The other tables are nautical. 94 REPORT — 1873. Croswell) 1791. T. I. Log secants, half log secants, and half log sines, viz. log sec X, ^ log sec x and ^ log sin x, to every minute of the quadrant, to seven places, the last two being separated by a comma for the convenience of those who outy requii-e five places ; semiqiiadrantally arranged : no differences. The table, as headed in the book, implies that the tabular results are natural ; but they are as above. T. V. Proportional logarithms for every second to 3°, to 4 places : the same as T. 74 of Rapek. T. XIll. Small table to convert arc into time. The other tables are nautical. De Decker, 1626. T. I. Ten-figure logarithms of numbers to 10,000, with characteristics and differences. T. II. Logarithmic sines and tangents, to seven decimals, for every minute, from GuNTER 1620 (§ 3, art. 15). These tables were always assigned to Vlacq till, in the course of the pre- paration of this Report, it came to light that De Decker was the author, Vlacq having only rendered some assistance. For the history of them, as well as for their connexion with ' Tables des Logarithmes pour les nombres d'un a 10,000 composes par Henry Brigge,' Gouda, 1626, and the tables in Wells's ' Sciographia,' 1635, see Phil. Mag., October and December (Supp. No.), 1872, and May, 1873. Degen, 1824. T. I. Log ^^ (1 . 2 . 3 . . . . .r) is given from .v = 1 to x= 1 200, to 18 places. The complement of the logarithms from 100 is also added if the characteristic be less than 100 — if not, the complement from 1000 or 10,000 ; thus log (1.2.... 69)= 98-233. . . . , and the complement is 1-766 . . . . ; log (1.2 70) =100-078 , and the complement is 899-921 The first portion of this table is reprinted by De Morgan, to 6 places, in the ' Ency- clopasdia Metropolitana ' (§ 3, art. 25). T. II. The first hundred multiples of the modulus -434 . . . , to 30 places. T. III. The first nine multiples of log 2, log 3, log 5, log 0, log 7, log 11, log 12, log 13, log 14, log 15, log 17, log 18, log 19, log 21, log 22, log 23, log 24, log 26, log 28, and log 29 (Briggian). The other tables consist of formulge &:c. There is a full introduction. [De Morgan] 1839. [T. I.] Five-figure logarithms to 10,000 (arranged consecutively, and not as in seven-figure tables), with differences, and degrees corresponding to the first number in each column. [T. II.] Logarithms from 1001 to 1100, to 7 places. [T. III.] Log sines, cosines, tangents, and cotangents to every minute, to 5 places, with differences. [T. IV.] Log sines for every second of the first nine minutes, and also for every tenth of a minute in the first degree. [T. v.] A small table of constants ; most of them taken from Babbage. [T. YL] Log (1 . 2. 3. . . .a'), from a:=6 to .r=25, at intervals of unity, and thence to 265, at intervals of 5, these last three tables being also to 5 places. The tables are beautifully printed, and are practically free from error. Prof. Wackerbarth states ('Monthly Notices of the Royal Astronomical Society,' April 1867) that he finds the only error in the work to be among the constants on p. 213, line 5, where 2-718281829 should be 2-718281828, the following figure being 4. There is no name on the titlepage ; but it is well known that the tables were prepared by De Morgan, and they are always spoken of by his name. They were examined by Mr. Farley of the Nautical- Almanac Office. 0\ MATHEMATICAL TABLES. 95 De Prasse, 1814. [T. I.] Five-figure logarithms of numbers to 339 (with characteristics), and thence to 10,000, arranged as is usual in seven- figure tables. When the fifth figiire has been increased it is printed in different type. The change in the line is denoted by an asterisk prefixed to the third figure of all the logarithms affected. [T. II.] Log sines and tangents for every minute to 5°, and thence for every ten minutes to 85°, when the intervals are again one minute to 90°, to 5 places. TV and e, and nine multiples of the modulus and its reciprocal are given on the last page. The price is one franc. A short review of this work, reprinted from the ' Gottingische gelehrte Anzeigen,' Dec. 19, 1814, wiU be found on p. 243 of t. iii. of Gauss's ' Werke.' On pp. 241-243 is also reprinted a review of the original edition (Leipzig), from the same ' Anzeigen ' for May 25, 1811. Dodson, 1747. T. XVII. Least divisors of numbers to 10,000 (mul- tiples of 2 and 5 omitted) . T. XVIII. Primes from 10,000 to 15,000. T. XIX. Square and cube roots (to 6 places) of numbers to 180. T. XX. Combinations up to the combination of 34 things, 29 together : a table of double entry. T. XXI. Powers of 2 to 2^" &c. T. XXII. The first 20 powers of the 9 digits. T. XXIII. Permutations, viz. 1 . 2. . . ..r, to a;=:30. T. XXV. Circular measure of 1°, 2°, 180° ; of 1', 2', 60' ; of 1" 60" ; and of 1'" .... 60"' : to 7 places. T. XXVI. Versed sines of arcs, and the areas of the segments included by those arcs and their chords to every 15' of the quadrant, to 7 places, with differences. • T. XXVII. The first 9 multiples of 12 constants (viz. ;r, - , -/ ^, &c ) ■K 4 47r '^' to 7 places. T. XXVIII. Table of polygons, giving any three of the four quantities, length of side, radius of inscribed circle, radius of circumscribed circle, area when the fourth is given=l, for polygons of less than 13 sides, to 7 places.' T. XXIX. Table of regular solids, giving any four of the five quantities, side, radius of circumscribed sphere, radius of inscribed sphere, superficies, solidity, when the fifth is given=l, to 7 places, for the 5 regular solids. T. XXXII. Seven-figure logarithms to 10,000, with differences. T, X XX III. Antilogarithms, viz. numbers to logarithms from -0001 to •9999 at intervals of -0001, to 7 places. T. XXXIV. Log sines and tangents for every minute of the quadrant, to 7 places, with differences ; but between 0° and 2° the difierences between the logarithms of the arcs and the logarithms of the sines and tangents of those arcs are given instead. T. XXXV. The number of seconds contained in any number of minutes less than 2°. T. XXXVI. Logistic logarithms, viz. log 3600'— log x from x=l to a^=4800^ ( = 80™) (argument expressed in minutes and seconds), to 4 places. T. XXXVII. Neper's hf/aritJims. The table, however, is really one to con- vert common into hyperbolic logarithms, and is in fact, when so regarded the first 1 000 multiples of the reciprocal of the modulus, viz. 2-302 . . . , to 6 places T. XXXVIII. Products to 9 x 9999. There are, besides, very many other tables of all kinds, astronomical com- mercial, &c. : we have described all the mathematical ones. ( 96 REPORT— 1873. Domkey 1852. T. XXX. Quadrate der Minuten des Stundenwinkels, viz. '^'"'"fioj froma;=l to cr=15, and from y = l to y=QO, to one decimal place; thus corresponding to 8' 20" the table has 69-4; for 8' 20" = 8^ = 8-33 . . . , and its square, retaining one decimal place, is 69-4. T. XXXII. Six-figure logarithms to 100, and from 1000 to 10,000, with differences : aU the logarithms written at fuU length. T. XXXIII. Log sines, tangents, and secants to eveiy quarter point, to 6 places. T. XXXIV. Log sines and tangents for every second, for the first two degrees, to 6 places : all the logarithms written at length. T. XXXV. Log sines, tangents, and secants, to every minute of the quadrant (arguments also expressed in time), with differences, arranged semi- quadrantally : all the logarithms written at length. T. XXXVI. Natural sines to every minute of the quadrant, to 6 places, arranged quadrantally. T. XXXVII. Logarithmen der halbverjlossenden Zeit, viz. log cosec x from .^=0'' to 07 = 3'' 59"" 55^ at intervals of 5', to 5 places, with proportional parts for seconds. T. XXXVIII. Logarithmen der Mittelzeit, viz. log 2 sin x, from ir=0'' to.t'=3'' 59" 55' at intervals of 5', to 5 places, with proportional parts for seconds. T. XXXIX. Logarithmen des Stundenwinkels, viz. log versed sine x, from x=0^ to x=7^59"' 55' at intervals of 5', to 5 places, with proportional parts for seconds. T. XL. Proportional logarithms for every second to 3°, to 4 places ; the same as T. 74 of Eaper. T. XLVII. and XL VIII. occupy one page, and are for the conversion of arc into time, and vice versa. The other tables are nautical. In all the tables the logarithms are written at full length ; the type is thin and very clear, the figures having heads and tails. T. XXX. was calculated from this work; T. XXXII., XXXIIL, and XXXV.-XL. were taken from Norie's ' Epitome of Navigation,' (they are Maskeltne's tables ; but see Bowditch, 1802, T. XIII.) and T. XXXIV. from Callet. On the accuracy of this work see the tract of Gernerth's referred to in § 3, art. 13 (p. 55). There was a second edition in 1855 (Gernerth). Donn, 1789. T. I. Seven-figure logarithms to 10,000, with differences. T. II. Log sines and cosecants to every quarter point, to 7 places. T. III. Log sines and tangents and natural sines for every minute of the quadrant, to 7 places. T. IV. Log I elap. time, mid time, and rising (see explanation of the terms under T. XVI. of Maskelxne, 1802), for every half minute to 6'', to 5 places. T. V. Log versed sines and natural tangents and secants for every 10' of the quadrant, to 4 places. The other tables are nautical. We have also ' The British Mariner's Assistant, containing forty Tables . . ' London, 1774, 8vo (352 pp. of tables), the tables of which are the same as those described above. Douglas, 1809. [T. I.] and T. I. Supplement, and T. II. Supplement. Logarithms of numbers to 10,999, and from 100,000 to 101,009, to 7 places (without differences). ON MATHEMATICAL TABLES. 97 [T. II.] Log sines, tangents, and secants for every minute of the quadrant^, to 7 places (without difl'ereuces). [T. III.] 2^atural sines, tangents, and secants for every minute of the quadrant, to 7 places (without differences). [T. IV.] Natural and log versed sines to every minute, from 0° to 180°, to 7 places (without differences). T. III. Supplement. Table to convert sexagesimals into decimals. It gives 1", 2", 4" . . . 58", 1', 1' 1", 1' 2", 1' 4". . . 1' 58", 2' ... 2' 58", &c. to 60', expressed as decimals of 60', to 4 places. T, IV. Supplement. Logarithms of numbers from 1 to 180, to 15 places. Ducom, 1820. T. VII. Proportional logarithms for every second to 3°, to 4 places ; same as T. 74 of IIapee. T. IX. Log sines and tangents for every second to 2° ; then follow log cosines and cotangents for every 10" to 2^; and then log sines, cosines, tangents, and cotangents from 2^ to 4-5°, at intervals of 10", to 6 places. Proportional parts are added for the portion where the intervals are 10". T. XIX. Natural sines for every minute of the quadrant, to 6 places. T. XX, Parties proportionnelles for interpolating when the tabular result is given for intervals of 24'', viz. g^*( (expressed in hours, minutes, and seconds), where x is 1™, 2™, . . . . 60™, and, in the first table, y is 1'', 2", 24\ and in the second 1"", 2™, 60'". T. XXI. Six-figure logarithms of numbers to 10,800, with corresponding minutes and seconds : logarithms printed at full length ; no differences. The other tables are nautical &c. The tables form the second part of the work. It may be noticed that, in the remarks on T. XIX. (p. xiv), the versed sine of x is erroneously defined as if it were 1 — sin .r. Dunn, 1784. [T. I.] Six -figure logarithms to 10,000. The arrangement is the same as is usual in seven-figure tables ; only instead of the numbers 0, 1, 2, . . . . 9 running along the top line, they are printed 0-00, 100, 2-00, .... 9-00, which gives the table the appearance of being arranged differently. [T. II.] Log sines, tangents, and secants to every minute of the quadrant, to 6 places. At the foot of each page is a small table, giving the differences (for the sine and tangent) for an interval of 60" in the middle of the page, and their proportional parts for 50", 40", 30", 20", 10", 9", 8", 7", 6", 5", 4", 3", 2", 1". At the end is a table of the differences of the log sines, tangents, and secants for every 10'. Dupuis, 1868. T. I. & II. Seven-figure logarithms from 1 to 1000, and from 10,000 to 100,000. Proportional parts to tenths, viz. multiples with the last figure separated by a comma, are added. (The separation of the last figure is an improvement on the simple multiples given in Sang, 1871, and others, as the table can be more readil}'- used by those accustomed only to proportional parts true to the nearest unit.) S and T (§ 3, art. 13) arc given at the bottom of the pages at intervals of 10". Dupuis states in the preface that his intention had been that the table should extend to 120,000, and that accordingly he had calculated the last 12,000 logarithms by differences, but at the request of a number of professors he stopped at 100,000. "\Vc venture to think he would have acted more wisely if he had not listened to the professors*; but the matter is of no consequence now, as Sang, 1871, extends to 200,000, * Several of tbe ordinary seven-figure tables (Babbage, Callet, Hulsse's Vega, and many otliers) extend to 108,000, and the last 8U00 logarithms are given to eight places. 1873. H 98 REPORT— 1873. T. III. Hyperbolic logarithms to 1000, to 7 places. T. IV. & V. First hiuiidred multiples of the modulus and its reciprocal, to 7 places. T. YI. & VII. Log sines and tangents for every second to 5°, to 7 places, ■with negative characteristics (viz. 10 not added). T. VIII. Log sines, tangents, cotangents, and cosines (arranged in this order) from 0° to 45° at intervals of 10", with negative characteristics, to 7 places ; with diiferences and proportional parts, as before, to tenths. T. IX. Circular measure of 1°, 2°, . . . , 180°, 1' . . . . 60', 1" . . . . 60", to 7 places. T. X. (reduction des parties de I'equateur en temps) ; hours and minutes (or minutes and seconds) of time in 1°, 2°, 360° (or 1'. .. . 300'), and seconds of time in 1", 2", .... 60", to 7 places ; then foUows an explanation of the use of the tables. This is the only work we can call to mind in which negative characteristics (with the — sign printed over the figure) are given throughout ; and to the mathematical computer such are preferable to the ordinary characteristics inei-eased by 10. Also the edges of the pages of T. VI.-VIII. are red (the rest being grey), which facilitates the use of the tables. It is curious that it never should have occurred to any editor or publisher of a collection of tables to colour the edges of the pages of the separate tables difi"erently, and print thereon also their titles, as is done with the different businesses &c. in the Loudon Post-OfRce Directory. Dupuis was also the editor of the 1862 edition of Callet ; and the titles of several small tables of logarithms that we have not seen are advertised in this work, viz, : — (1) an edition of Lalande's five-figure tables, with Gaussian logarithms added, &c. ; (2) an 18mo book of four-figure tables ; and (3) logarithmic and antilogarithmic tables to 4 places, for the use of physicists, giving log (1 -f at) for the calculation of dilatations &c. [Encke, 1828.] [T. I.] Four-figure logarithms to 100 (with characteris- tics and differences), and from 100 to 1009. [T. II.] Log sines, tangents, cotangents, and cosines for every 4' from 0° to 10°, and thence to 45° at intervals of 10', to 4 places, with dif- ferences. [T. III.] Gaussian logarithms ; B and C are to 4 places, for argument A, from A=-00 to 1-80 at intervals of -01, and thence to 4-0 at intervals of -1, with differences. Encke's name is written on the Royal Society's copy of these tables ; and they are also spoken of as Encke's by De Morgan. They are reprinted in 'WAENSTOEFi-'s ScHUMACHEE, 1845 (§ 4). Everett [1866]. Two cards (one of which, unfolded, is equal in size to three folio pages, the other, which is equal in size to one, being perforated), in a cover. This very frequently gives rise to errors, as the computer who is accustomed to tlu-ee leading figures common to the block of figures is liable to fail to notice that in this part of the table there are four ; and on this account a figure (the fourth) is sometimes omitted in taking out the logarithm. It is therefore often desirable to ignore the con- tinuation of the table and only use the portion below 100,000. The extra logarithms are thus not always an advantage ; and it is on the face of it inconvenient that some of the tabular results should be given to 7 and others to 8 places. When tables of logarithms are placed in the hands of common computers, it is as a rule better to forbid the use of the portion beyond 100,000 ; and it may have been some considerations of this nature that induced M. Dupuis to take this number as his limit. But there is no objection that we can see against giving the logarithms beyond 100,000 to 7 places (as in Sano, 1871) ; aad whenever this is done, the continuation is found very useful. ON MATHEMATICAL TABLES. 99 These cards correspond to the fixed and movable portions of a slide-rule 100 inches long. A few small tables of cube roots, sines, &c. are printed on one of the cards. Prof. Everett (to whom we applied for information with re- gard to the date of the table) gives the following brief description — " Two cards, one of them cut like a grating, equivalent to the two pieces of a slide- rule;" and adds "that in the first edition [which is the one we have described] one of the cards had a pair of folding leaves attached to it, but these merely contained subsidiary tables and directions, and were quite unessential. In the next impression the two essential cards and the two cards with subsidiary tables and directions were all detached from each other." A description of the table is given in the Phil. Mag. for November 18G6. Parley, 1840. [T. I.] Six-figure logarithms to 10,000 (the line is broken when the change occurs in the third figure) ; followed by the loga- rithms of numbers from 1001 to 1200, to 7 places. [T. II.] Log sines and tangents for every minute of the quadrant, to 6 places, with difi^erences for 100". [T. III.] Log sines from 0° to 2° at intervals of 6". There are also a few constants and some formulae. Parley, 1856. This very fine table of versed sines contains : — [T. I.] Natural versed sines from 0° to 125° at intervals of 10", to 7 places, with proportional parts throughout. [T. II.] Log versed sines from 0° to 135° at intervals of 15", to 7 places, with difi"erences throughout. The arguments are also given in time, the range being from 0*" to 9^ to every second. A short preface by Mr. Hind states that the table was prepared by Mr. Farley, of the Nautical- Almanac Office, in 1831, and the manuscript pre- sented by him to Lieut. Stratford, the then superintendent. The manuscript having been in use for 25 years, and having become dilapidated, it was *' deemed the most economical course to print it." It is added that the last figure cannot be relied on, though it is probably very rarely in error by more than a unit. These, the most complete tables of versed sines we have seen, are beauti- fully printed, in the same type as the Nautical Almanac. Faulliaber, 1G30 (' Ingeuieurs-Schul '). The copy we have seen of this book (viz. that in the British Museum) contains no logarithms, though it must evidently have been intended to accompany some tables. In the Brit- Mug. copy the work is bound up (in a volume containing four tracts) after the two described below and attributed by us to Faulhaber. Murhard gives the full titles of this work and of the next two, and marks them as having come under his eye ; he does not, however, assign the two tables to Faulhaber. Hogg, who also gives the titles of the three works, attributes them all to Faul- haber. He adds, speaking of the tables, that they are also contained in the ' Ingenicurs-SchuL' This is no doubt correct; for, as noted below, some errors in the latter work are given at the end of the Canon. It seems therefore certain that Faulhaber was the editor of the tables. It may be mentioned that both Eogg and Murhard agree in describing the ' Logarithmi ' and the ' Canon ' as parts of the same work, so that most likely they were never issued separately. Hogg gives the date of the ' Ingenieurs-Schur as 1731, which must be "a misprint for 1631; the copy before us is dated 1630, agree- ing with Murhard. A lengthy account of Faulhaber and his works wiU be found in Kiistner's ' Geschichte.' Sec also Schcibel, ' Math. Biicherk.' B. 2. p. 39. k2 100 REPORT — 18r3. [Faulhaber] 1631 ('Logarithmi'). Seven-figure logarithms of numbers from 1 to 10,000, arranged in columns (three to the page), with charac- teristics, xis there arc 3 coUimns, there are 99 logarithms on each page. The printing is imperfect, the types having here and there become displaced, so as to leave no mark. There are some errata on the last page, headed " Typographus Lectori S." See above, Faulhaber, 1630 (' lugenieurs- Schul'). [Faulhaber] 1631 ('Canon'). Logarithmic sines, tangents, and secants for every minute of the quadrant, to 10 places (semiquadrantally arranged); no differences. Taken from Vlacq, 1628. The table is followed by 8 pages of errata in the Frankfort 'lugenieurs-Schul,' in the logarithms of numbers, and in the ' Canon.' Except perhaps Norwood, ] 631, this is the first reprint of Vlacq's corrected ' Canon ' (1628), the previous writers haviug copied GuNTER (1620). Rogg gives place and date as Nuremberg, 1637 ; but the copy before us is not so. See above, Faulhaber, 1630 (' Ingenieurs- Schul'). Filipowski, 1849. T. I. Antilogarithms. The numbers (to 7 figures) are given answering to the logarithms as arguments, the range being frona •00000 to 1-00000 at intervals of -00001. The arrangement is exactly the same as in ordinary seven-figure tables of logarithms ; and the table occupies 201 pages. The proportional parts are given to hundredths (viz. 100 pro- portional parts of each difference are given); and the change of figure in tho middle of the line is denoted by two dots (thus, 0) placed over the fourth figure of all numbers affected ; and when a final 5 has been increased it is printed Y. The first 3 figures in the number arc alwaj-s separated by a space from the block of figures. . T. II. Gaussian logarithms, arranged in a new way. Let A=log x and X=log (.f-l- l)(so that 10^ =10^^ + 1), then on the first page of the table (p. 203 of the book) we have A given to 3 places for argument \ from \= -00000 to •00449 (which last corresponds to A = 8-017), at intervals of -00001. On the succeeding 16 pages we have ,\ as a tabular result for argument A from A= 8-000 to 13-999, at intervals of -001, to 5 places. Since log (a + 6) =log b-\- log | ,- -|- 1 j, and log {a—b)=log 6-1- log (J-i), it is clear that the rules are very simple and uniform, viz. log a and log b being given (6 < « suppose), we take log « — log 6 as argument, and enter the table at the A or \ column, according as we want log a-\-b or log a — b, and add the tabular result to log b. In this table also the notations 0, V, &c. are used, as well as another in which a wavy line runs down by the side of the logarithms whose leading figures have changed. This method of marking is only possible when the tabular results appear one under the other. The figures are throughout neat and clear, having heads and tails ; and the copy before us is printed on green paper^ of a pleasant colour. In many places there is a parsimony of figures, which we dislike extremely ; thus there occur 44, 5, 6 as headings for 44, 45, 46, and or for 10 &c. A list of 36 errors affecting the first 8 figures of Dodson"s Canon (1742) is given, and in- troduced by the remark, " The following is a list of errors as detected, by means of our table, in the first 8 places of Dodson's Anti-Logarithmic Canon, in addition to those corrected v.ith tlie author's own hand." These words im- ON MATHEMATICAL TAULKS. 101 ply that Mr. Filipowski's table was the result of an independent calculation ; or at all events they ought not to have been ■written unless such had been the case. It is, however, nowhere stated in the preface that the table was calculated anew ; and we may therefore assume that it was copied from Dodwn, after examination (which would not have been difficult, as a mere verification by differences would have sufficed). In a letter by Mr. Peter Gray, in the ' Insurance Record ' for Juue 9, 1871, there are given two errors in Dodson which also occur in Filipowski, affording additional evidence that the tables of the latter were not calculated independently ; and, this being so, Dodson has not been treated fairly, as Mr. Pilipowski should have acknowledged the obligations he was under to his table. In the same letter Mr. Gray gives three other errors in FUipowski (1st edit.) ; and it is to be in- ferred from other passages in the letter that a second and a third edition, *' corrected," have been published. Mr. Gray proceeds : — " but he [Fili- powski] has never, so far as I know, given a list of the errors contained in the first and second, and corrected in the third," an omission on which he strongly (and most justly) animadverts. See Shortrede (1849). De Morgan has stated that no antilogarithmic table was published from Dodson (1742) till 1849 ; but this is only true if Shortrede's tables of 1844 be ignored ; for which there is no sufficient reason, as thej' were published and sold in that year, and copies of the 1844 edition are contained in all good libraries. Galbraith, 1827. T. II. Six-figure logarithms of numbers to 10,000, with proportional parts on the left-hand side of the page. This table is headed " Logarithms of numbers to 100,000." T. IV. Log sines, tangents, and secants to every quarter point, to 6 places. T. V. Log sines, tangents, and secants to every minute of the quadrant (arguments expressed also in time, the intervals being 4^), with differences, to 6 places. T. VI. N'atural sines, tangents, secants, and versed sines to every degree of the quadrant, to 6 places. T. IX. Diurnal logarithms : proportional logarithms for every minute to 24'' (viz. log 1440— log a;) from x—1 to .r=1440 (expressed in hours and minutes), to 5 places. T. X. Proportional logarithms for every second to 3°, to 5 places. Same as T. 74 of Paper, except that 5 instead of 4 places are given. T. LXIII. A few constants. The other tables are nautical. There are a few small tables in the introduction that may be noticed, viz. : — T. XI. and XII. (p. 113), to express hours as decimals of a day, convert lime into arc, &c. ; T. XV. (p. 141), of the areas of circular segments (same as in T. XIII. of Hantscul, but to hundredths only, and to 5 places) ; and T. XVI., table of polygons (as far as a dodecagon), giving area, and radius of circumscribing circle for side=unity, and factors for sides, viz. length of side for radius = unity ; there are also one or two small tables for the mensuration of solids. Galbraith and Haughton, 1860. [T. I.] Five-figure logarithms to 1000, arranged in columns. This is followed by a small table to convert common into hyperbolic logarithms, and vice versa. [T. II.] Five-figure logarithms from 1000 to 10,000, M'ith proportional parts. [T. III.] Log sines and tangents to every minute of the quadrant, to 5 places, with differences. [T. IV.] Gaussian logarithms. B and C arc given for argument A, from 102 REroiiT — 1873. A = -OpO to A=2-000 at intervals of -001, thence to 3-40 at intervals of -01 and to' 5 at intervals of -l to 5 places, with differences. This table is followed by a page of constants. Gardiner, 1742. [T. I.] Seven-figure logarithms to 1000, and from 10,000 to 100,100, with proportional parts ; the change of the fourth figure in the line is not marked ; the first three figures of the logarithm are sepa- rated from the block of figures bj' a point, which is very clear. [T. II.] Log sines to every second to 1' 12", to 7 places, without diflTer- enees; and log sines and tangents throughout the quadrant at intervals of 10", to 7 places, with differences. [T. III.] Four-figure logistic logarithms, viz. log. 3600"— log x from .t'=0 to .r=4800" (=80') at intervals of 1". [T. IV.] Twenty-figure logarithms to 1000, thence of odd numbers to 1069, and of primes &c. to 1143. [T. v.] Twenty-figure logarithms of numbers from 101,000 to 101,139, with first, second, and third differences. [T. VI.] Anti-logarithms, viz. numbers to logarithms from -00000 to •00139 at intervals of -00001, to 20 places, with first, second, and third dif- ferences. A list of errata is given in the Prench reprint described below ; and 69 errors are pointed out by Hution on p. 342 of the edition of 1794 (and no doubt in other editions) of his mathematical tables. The list given in the edition of 1822 (the last published in Hutton'e lifetime) is much fuller. De Morgan speaks of Gardiner as "rare, and much esteemed for accuracy;" and its rarity in 1770 is the reason assigned by the French editors for the neces- isity of reprinting it. Gardiner (Avignon Eepriut, 1770). The reprint is so similar to the ori- ginal edition that it is only necessary to point out the differences. [T. I.] is the same ; but in [T. II.] the log sines are given at intervals of 1" as far as 4°, and a similar table of log tangents is added ; they were taken from a manuscript calculated by ilouton, bequeathed by him to the Academy of Sciences, and lent to the editors by Lalande. Also in the original edition, in the second portion of this table, viz. that giving the functions at intervals of 10", the parts common to both are repeated ; but this is not done in the reprint, in which therefore there is a table of log cosines and cotangents only, from 0° to 4°, at intervals of 10", the sines and tangents being given in the previous portion. [T. III., v., and VI.] are unaltered ; but [T. IV.] proceeds by odd numbers to 1161. One fresh table is added, viz. [T. VII.], giving hyperbolic loga- rithms from 1-00 to 10-00 at intervals of -01, to 7places, and also log^ 10", . . . 10'. Mouton's manuscript also gave log cotangents and cosines to every second of the first four degrees ; but the former are so easUy deducible from the tan- gents, and the latter vary so slowly, that their publication in e.vtenso seemed uu- ueccssarj\ A page of errata at the end of the book contains errors in Vlacq (1628), in Gardiner (1742), and in the French reprint itself (1770), the last having been published in the ' Connaissance des Temps ' for 1775. As the ' Connaissance des Temps' could not have been published as much as five years in advance, it is clear either that some copies of the French reprint were published subsequently to 1770, although retaining that date on the titlcpage, or that this page was circulated separately and bound up afterwards with the work. We have examined two copies, in one only of which this errata-pago appears. No editors' names aj»pcar ia the work i but Lalande (Bibliog.Astron. p. 516) ON MATHEMATICAL TABLES. 103 says that this edition was edited by Pere Pczenas, Pcre Dumas, and Pero Elauchard, and adds that ho has given an errata-list in the ' Connaissanco des Temps ' for 1775. On Dumas, mathematician of Lyons, who was La- lande's first master, he gives a reference to the ' Journal des Savants,' No- vember 1770. Tlie edition is very commonly known by the name of Pezonas. A good deal about Pezenas will be found in Delambre's ' Histoire de I'Astronomic,' pp. 368-386. He was born at Avignon in 1692, and died in 1770. The French edition is even better printed than the original, but is not quite so accurate. A list of 85 errors is given by Hutton on p. 343 of his mathematical tables in the edition of 1794, while he discovered only 69 in the original edition; more complete lists are to be found in the later editions. Graesso (' Tresor') says that there was a reprint of Gardiner in octavo at Florence by Canovai and Ricco. *Gardiner (Paris edition, 1773). Hogg gives the title of a Paris edition of Gardiner, viz. 'Tables des Logarithmes de Gardiner, foL, Par. Chez Sail- lard et Nyon, 1773,' which he takes from the * Journal litterairo do Berlin,' t. vii. p. 318 ; but the fact that Lalande does not mention it seems to him very suspicious : we have seen no other reference to it, and agree with Hogg. Garrard, 1789. This work contains only traverse and meridional part tables. It is referred to here, as its title would imply that it was included in the subject of the Report. Gordon, 1849. T. IX. Log sinea, tangents, and cosecants for every minute from 6° to 90°, to 4 places. T. X. Proportional logarithms for every second to 3°, to 4 places : same as T. 74 of Rapbr. • . T. XI. Small table to convert space into time. T. XVII. Half-sines and half-cosines, viz. halves of natural sines for cvcrj' minute of the quadrant to four places, reckoned as seconds for the purpose of adapting them to the table of proportional logarithms : thus, cor- responding to 12° 40' we find as tabular result 18' 16" ; for the number of seconds in this anglc = 1096, and i sin 12° 40'=-1096 . . . T. XVIII. Logarithms of the meridian distance, viz. log (|- vers sin x), from .^=0'' to x=7^ 59'" 55^ at intervals of 5% to 4 places. T. XIX. Proportional logarithms for every minute to 24'', viz. log 1440 — log.r from x=l to a'=1440, to 4 places (arguments expressed in hoiu'S and minutes). T. XXI. Proportional logarithms for one hour, viz. log 3600— log ,v from cc=l to .r=3600, to 4 places (arguments expressed in minutes and seconds). The other tables are nautical. Gregory, Woolhouse, and Hann, 1843. T. VIII. Proportional logarithms for every second to 3°, to 4 places ; same as T. 74 of Rapek. T. IX. Log sines, tangents, and secants for every minute of the quadrant, to 5 places. T. X. Natural sines to every minute of the quadrant, to 5 places. T. XI. Five-figure logarithms from 1000 to 10,000, with proportional parts. T. XII. Proportional logarithms for every minute to 24'', to 4 places, viz. log 1440— logo? from a;=l to 1440 at intervals of unity (arguments ex- pressed in houi's and minutes). The other tables are nautical. ]0i REPORT — 1873. Griffixi, 1843. T. 16. Log sines, tangents, and secants to every quarter point, to 6 places. ■ T. 17. Six-figure logarithms of numliers to 100, and from 1000 to 10,000, to 6 i:»laccs, witli ditferences. T. 18. Log sines, tangents, and secants to every minute of the quadrant (arguments expressed also in time), to 6 places, with differences for the sines and tangents ; arranged semiquadrantally. T. 19. Natural sines to every minute of the quadrant, to 6 places, without differences. T. 41. Proportional logarithms to every second to 3°, to 4 places ; same as T. 74 of Raper. The logarithms are in all the tables printed at full length. The other tables are nautical. Gruson, 1832. T.I. Seven-figure logarithms to 10,000 : no differences. The change in the line is marked by a difference of type in all the logarithms affected. In three or four parts of the book this table is stated to extend to 10,100, but the limit is as above ; and there is no possibility of a page having been torn out, as the next table is printed on the back of the page ending with the number 9999. T. II. & III. Squares and cubes of all numbers from 1 to 1000. T. IV. & V. Square and cube roots of all numbers from 1 to 1000, to 7 l)laces. T. VI. Circular measure of 1°, 2°, 3° . . . 360°, of 1', 2', . . . 60', and of 1", 2", . . . 60", to 7 places. T. VII. Natural and log sines, cosines, tangents, cotangents, secants, and cosecants, to 7 places, with differences from 0° to 5° at intervals of 1', and thence to 45° at intervals of 10'. The book was intended for schools. Hantschl, 1827. T. I. Five-figure logarithms (written at full length) of numbers from 1000 to 10,000. T. II. Log sines for every 10 seconds from 0° to 90°, to 6 places. T. III. Log tangents for every 10 seconds from 0° to 90°, to 6 places. T. IV. Ten-figure logarithms of primes to 15,391. T. V. Natural sines, tangents, secants, and versed sines for every minute of the quadrant, to 7 places ; arranged semiquadrantally. T. VI. Hyperbolic logarithms of numbers to 11,273, to 8 places. T. VII. Least divisors of numbers to 18,277 (multiples of 2, 3, 5, and 11 excluded). T. VIII. Squares, cubes, square and cube roots (to 7 places) to 1200. T. IX. <^-^), . . . ^(^-^^y--0^-5) from n=0 to n=l-00 at intervals of "01, to 7 places. T. X. Circular measure of 1°, 2°, 3°, ... 180°, of 1', 2' . . . 60', and of 1", 2" . . . 60", to 15 places. T. XI. The first nine multiples of ""' I' I' i' h S' K-)'' W' ""^ (e)"'' to 24 or 21 places. T. XII. Small table to express minutes and seconds as decimals of a degree. T. XIII. Areas of segments of circles for diameter unity to 6 places : the ox MATHEMATICAL TABLES. 105 versed sines are the arguments ; and the table proceeds from '001 to -500 (of the diameter). The table may therefore be described as giving ^(2^— sin 2d) from ^(1— cos 0) = -001 to -500 at intervals of -001. A few constants are then given to a great many places ; and the last page (T. XIV.) is for the calculation of logarithms to 20 places. The work is clearly printed. Hartig, 1829. The tables are of so commercial a kind that only one or two deserve notice here. The first (T. I.) is for computing the contents of planks &c., the thickness and breadth being given in Zolle and the length in Fusse, and may be described as a sort of duodecimal table, as the Kubik-ZoU =; J^ Kubik-Fuss, and the Kubik-Linie = J^ Kubik-ZoU. Thus for arguments 3 Zoll, 13 ZoU, and 5Fusswehave 1 F. 4 Z. 3 L. as result; ioT j\x\-lx5=\^^=l + -Jj + j^^. The arguments are : — (thickness) 1 ZoU to 9 Zoll at intervals of i Zoll ; (breadth) 1 Zoll to 18 Zoll at intervals of 1 Zoll; (length) 1 Fuss to 60 Fuss at intervals of 1 Fuss. Another table (T. II.) is of the same kind, only intended for blocks &c. ; BO that the thickness is greater, and the result is only given in fractions of a Kubik-Fuss. T. III. contains volumes of cylinders for diameter (or circumference) of seciion and length as arguments ; expressed as in T. I. and II. The money- tables can have no mathematical value, as the Thaler = 30, 24, or 90 Groschen, &c. T. X. is for the calculation of interest. The simple-interest tables (T. A) are too meagre to be worth description. T. B and C may be described as giving the compound interest and present value of £1 for any number of years up to 100 at 3, 4, 5, and 6 per cent, per annum, viz. (i + mj -* (i + m)' to 6 decimal places. Other tables of this kind that we met with have not been noticed ; the title of one such is given under Jahn, 1837. Hassler, 1830. [T. I.] Seven-figure logarithms of numbers from 10,000 to 100,000, with proportional parts. The line is broken for the change in the third figiire, as in Callet. [T. II.] Log sines and tangents for every second of the first degree, to 7 places. [T. III.] Log cosines and cotangents for every 30" of the first degree, to 7 places,, with differences. [T. IV.] Log sines, cosines, tangents, and cotangents, from 1° to 3°, at intervals of 10", with difierences, and from 3° to 45°, at intervals of 30", with differences for 1 0", to 7 places. [T. v.] Natural sines for every 30" of the quadrant, with differences for 10", to 7 places. Copies of this book were published with Latin, English, French, German, and Spanish introductions and titlepages (the titles will be found in the list at the end of the Eeport). The tables are the same in all ; and the special titlepages for each table have the headings in the five languages. The Eoyal Society's library contains the Latin copy perfect, and the introduc- tions in the four modern languages boimd together in another volume, pre- sented to the Society by the author. At the end of the latter volume is pasted-in a specimen page of the table, set up with the usual even figures ; 106 REPORT — 1873. and the author has written on the back, " This sheet proves that, with the usual form of figures of the same size as those used in the tables, they woidd not have been distinctly legible." The figures actually used are very thin, and have large heads and tails, resembling somewhat figures made in writing ; and a comparison of the specimen and a page of tho tables shows very clearly the superiority of the latter in point of distinctness. The words in minima forma are quite justified, as we do not think it would be possible to make the tables occupy less room without serious loss of clearness. All that is usually given in a page of seven-figure logarithms is here contained in a space about 3 in. by 5 in. ; and yet, owing to tho shape of the figures, every thing is very distinct. The author says on the titlepage, " purcjaUe ah erroribus prceccdeniium tabulariim ;" but the last figure of log 52943 is ])rinted 6 instead of 5. There is also another last-figure error. Sec ' Monthly Notices of the Eoy. Ast. Soc.,' March 1873. A short I'eview of this work by Gauss appeared in the ' Gottingische ge- lehrte Anzcigcn,' March 31, 1831 (reprinted ' Werke,' t. iii. p. 255). Henrion, 1026. [T. I.] Logarithms to 20,001, to 10 places, with interscript differences (characteristics not separated from the mantissa)), copied from Briggs, 1624, [T. II.] Log sines and tangents for every minute, to 7 places (charac- teristics unscparated from the mantissae), taken from Guntee, 1620. Hen- BION had calculated some logarithms himself when he received Beiggs's work (see PhU. Mag., Supp. No. Dec. 1872). The copy of Heneion wo have seen is in tho Brit. Mus. The full titlepage is given in § 5. Heutschen (Vlacq), 1757. [T. I.] Natural sines, tangents, and secants, and log sines and tangents to eveiy minute, to 7 places (arranged on what De Morgan calls the GeUibrand model) (180 pp.), and [T. II.J logarithms of numbers to 10,000, to 7 places, arranged in columns (lOO pp.). A former edition of 1748 is spoken of in the preface ; and it is stated that the tables were compared with the editions of Vlacq, Leydeu, 1051, the Hague, 1665, and Amsterdam, 1673. The type is very bold and clear, much easier to read than in most modern tables. This is one of the numerous series of small tables known by the name of Vlacq, and is described here because it is not mentioned by De Morgan ; small editions hke the present are so difficult to meet with that it is desirable to notice them whciiever any are found. Hobert and Ideler, 1799. [T. I.] Natural and log sines, cosines, tan- gents, and cotangents for the quadrant, divided centesimally; viz. these func- tions are given for arguments from -00001 to -03000 of a right angle at in- tervals of -00001 of a right angle, and from -0300 to -5000 of a right angle at intervals of -0001, to 7 places, with differences. Expressed in grades (cen- tesimal degrees) &c., the arguments proceed to 3" at intervals of 10", and thence to 50" at intervals of V. The manner of calculation of the table is fully explained in the introduction ; and this adds much to the value of the work. Several of the fundamenta were calculated to a great many places. Two or three constants are given on p. 310. B. Table of natural sines and tangents for the first hundred ten-thousandths (viz. for -0001, -0002 &c.) of a right angle, to 10 places. C. Four tables, expressing (I.) 1°, 2°, 3°,. . . .89°, (II.) 1', 2',. . . .59', (111.) 1", 2",. . . .59", (IV.) V", 2'",. . . .59'", aU as decimals of 90°, to 14 places. D. Three tables to express (I.) hundredths, (II.) thousandths, (III.) ten- thousandths of 90°, in degrees, minutes, and seconds (sexagesimal). ON MATHEMATICAL TAULES. 107 E. Four tables to express (I,) hours, (II.) minutes, (III.) sccouds, (IV.) thirds, as decimals of a day. F. Small table to express decimals of a day, in hours, minutes, and seconds. G. Circular measure of "1, -2, . . . . '9, 1-0, of a right angle, to 44 places. [T. III.] Logarithms of numbers to 1100, and from 999,980 to 1,000,021, to 36 places. The work concludes with two remarkable lists of errata found in the course of the calculations, viz. 381 errors in the trigonometrical tables of Callet, all of which, with one exception, affect only the last figure by a unit, and 138 similar errors in Vega's ' Thesaurus,' 1794, The errors in CaUet have, we presume, been corrected in the later tirages. Houel, 1858. T. I. Five-figure logarithms of numbers to 10,800 with the corresponding degrees, minutes and seconds, and proportional parts. The constants S and T (see § 3, art. 13) arc given at the top of the page ; then follows a page of small tables for the conversion of degrees, minutes, &c. T. II. Natural and log sines, tangents, and secants to every minute of the quadrant, to 5 places, with jiroportional parts. T. III. Gaussian logarithms. The addition and subtraction tables are sepa- rated, as in Zech (§4). In the first B is given for argument A, from A='000 to 1-650 at intervals of -001, thence to 3-00 at intervals of -Ol, and thence to 5-0 at intervals of -1. In the second B is given for argument C, from C=-3000 to -4800 at intervals of -0001, thence to 1-500 at intervals of -001, thence to 3-10 at intervals of -01, and to 5-0 at intervals of 4, with pro- portional parts : all to 5 places. These tables are followed by the first hun- dred multiples of the modulus and its reciprocal, to 8 places. T. IV. Tables to calculate logarithms to 8 places &c. T. V. (one page). To calculate logarithms to 20 places. T. VI. A page of four-figure logarithms to 600, and of three-figure anti- logarithms. T. VII. Least factors of composite numbers (not divisible by 2, 3, 5, or 11) up to 10,841. T. VIII. A page of constants. [We have since obtained a " nouvelle edition, revue et augmentee," Paris, 1871, pp. 118 and introduction xlvi.] Hiilsse^s Vega, 1840. T. I. Seven-figure logarithms to 1000, and from 10,000 to 108,000, with proportional parts ; the change in the line is denoted by a small asterisk prefixed to the fourth figure of all the logarithms affected. The portion from 100,000 to 108,000 is given to 8 places. One page at the end is devoted to a small table to convert common into hyperbolic seven- figure logarithms, and vice versa. T. II. Log sines, tangents, and arcs (all equal) to every tenth of a second to 1' ; log sines and tangents from 0° 0' to 1° 32' to every second ; log sines, cosines, tangents and cotangents for every ten seconds from 0° to 6°, and for every minute to 45° : all to 7 places. AVhen the intervals arc 10" or 1', differences for. a second are added: the logarithms are written at length. The table is followed by a page giving the circular measure of 1°, 2°, . . . . 10°, and thenco by tens to 360°, of 1', 2', . . , . 60', and of 1", 2", .... 60", to 11 places. T. III. Natural sines and tangents to every minute of the quadrant, to 7 places, with differences for 1". T. IV. Chord-table to radius 500, viz. lengths of semichords of arcs from 0° to 125° at intervals of 5', to 6 laces, for radius unity. \i.e. sm -, I 108 REPORT— 1873. This tabic is followed by 2 pages of tables for the conversion of centesimals into sexagesimals &c. T. V. All prime divisors of numbers to 102,000 (multiples of 2, 3, and 5 excluded), and primes from 102,000 to 400,313. T. VI. Hyperbolic logarithms of numbers to 1000, and of primes from 1000 to 10,000, to 8 places. This is followed by powers of 2, 3, and 5 to the 45th, 36th, and 27tb respectively. T. VII. Powers of e and their logarithms, viz. e' and log ^^e^, from .r=*01 to .r=10 at intervals of "Ol, to 7 figures and 7 places respectively. T. VIII. Square and cube roots of numbers to 10,000, to 12 and 7 places respectivelj''. The table is followed by a page of coefficients, such as ^ — ~\ 1 1.3 5 — T~7;) 9~~i — -■> <i^c., to 10 places, and their logarithms to 7 places. T. IX. Power-tables. A, the first 11 powers of numbers from -01 to 1*00 at intervals of -Ol, to 8 places. B, the first 9 powers of numbers from 1 to 100. C, squares and cubes from 1 to 1000. D, the first hundred powers of I'Ol, 1-02, 1-025, 1-0275, 1-03, 1-0325, 1-035, 1-0375, 1-04, 1-045, 1-05, 1-06, to 6 places. E, the first hundred powers of the reciprocals of these numbers, to 7 places, r, the sums of the powers in D : this table therefore gives x-\-x'^ ■\- . . . ..r" (-fe?) for the values of x written down under D, and for n = 1, 2, 3, . 100. G stands in the same relation to E thatF does to D. The tables from D to G Avere calculated for their use in computing interest &c. T. XII. An extended table of Gaussian logarithms. It gives B from A = •000 to A= 2-000 at intervals of -001, from A = 2-00 to A = 3-39 at intervals of •01, and thence to A = 5-0 at intervals of -1, to 5 places. There are also given, be- sides, other quantities for the same arguments, viz. C (=:A + B), D (=B4-C), E ( = A-|-C), and F (=B— A), all to 5 places, with difierences and propor- tional parts (of two kinds) for B and C. ^( ^ W Cc(oO^—\'\ ( X Of T. XIII. Interpolation table, viz. '-^^-^ — -, . . ^— — , o ft > ^^°™ a; = '01 to .r=l-00 at intervals of -01, to 7 places ; then follows a page of constants. There are, besides, mortality tables, very complete tables of mea- sures and weights of different countries, &c. The table of 12-place square roots was published' here for the first time : it was calculated by Hensel in 1804. The seven-place cube roots, the chord-table, and the new columns of the Gaussian table were calculated by Dr. Michaelis, of Leipzig. The author draws attention to the fact that the last figures in T. VIII. and XII. are given correctly. It is a matter of sufficient interestto note here that, though the work is called an edition of Vega, it contains one error from which the other tables known by the name of Vega and published subsequently to his folio of 1794 were free. In Vlacq (1628), log 52943 was printed 7238085868 instead of 7238085468, and the error was first pointed out and corrected by Vega in his folio of 1794. All the seven-figure tables, therefore, from 1628 to 1794 (and several of the subsequent tables also), have 7238086 instead of 7238085 ; but Vega's smaU editions (the ' Manuale ' and ' Tabuloe ') have the logarithms correctly printed. In Hulsse's edition, however, the error is reproduced afresh, and the last figure is printed 6. It follows therefore either that Hiilsse did not reprint Vega's table, or that, if he did, he noticed the discrepancy, and decided in favour of tlie erroneous value. The slight suspicion thus cast on these tables is unfor- ON MATHEMATICAL TABLES, 109 tunate, as they form a most valuable collection, and are supplemental to Callex. Wc have seen advertised a second edition (1849) ; and Zkch's tables (see Zech, 1849, § 3, art. 19) are extracted from it. The last-figure error noticed above is the only one of the hereditary Vlacq's errors that appears in the table of tlie logarithms of numbers ; so that but for this curious blunder the present work would have been, we believe, the first to be free from errors of this class (see ' Monthly Notices of the Roy. Ast. Soc' March, 1873). Some remarks by Gauss on T. XII. appear in t. iii. pp. 255-257 of his ' Werke.' Hutton, 1781 (products and powers of numbers). [T. I.] Products to 1000 X 100 (pp. 51). [T. II.] Squares and cubes of numbers from 1 to 10,000 (pp. 54-78). [T. III.] Squares of numbers from 10,000 to 25,400 (pp. 78-100). [T. IV.] Table of the first ten powers of numbers from 1 to 100. Two eiTors (viz. the last three figures of 81' should be 401, not 101, and the last three of 98^ should be 672, not 662) are pointed out by the reporter in the Philosophical Transactions, 1870, p. 370. The remaining three pages of the book are devoted to weights and mea- sures &c. The table is closely printed; and some of the pages contain a great many figures, as there are a hundred lines to the page. De Morgan states that the table has not the reputation of correctness ; and the charge is no doubt true, as, besides the two errors noted above (both of which we found on the only page we have used), it is to be inferred from Barlow's intro- duction to his tables that he found errors ; he did not, however, publish any account of them. Hutton, 1858. T. I. Seven-figure logarithms to 1000, and from 10,000 to 108,000, with proportional parts for all the differences. The change in the line is denoted by a bar placed over the fourth figure of all the logarithms affected. T. II. Logarithms to 1000, and thence for odd numbers to 1199, to 20 places. T. III. Logarithms from 101,000 to 101,149, to 20 places, with first, second, and third differences. T. IV. Antilogarithms, viz. numbers to logarithms from -00000 to •00149 at intervals of -00001, to 20 places, with first, second, and third differences. T. V. Hyperbolic logarithms from 1-01 to 10-00 at intervals of -01, and for 10^. . . .10', to seven places. T. VI. Hyperbolic logarithms to 1200, to seven places. T. VII. Logistic logarithms, viz. log 3600" -log .r, from x=l" to .r= 5280" (=88') at intervals of 1", to four places, the arguments being ex- pressed in minutes and seconds. T. VIII. Log sines and tangents to every second of the first two degrees, to seven places ; no differences. T. IX. Natural and log sines, tangents, secants, and versed sines for every minute of the quadrant, with differences, to seven places, semiquadrautally arranged. The natural functions occupy the left-hand pages, and the loga- rithmic the right-band. In both these last two tables the logarithms are all written at full length. T. XI. Circular arcs, viz. circular measure of 1°, 2°, . . . .180°, of 1', 2' 00', of 1" 60", and of 1'" to 60'", to seven places. T. XII. Proportional parts to hundredths of 2-302 , the reciprocal of the modulus. 110 REPORT~1873. Some constants are given in T. XX. ; the other tables consist of a traverse table, formulae, &c. The edition described above is one of those edited by Olintlius Gregory, and is the last we have met with. The first edition was published in 1785, the second in 1794, the third in 1801, the fifth in 1811, and the sixth, the last published in Hutton's lifetime (he died 1823), in 1822. We have compared the first, second, and sixth editions, and that of 1858 described above. The first two are nearly identical, so that we need only notice the diff'erences between the tables of 1785^ 1822, and 1858. In both the two former of these editions T. I. only extends to 100,000 ; and while in that of 1785 the change of figure in the line is not marked at all, in that of 1822 the fourth figure in the first logarithm aflfected only is marked. T. II. is the same in the 1822 edition, but it ends at 1161 instead of 1199 in that of 1785. T. III. in 1785 ended at 101,139, and is extended to 101,149 in both the other editions, as also did T. IV. originally end at '00139. In the edi- tions of 1785 and 1822 occur two tables that were left out by Gregory in 1830 and in succeeding editions, viz. T. 5, giving logarithms of all numbers to 100, and of primes from 100 to 1100, to 61 places, and T. 6, giving the logarithms of the numbers from 999,980 to 1,000,020, to 61 places, with first, second, third, and fourth differences. T. VI., of hyperbolic logarithms, ap- pears in the edition of 1822, but not in that of 1785. T. VII. extended only to 80' in 1785. To all the first six editions is prefixed Hutton's introduction, containing a history of logarithms, the difi"erent ways in which they may be constructed, &:c. This very valuable essay was omitted by Gregory in the seventh (1830) and subsequent editions (on account of its being rather out of place in a col- lection of tables), and with some reason. In the 1785 edition it occupied 180 pp., 55 pp. of which are the " Description and Use of the Tables." This portion Gi'egory retained ; and in the 1858 edition it occupied 68 pp. The whole work was reset in the later editions, published in Hutton's lifetime, the chief additions, as we infer from the preface, having been made in the fifth (1811) edition. On the last page of the 1822 edition are some errata found in Callet (1783, 1795, and 1801), and also in Taylor (1792); the lists of errors in GARDrNER (London and Avignon) are also more complete than in the earlier editions. Hutton's tables were the legitimate successors of Sheewin's, and bring down to the present time one of the main lines of descent from Vlacq (see Sherwin, § 4). Inman, 1871. [T. I.] Logistic logarithms, viz. log 3000'— log a* from .v = 2 to A' =3600^ (=60™) at intervals of 2% to 5 places. Arguments expressed in minutes and seconds. [T. II.] Proportional logarithms, viz. log 10800"— log .r to every second to 3° (same as T. 74 of Raper, only to 5 places instead of 4), preceded by a jjage giving the same for every tenth of a second to 1'. [T. III.] Log sines at intervals of 1" to 50', to 6 places. [T. IV.] Log sines, tangents, and secants at intervals of 1^ to S^ (argu- ments also given in arc, the intervals being 15"), to 6 places : the table is followed by a page of proportional parts for use with it. [T. v.] g log liaversines, viz. g log semi- versed sines = log sin '-, from a=0° to 15° at intervals of 15", thence to 60° at intervals of 30", and thence to 180° at intervals of 1', to 6 places (arguments also in time). Nole. — In several instances in this table ' is misprinted for ". [T. VI.] Log havenines, Same as previous table, except that 2 log sin ON MATHEMATICAL TABLES. Ill ■- is the function tabulated; so that all the results are double those in [T. Y.], and that the intervals are 15" up to 135°, and then 1' to 180°. [T. YII.] Six-figure logarithms to 1000, and from 1000 to 10,000 in de- cades, with proportional parts. [T. VIII.] Natural versed sines to every second (of time) to 36"", to 6 places. [T. IX.] Natural versed sines to every minute (of arc) to 180°, to 6 places, witli complete proportional parts for every second up to 60". The other tables are nautical. The paging of the book runs at the top of the pages to 216, and thence at the bottom to 275 ; it then recommences at the top at p. 217. This is no doubt caused by [T. Y., YI.] having been introduced in this edition only. \Ye have seen the original work, ' Nautical Tables designed for the use of British Seamen, by James Inman, D.D. London, 1830' (400 pp. of tables), but have not compared the two together : except for the " haversines," how- ever, the tables seem to be nearly identical in the two editions. Inraan's ' Navigation and Nautical Astronomy ' (2nd edit.), Portsea, 1826, contains no tables. Irsengarth, 1810. These are merely land tables, and the units (Ruthe, Fuss, &c.) are so special that they do not appear to possess any mathema- tical value. John, 1837. Vol. I. Six-figure logarithms to 100,000 ; the change in the hne is denoted by a dagger (f) prefixed to the fourth figure of all loga- rithms affected. There are no proportional parts on the page ; but they are given in a separate table at the end. Vol. II. Logarithmic sines and tangents for every second of the first degree ; log sines and tangents for every third second of the quadrant (semi- quadrautally arranged) : aU to 6 places. Proportional parts are given in the extreme right and left columns of the double page for every twentieth of the three-second interval. The introductory matter is both in German and Latin. We rather like the paper on which the second volume is printed ; though not of a good quality, it is thick and stiff, and of a brownish colour, so that the book could be, we think, used for a long time at once without injury to the eye : the first volume (in the copy before us) , however, is printed on paper of the soft, flaccid kind common in German books. The author was led to publish his tables by observing that nearly all those in use were either five- or seven-figure tables. We have seen, by the same author, 'Tafehi zur Berechnnng fiir Kubik- Inhalt (Src.,' 2nd edit., Leipzig, 18-17; but the tables are commercial (argu- ments expressed in ZoUe, Ellen, &c.), and do not need notice here. Kerigan, 1821. T. YIII. Log sines for every second to 2°, and thence, at intervals of 5", to 90°, to six places ; in this latter part of the table pro- portional parts for seconds are added, so that the table practically gives log sines to every second ; arranged quadrantaUy. The logarithms are aU printed at length. T. IX. Natural sines from 0° to 90° at intervals of 10", to six places ; no difterences ; the sines written at length. T. X. Six-figure logarithms from 1000 to 10,000, with proportional parts ; arranged as is usual in seven-figure tables; the change in the line is marked by the ciphers after the change in the third place being filled in, so as to render them black circles. 112 REPORT— 1873. T. XI. Logarithmic Rinincf, viz. log versed sines from 0'' to S'' at inter- vals of 5", with jjroportional parts to seconds, to 5 places : the logarithms are ■written at length. T. XII. Proportional logarithms for every second to 3°, to four places ; same as T. 74 of Raper. T. XIII. Small table to convert arc into time : the other tables are nautical. Kbhler, 1832. [T. I.] Five-figure logarithms to 10,000, arranged con- secutively in columns, with differences and characteristics; the degrees, min- utes, tfec. for every thirtieth number are added. [T. II.] Log sines and tangents for every minute of the quadrant, to five places, with differences. [T. III.] Gauss's table (§ 3, art. 19) ; viz. B and C are given for argument A from -000 to 2-000 at intervals of -001, thence to 3-40 at intervals of -01, and to 5 at intervals of "1, to five places, with differences. There are besides a few constants; the introduction is in French and German. Kohler, 1848. [T. I.] Seven-figure logarithms to 1000, and from 10,000 to 108,000 (this last 8000 being to 8 places), with differences and proportional parts ; the change in the line is denoted by a bar placed over the fourth figure of aU the logarithms affected. The constants S and T (§ 3, art. 13) and the variation are given at the top of the page, as also is the number of degrees, minutes, &c. corresponding to every tenth number. At the end are the first hundred multiples of the modulus and its reciprocal to 8 places, and a small table to convert arc into time. [T. II.] Gaussian logarithms : B and C are given to.5 places (with differences) for A = -000 to 2-000 at intervals of -001, thence to 3-40 at intervals of -01, and to 5-0 at intervals of -1 (same as Gauss's table 1812, § 3, art 19). [T. III.] Briggian logarithms of primes from 2 to 1811, to 11 places, fol- lowed by 2 pages of constants, some weights and measures, &c. [T. IV.] Log sines, tangents, and arcs (all equal) for every second to 1' ; and log sines, cosines, tangents, and cotangents for intervals of 10" to 10°, and thence for intervals of 1' to 45°, to 7 places, with differences for one second. [T. v.] Circular measure of 1°, 2°. . . . 100°, 110°. . . .300°, 330°, 3G0°, of 1', 2' 60', and of 1", 2" 60", to 11 places. Then follow some for- mulai, and we come to the second part of the work, ' Mathematische Tafeln, die oft gebraucht werden,' containing : — T. I. Hyperbolic logarithms (to 8 places) of numbers from 1 to 1000, and of primes from 1000 to 10,000. T. II. The first 45, 36, and 27 powers of 2, 3, and 5 respectively. T. III. (^ from .x- = -01 to 10-00 at intervals of -01 to 7 figures. T. IV. The first ten powers of numbers from 1 to 100. T. V. Squares of numbers from 1 to 1000. T. VI. Cubes of numbers from 1 to 1000. T. VII. Square and cube roots (to 7 places) of all numbers from 1 to 1000. T. VIII. Factor tables, giving all divisors of all numbers not prime or divisible by 2, 3, or 5, from unity to 21,524. T. IX. To express minutes and seconds as decimals of a degree &c. T. X. Binomial-theorem coefiicients, viz. x, ' ^" ~ ' , &c. ...',' A .~-r » from .r=-01 to 1-00 at intervals of -01, to G places. ON MATHEMATICAL TABLES. 113 1.3.5 1 T. XI. Decimal values of certain coefficients, such as — , , 2.4.0.7 2 . 4 . r>' 1.3 -, «S:c., with their logarithms. There arc 40 in all ; and the talkie 2.4.().7' occupies one page. A reward of a louls d'or was offered for every error found in the first edition ; all the errors so found are corrected in the second, here described. Lalande, 1805. [T. I.] Five-figure logarithms of numbers from 1 to 10,000, arranged consecutively in columns, with differences. [T. II.] Log sines and tangents for every minute of the quadrant, to 5 places. An explanation of 34 pp. is prefixed. Lalande, 1829. [T. I.] Seven-figure logarithms to 10,000, arranged in columns with characteristics and differences ; the number of degrees, minutes, &c. for the first number in each column (viz. for every thirtieth number) is given at the top. [T. II.] Log sines and tangents for every minute of the quadrant, to 7 places, with differences. Lambert, 1798. T. I. Divisors of all numbers up to 102,000 not divi- sible by 2, 3, or 5. If the number is the product of only two prime factors, then the least only is given ; but if of more than two, the others are given, except the largest. The table tlierefore gives all the simple factors except the greatest. The letters/, rj, Ji, &q. are used for 11, 13, 17, &c. (as explained on p. xviii of the introduction), not only because they occupy less room, but also because they can be placed in contact without risk of mistake; the least factor, however, is always written at length. T. II. Ahaciis numeroi-um jyrimorum, viz. first 10 multiples of all the primes up to 313. T. III. Seven products, each of seven consecutive primes, from 7 to 1 73. T. IV. List of the three-figure endings that squares of odd numbers admit of. T. VI. Primes from 1 to 101,977. T. VIT.-IX. Powers of 2 to 2'°, of 3 to 3'", of 5 to 5'^ T. XL e-^- (to 7 places) for .r=-l, -2, . . . -9, 1, 2, . . . 10. T. XIII. & XV. Hyperbolic logarithms (to 7 places) of numbers from 1 to 100, and from 1-01 to 10-00 at intervals of -01, respectively. T. XIV. & Xyi. contain loge 10, 10^ . . . 10'", to 7 places, and log, 2, 3 ... 10, and ^ rrr, to 25 places. lege 10' ^ T. XVII. Tables of numbers of the form 2". 3"'. 5''. 7' arran"-ed in order up to 11,200. T. XXIIL Circular measure of 1°, 2°. . .100°, 120°, 150°, 180° . 3G0°, of 1', 2'. . .10', 20'. . .60', and of 1", 2". . .10", 20". . .60", to 27 places. T. XXIV. 0=lOOOO"m; ((,, <j>- . . .0" expressed in terms of m (in circular measure), to 16 places, and sin f, cos f expressed in terms of hi with decimal coefficients, to 18 places. Also n-, log tt, -, Vtt, &c. to a good many places. T. XXV. Natural sines to every degree and their first 9 midtiples, to 5 places. T. XXVI. Sines, tangents, and secants, and log sines and tangents to every degree, to 7 places. T. XXIX. Table for facilitating the solution of cubic equations, viz. «= ±(.r— .r') from a;=-001 to 1-1.55 at intervals of -001, to 7 places. 1873. I 114 REPORT 1873. T. XXXII. Functiones Jiypcrlolicce circularibus analogs. Q, q being a rectangular hyperbola, centre C, P C Q is tlie so-called angulus imnscendens = say, 5 C Q the angulus communis =\{/ say; jj ^ is the hyperbolic sine, C p the hyperbolic cosine, and C g Q the sector ; so that if the hyperbola be .^■'^ — ?/- = ], .r=sec and ?/=tan <p. Tlie argnment is 0, and proceeds from 0° to 90° at intervals of 1° ; and the table gives the sector, y, x, log y, log .r, tan i^, log tan i// and ;//, all ex- cept the last to 7 places, and the last to one decimal of a second. T. XXXV. & XXXYI. Squares and cubes of numbers from 1 to T. XXXVII. ^+l)(^+2) 1".2:3 Figurato numbers .(a:-fll 1.2.3... 12 (first 12 series), viz. x, from a'=l to 30. T. XL. First 11 powers of -01, -02, -03. . .1-00, to 8 places. 1000. x{x^^) their T. XLIV. Coefficients of the first 16 terms in (l-j-.r)^ and (l + .r)''^, accurate values being given as decimals. Besides the above, T. XIX. gives sin 3°, G° . . .89° in radicals, and T. XLII. the first 6 or 9 convergents to \/2, >^3, i^5...>i/12 as vulgar fi-actions. The other tables contain formulte &c. The work is edited by Felkel, who has prefixed a Pra'fatlo Interpretis of xi pp., giving a description of his (Felkel's) tables of divisors &c. ; and there is also added at the end an account of his proposed scheme of tables in rela- tion to the theory of numbers. About Felkel, see Felkel, 1776, § 3, art. 8. The titlepage states that this is a translation from a German edition. The original was entitled ' Zusatze zu den logarithmischen tind trigonometrischcn Tabellen," and was published in 1770 ; or, at all events, De Morgan's descrip- tion of the contents of this latter work, which we have not seen, agrees, as far as it goes, almost entirely with the ' Supplementa ' &c., which De Morgan had heard of, but not seen. The introduction to the latter shows signs of having been amplified by Felkel. Lax, 1821. T. XIV. Proportional logarithms, viz. log 10800" — log .r from .r=0" to 07=10800" (=3°) at intervals of 1" (the arguments being expressed in degrees, minutes, and seconds), to five places. On the first page, however, which extends to 10', only two, three, or four places arc given cor- rectly, the number being filled up to five by adding ciphers ; facing 0° 0' 0" there is given 4-88. . instead of — oo . T. XVII. Xatural versed, suversed, coversed, and sucoversed sines, viz. 1 — cos x and 1 + cos .r for every minute of the quadrant, to six places, with proportional parts for 1", 2".. .60", so that the tabular results can be taken out very easily to seconds. It may be observed that of tbe double columna ON MATHEMATICAL TABLES. 115 headed ' and " the first refers to the argument and the second to the propor- tional parts. This table occupies pp. 57-80 of the book. T. XVIII. yix-figurc logarithms to 15,500, with proportional parts at the foot of the page to twentieths for the portion beyond 1000. The table is so arranged that all the logarithms are given at full length, though this is not the case with the numbers ; for example, to find the logarithm of 15184 we seek 15150 at the head of the column, and line 34 in the column : this defect might have been partially remedied by the introduction of another, column at the right-hand side of the page containing the numbers 50, 51 . . . 99. The other tables, 22 in number, are nautical. Iiynn, 1827. T. Z. (pp. 244-283). A sexagesimal proportional table, exhibiting at sight, in minutes, seconds, and tenths of a second, the fourth term in any proportion in which the first term is 60 minutes, the second term any number of minutes under 60 minutes, and the third term any number of minutes and seconds under 10 minutes. If the second term is not an exact number of minutes the table can still be used, though two operations are ceil required. The table may be described as giving ^, in minutes, seconds, &c., X (running down the column) being 1', 2' . . . 60', and y (running along the top lines) extending to 10' at intervals of 1". T. E. (pp. 288, 289). Proportional logarithms for every minute to 24", viz. log 1440'"— log. r, from .r=l" to a'=1860'" (=31'') at intervals of unitj% tlie arguments being expressed in hours (or degrees) and minutes, to four places ; the other tables are nautical. Mackay, 1810 (vol. ii.). T. XLI. Natural versed sines for every ten seconds to 180°, to six places. T. XLV. Six-figure logarithms of numbers to 100, and from lOOO to 10,000, Avith differences; the logarithms written at length. T. XLVI. Log sines to every ten seconds of the quadrant, to six places. T. XL VII. Log tangents to every ten seconds of the quadrant, to six places. T. XLVIII.-L. To find the latitude hi/ doicble altitudes of the sun or stars and the elapsed lime. The first and second of these tables give log cosec .^• and log (2 sin x) from ,r=0'' to .r=3'' 59"^ 50' at intervals of 10' ; and the third gives' log versed sines to 7'' 59'" 50' at intervals of 10', all to five places, the logarithms being written at length. These tables were copied, according to the author (see note, vol. ii. p. 31), from the second edition (1801) of this work without acknowledgment into Norie's ' Epitome of Xavigation.' T. LI. Proportional logarithms to every second to 3°, to four places ; same as T. 74 of Eater ; the other tables are nautical. The table of natural versed sines was calculated for this work, and ap-. peared in the first edition (1793) ; it has since, the author states, been fre- quently copied (see note, vol. ii. p. 13). Maseres, 1795. This is a collection of reprints of tracts, and, among others, of "An Appendix to the English Translation of Ehonius's German Treatise of Algebra, made by Mr. Thomas Brancker, M.A., ... At London, in the year 1068 " And on pp. 367-416 is given "Thomas Brancker's Table of lucomposit or prime Xumbers, less than 100,000," viz. least factors of all numbers up to 100,000 not divisible by 2 or 5. On p. 306 is a rather long list of errors in the table (we sui)posc Maseres reprinted verbatim from his copy, as some of the errata are corrected and some are not), and also some errors in Guldinus, Schooten, and llhonius. The table is preceded (pp. 364, 365) bv ' A Tarriffa, or Table, of all Incomposit or prime numbers less than V100,000, multiplied by 2, 3, 4, 5, 6, 7, 8, 9." i2 116 UEPoiiT— 1873. On pp. 591, 592, T. XIX. of Dodson's ' Calculator,' 1747 (viz. square and cube roots of numbers less than 180, to 6 places), is reprinted ; and on pp. .595-004 are reciprocals (to 9 places) and square roots (to 10 places) of numbers from 1 to 1000, reprinted (as Maseres states in the preface) from vol. iv. of Button's 'Miscellanea Mathematica ' (1775, 4 vols. 12mo). Maskelyne (Requisite Tables), 1802. T. XV. Proportional logarithms for every second to 3°, to 4 places ; same as T. 74 of Rapek. T. XVI. For computing the latitude of a ship at sea, &e. The ai-guments run from 0'" to 6'' at intervals of 10''; and there are three columns of tabular results headed Log i Elap. time. Log Mid. time, Log rising, which give respectively log cosec .r, log (2 sin a-), and log vers sin .r, to 5 places ; the lor/ rising is also continued for arguments from 6** to 9'' at the same intervals. This table, modified in form &:c., is reproduced in Mackat, Domke, &c. (see § 3, art. 15, p. 68, and Boavdixch, 1802), and is sometimes called by Maskelyne's name. T. XVII. Natural sines to every minute of the quadrant, to 5 places, T. XVIII. Five-figure logarithms of numbers to 10,000. T. XIX. Log sines, secants, and tangents to every minute of the qua- drant, to 5 places; the sines are given to 6 places, the last being separated from the rest by a point ; the other tables are nautical. Maskelyne's name does not appear on the titlepago to these tables ; but the preface is signed by him. Appendix to the Third Edition. T. I. Natural sines to every mimite of the quadrant, with proportional parts for seconds. T. II. Natural versed sines for every minute to 1 20°, with proportional parts for seconds. T. III. Logarithms of numbers to 1000, arranged consecutively, and printed in groups of five; and thence to 100,000 grouped in decades, with proportional parts for each decade by its side. All the tables in the Appen- dix are to six places. Copies of the Appendix were circulated separately. Minsinger, 1845. [T. I.] Seven-figure logarithms to 100 and from 1000 to 10,000, with proportional parts at the foot of the page ; the sixth place is separated by a comma from the seventh, for convenience if the table is to be iLsed to six places. The change in the lino is denoted by an asterisk attached to all the logarithms affected. [T. II.] S(piares, cubes, and square and cube roots (to 6 places) of all numbers from 1 to 100, and squares and cubes only of numbers from 100 to 1000. Then follow a few constants and [T. IV."] primes to 1000. Moore, Sir Jonas, 1G81. [T. I.] Seven- figure logarithms to 10,000 (arranged as is now usual), Avith diftereuces : the proportional parts [T. II.] are given by themselves at the end, and occupy 22 pp. This may bo regarded as a separate table, containing proportional parts (to tenths) of numbers from 44 to 4320— the interval being 2 to 900, 3 to 999, 4 to 1415, 5 to 2000, and 10 to 4320. [T. III.] Natural and log sines, tangents, and secants to every minute of the quadrant, to 7 places (scmiquadrantally arranged), without differences. It may be remarked that many of the N's at the top of the columns are imperfectly printed, and appear like V's ; thus N. tangent is often printed V. tangent, [T. ly.] (pp. 202-351). Natural and log versed sines from 0° to 00° to every minute, to 7 places. De Morgan says that this is the first appearance of this table in England. The other tables relate to navigation, geography, itc. [Moore, Sir Jonas, 1681] (Versed sines). Natural and log versed sines to every minute of the quadrant, to 7 places, scmiquadrantally arranged. ON MATHEMATICAL TABLES. 117 The copy of this tract before us (which is bouiul up in a volume with several otlicrs, and belongs to the Cambriclge University Library) is clearly either a separate reprint or merely a table torn out from some larger work. The paging runs from 262 to 351 : at the beginning there is a plate, the size of the page, of a, person observing with a sextant, and the words " between page 248 and 249 " below in the left hand-corner, and at the end a diagram with a movable circle and pointer, headed " The fore part of the Nocturnall or side held next the face in time of obser\'ation," and " between page 254 and 255 " below. On examination we find the table is [T. IV.] of Sir Jonas Mooee's ' Systeme of the Mathematicks,' 1681, just described. The engravings do not, however, appear to be taken from cither volume of this work. It is very likely that this table was merely torn out from the work, and was never published separately ; still as, according to De Morgan, this is the first appearance of such a table in England, it is not improbable that copies may have been in request, and therefore issued separately. J. H. Moore, 1814. T. III. Log sines, tangents, and secants to every quarter-point, to 5 places. T. IV. Five-figure logarithms of numbers to 10,000. T. V. Log sines, tangents, and secants for every minute of the quadrant, to 5 places, T. XXIII. Log 2 elapsed time, mid. time, and rising (for explanation of these terms see T. XVI. of Maskeltxe, § 4) for every 10' to G^, except the last, Avhich is to 9^, to 5 places. The tables are separated as in Mackay. T. XXIV. Natural sines for every minute of the quadrant, to 5 places. T. XXV. Proportional logarithms for every second to 3°, to 4 places ; same as. T. 74 of Eaper. AVe have seen the 18th edition (1810), which is identical with that above described, an edition of 1793, and the 9th edition (1791) (the last two not edited by Dessiou). All contain the tables described in this account (though the order is diflerent), except that the tables in T. XXIII. are not separated; the log rising is only given to 6'', and the intervals also 30', in the two eai'lier editions. Three out of the four editions contain different portraits of the author. Mtiller, 1844. [T. I.] Five-figure logarithms of numbers from 1000 to 1500, and four-figure logarithms from 100 to 1000. [T. II.] Table of Gaussian logarithms in a somewhat modified form, viz. S and U to 4 places, from A=-0000 to -0300 at intervals of -0001, thence to -230 at intervals of -001, and from -20 to 2-00 at intervals of -01, and thence to 4-0 at intervals of -1, with diff'ercnces ; where A = log X, a z=\ogfl + -\ and U = log ^ . .V [T. III.] Squares of numbers from to 1 at intervals of -0001, to 4 places, and quarter squares of numbers from to 2 at the same intervals, also to 4 places (intended for use in the method of least squares). [T. IV.] Four-place log sines and tangents for every second to 10', thence at intervals of 10" to 1°, thence at intervals of 1' to 4°, and to 90° at intervals of 10'. There are given also : — the circular measure (to 12 places) of 1°, 2° . . . 10°, 1' . . . 10' and 1" ... 10" ; 12 constants involving n ; natural sines and tangents to every half degree ; and a few three-figure logarithms. 118 REPORT— 1873. John Newton, 1658. [T. I.] Logarithms to 1000, to 8 places, and logarithms from 10,000 to 100,000, also to 8 places. A column is added to each page containing the logarithms of the differences, to 5 places. [T. II.] Log sines and tangents (semiquadrantaUy arranged) for every centesimal minute (viz. nine-thousandth part of a^-ight angle), to 8 places, with differences. [T. III.] Log sines and tangents for the first three degrees of the quadrant, to 5 places, the interval being the one thousandth part of a degree. Loga- rithms of the differences to 8 places are added. The trigonometrical tables are thus of the kind introduced by Briggs, and are partly centesimal (see § 3, art. 15, p. 64). This is the only extensive eight-figure table that has been published ; and it is also remarkable on account of the logarithms of the differences, instead of the differences, being given. It seems worth consideration whether, in the event of a republication of Vlacq, 1628, it would not be advantageous to replace the differences by their logarithms. It is usually most convenient, if many logarithms are to be taken out at one time, to interpolate for the last five figures in a ten- figure table by means of an ordinary seven-figure table ; but in other cases recourse is generally had to simple division, and the natural differences are best. The table would occupy too much space if both the diflerences and their logarithms were added ; and there is not much chance of two publi- cations ever being made, one with natural, and the other with logarithmic, differences. If the choice had to be made, the decision would probably be in favour of the simple differences as they are, though a good deal might be ■urged on the other side. A few errata are given at the end of the address to the reader, and a great many more on the last page ; the tables, however, reproduce nearly all Viacq's errors, which affect the first 8 places (see ' Monthly IS'otices of the Boy. Ast. Soc' March 1873), This was the first table in which the arrange- ment, now universal in seven-figure tables (viz. with the fifth figures run- ning horizontally along the top line of the page), was used. The change of the third figure in the line is not noted. The title of this work being the ' Trigonometria Britannica ' (printed * Britanica ' on the titlepage), it is often confounded with Beiggs's work of this name, Gouda, 1633 (§ 3, art. 15), from which it is derived. Also, as GeUibrand's name appears on the titlepage it is sometimes attributed to him in catalogues. In the Cambridge Univei-sity Library is a copy of this book, in which the titlepage and introduction are absent, "the first page being the titlepage to the tables, so that the work is anonymous. Whether some copies of the tables alone were published, or Avhether the copy in question is imperfect, we do not know. Norie, 1836. T. XXIII. Log sines, tangents, and secants to every quar- ter-point, to 7 places. T. XXIV. Six-figure logarithms of numbers to 10,000, with difterences. T. XXV. Log sines and tangents to every ten seconds to 2°, and log sines, tangents, and secants for every minute of the quadrant, to 6 jilaces, with differences. T. XXVI. Xatural sines for every minute of the quadrant, to 6 places. T. XXVII.-XXIX. To find the latitude by double altitudes and the elapsed time. Log i clap, time, middle time, and rising (for explanation of these terms see T. XVI. of Maskelyne, § 4) arc given at intervals of 5^ the two former to 6';, .and the last to <>, to 5 places, with proportional ON MATHEMATICAL TABLES. 119 : parts. The three tables aro sexmnitecl, as is now usual (see Mack ay, § 4, T. XLYIIT.). T. XXXI. Logarithms for Jimling the appcu'cat time or horary angle, viz. log -^ ~ ^^^ •^' ('= 2 log siii^) from a- = 0" to ,v = 0" at intervals of 5', to 5 places, with proportional parts for seconds. T. XXXIV. Proportional logarithms for every second to 3°; same as T. 74 of ILvrER. T. XXXVI. JS'atural versed sines to every minute of the quadrant, with proportional parts for every second of the minute-interval, to 6 ]jlaccs. The other tables are nautical. These tables also appear in Xoeie s ' Epi- tome of Navigation.' Norie (Epitome), 1 844. The tables are the same as in Korte's Xautical ■Tables just described ; they- are added after the explanatory portion, which occupies 328 pp. On the different editions, see Xorik's Epitome in § o. Norwood, 1631. Seven-figure logarithms to 10,000, and log sines and tangents to every minute, to 7 places, semiquadrantally arranged: of the latter we have seen separate copies under the title, " A triangular canon logarithmicall " (the title it has also in the work). The editions we have seen are : — ^first, 1631 ; second, 1641 ; thiixl, 1656 ; seventh, 1678. This was one of the first small tables in which the trigonometrical canon was derived from Vl.vcq, 1628, and not Guntee, 1620. Oppolzer, 1866. Eour-fignre logarithms, with proportional parts to 1000. A page of Gaussian logarithms, after Filipoavski, and u page of pro- portional parts. Log sines, cosines, tangents, cotangents to lO'' at intervals of 1', Avith diftcrences, and from 10° to 45° at intervals of 10', with difl^er- ences and proportional parts, all to 4 places. Oughtred, 1657. [T. I.] Sines, tangents, and secants (to 7 idaees) and log sines and tangents (to 6 places) for every centesimal minute ( = tro'Tnr °^" * right angle) of the quadrant. Sines, tangents, and secants on the left-hand page of the opening, and cosines, cotangents, and cosecants, &c. (though not 60 called or denoted) on the right-hand jjage. [T. II.] Seven-figure logarithms of numbers from 1 to 10,000, followed by a ' Tabula differentiarum ' for the sines and tangents. In an appendix at the end of the book it is explained that the logarithmic sines and tangents were intended by the author to consist of seven figures after the index, but that " the seventh figure was unhappily left out." This is also referred to in the dedication. Ozanam, 1685. Natural sines, tangents, and secants, and log sines and tangents, and logarithms of numbers to 10,000, all to 7 places. There ai'e 120 pp. of trigonometry &c. De Morgan points out that the tables are really Vlacq's, though his name is not mentioned, and takes occasion very truly to remark how many authors have considered that the merit of their books con- sisted in the trigonometry, and that the tables (which usually form by far the greater part of the work) were accessories of which no notice need be taken. . Parkhurst, 1871. This little book contains forty-two tables, with the last two of which this Report is not concerned. In describing briefly their contents, it will be convenient to mention first the tables which contain results most common in other works, such as logarithms &c., viz.: — ■ T. II., III., and IX. Logarithms fi'om 1 to 109, to 102 places. T. V. Multiples of the modulus -43429. ..from 10 to 96, to 35 places. T. XII. Logarithms of numbers from 1000 to 2199 at intervals of unity, 120 REroRT— 1873. from 2200 to 2998 at intervals of 2, from 3000 to 4995 at intervals of 5 ; all to 10 places (from Vlacq). T. XIII. Logarithms of numbers from 200 to 1199, to 20 places (from Callet). T. XIV. (continuation of T. XIII.). Logarithms of numbers from 1200 to 1399 at intervals of unity, from 1 400 to 2998 at intervals of 2, from 3005 to 4995 at intervals of' 10 ; all to 20 places. T. XVIII. Logarithms of primes from 113 to 1129, to 61 places (from Callet) . T. XX., XXI., XXII. A table of least divisors of numbers to 10,190, and, for certain divisors, to 100,000. Multiples of 2, 3, 5, 7, and 11 are excluded ; it is very inconveniently arranged, and is moreover imperfect. T. XXIII. Primes to 12,239. T. XXV. Keciprocals from 300 to 3299, to 7 places, arranged like an ordi- nary table of sevcu-figure logarithms. T. XXVI. Products of the numbers from 200 to 399 by the digits 1, 2 ... 9, and squares from 200' to 399". T. XXVII., XXVIII. A few logarithms and antilogarithms, to 3 places, and a similar small table to 4 places. T. XXX., XXXI. Xatural and log sines and tangents &c., to 4 places. T. XXXII. Binomial-theorem cocfHcicnts (the first six for indices from Tmity to 40), and squares from 1'^ to 200^. T. XXXIII., XXXIV. Multiplication table from 16 x 13 to 99 x 98, and multiplication table of squares from 16- x 13 to 99-x 98. T. XXXV., XXXVII., XXXVIII. Antilogarithms, logarithms to 8 places, and log sines. The other tables are : — T. IV. Logarithms of factors, 102 decimals. T. VI. Secondary multi- ples. T. A'll. Factors to 3 decimals. T. VIII. Logarithms of factors, 31 decimals. T. X. Factors to 61 decimals. T. XI. Log F, for logarithms to 10 decimals. T. XV., XVI., XVII. Logarithms to 20 decimals of factors. T. XIX. Constants derived from the modulus. T. XXIV. Log p, for addition and subtraction. T. XXIX. tSubtractiou logarithms. T. XXXVI. Factors. T. XXXIX., XL. Interpolations, Pessel's coefficients. Most of these tables are tabulated for their use in the calculation of logarithms by well-knoAvn methods. The arrangement of the work is most confused ; and it would be very difficult to understand from the author's description the objects of his tables. The paging of the book runs from 1 to 176; and this portion includes all the tables. Then Part 2 commences, and the pages are numbered afresh from 1 to 38. In Part 3 the pages pro- ceed from 1 to 27. Parts 2 and 3 are occupied with a description of the tables ; and the reader who wishes to luidcrstand the meaning of the nota- tion (which is often needlessly complex and confusing, to save the space of a few figures), &c.. is recommended to begin at Part 3, p. 5. It would take too much room, even if it were worth Avhile, to explain the tables in detail ; but it may be stated that the tables (for the constructiou of logarithms of factors) give the values of log (l+ '"^ Ymd log (l --'"' I for different values of vi and n \ 107 ° \ 10"/ to a great many places, as required in "VVeddlc's and similar methods. It will save the reader some trouble to mention that by " o m in the book is meant log ( 1 +T7t„), and by - >i^o m, — log / 1 — ,— 1. Generally ON MATHEMATICAL TABLES. 121 the m is loft out, -where it is thought the context prcveuts risk of mistake ; and instead of — n o jn there is sometimes written n om, and the lieading " cologarithm." The^Last page of the hook, headed (wrongly) Tahlc XXXIII., contains a very imperfect list of the abbreviations used. It is to be inferred from the Preface &c., that the book was set up and electrotyped by the author himself, Avho states that " it is probable that there is not now a single error in the whole table." The reward of a copj' of the book is also offered to the first finder of any important error under certain condi- tions. Parts of the book, in the cojiy before us, are very badly printed, so badly in fact that one or two images are wholly illegible ; and the tables are so crowded that we should think no one would use them who could procure any others that could be made to do as well. In fact the author's object seems to have been to crowd the greatest possible amount of tabular matter into the smallest space, without any regard to clearness. It is stated in the work that in the course of the printing, incomplete copies (some containing proofs almost illegible) were distributed to the author's friends ; and an advertisement on the cover states that copies containing proofs rejected in the printing may be had at different prices according to their completeness and the order of the tables. The book is printed phonetically ; and this adds to the awkwardness of the most confused, ba3ly printed, and ill-explained series of tables we have met with in the preparation of this Report. By issuing his tables in the form and manner he has adopted, the author has not done justice to himself, as several are the results of original calculation and are not to be met with elscAvhere. Pasquich, 1817. T. I. Five-figure logarithms to 10,000 (arranged consecutively in columns), without diff'erences. T. II. Log sines, cosines, tangents, and cotangents, from 0' to 56' at in- tervals of 10", thence to 1° at intervals of 20", and thence to 45° at intervals of l',.witli difterences for 1". Also squares of natural sines, cosines, tangents, and cotangents from 1° to 45° at intervals of 1', all to 5 places. De Morgan says, "This trigonometrical canon in squares is, we suppose, almost unique." T. III. Gaussian logarithms. B and C (same notation as in Gauss), to 5 places, with differences, for argument A, from A = -000 to A = 2-000 at intervals of -001, from A = 2-00 to A = 3-40 at intervals of -01, and from A = 3-4 to A = 5 at intervals of •!. This table is the same as that originally given by Gauss, 1812 (§ 3, art. 19). A iv\Y constants &c. are added in an Appendix. A lengthy review of this Avork by Gauss appeared in the ' Gottiugische gclehrte Anzeigen' for Oct. 4, 1817. It is reprinted on pp. 246-250 of t. iii. of his ' Werke.' Pearson, 1824. Yol. I. contains 296 large quarto pages of tables ; but only three pages come within the range of this lieport, viz.: — [T. I.], p. 109, a one-page table to convert space into time, and vice versa. [T. II.], p. 261, w^hich expresses 1°, 2°, 3° 360°, and 1', 2' 60' as decimals of the cir- cumference of the circle to 4 and 5 places respectively ; and [T. III.], p. 262, which gives the circular measure of 1°, 2°. . . .180°, of 1', 2'. . . .60' and of 1", 2".. ..60", to 8 places. The other tables are nautical, astronomical &c. Peters, 1871. [T. I.] pp. 16, 17. Himdredths, thousandths, ten-thou- sandths, hundred-thousandths and millionths of a day expressed in minutes and seconds. [T. II.] pp. 18, 19. For the conversion of arc into time, and vice versa. 122 REPORT— 1873. o [T. III.] pp. 20, 21. Lengths of circular arcs, viz. 1°, 2°, 3°.... 90 theuce to 115° at intervals of 5°, and to 360° at intervals of 10°, 1', 2' 60', and 1", 2". . . .60", expi-essed in circular measure, to 7 places. [T. IV.]. Interpolation tables. Tabic I. (p. 103) gives ''!^^^^^^^, xLv — l)(.v — i) , (x + lyi'lx — l)(,v—2) „ ^,„ , -, n. , — ^^ 7> ^- and ^ — ^_i^,- ^-y 1 from a?=-00 to ,v=l-00 at b 48 intervals of -01 — the first function to 5 places (with differences), and the second and third to -l places (without differences). It will be noticed that on writing 1 — x for .r, the first and third functions are unaltered, while only a change of sign is produced in the second. It is thus sufficient to tabulate them only from to -50, and to write the arguments down the column from 0-00 to -SO, and ujiwards from -50 to 1*00, attending to the sign of the second func- tion ; and this is accordingly the arrangement in the table. Tabic II. (pp. 1 04, 105) contains _., _v__^ — \ -24 ' 240 " ^'"'^^ ,v = 0-00 to ,v = 1-00 at intervals of -01, the first to 5 and the others to 4 places. The first two have differences added. [T. Y.] (pp. 106-150). Natural sines, tangents, and secants throughout the quadrant to every minute, to 5 places, without differefices. [T. YI.] (pp. 151-169). Table of squares to 10,000, arranged as in a table of logarithms, the last figures of the squares (which must be 0, 1, 4, 5, 6 or 9) being printed once for all at the bottom of the columns. The other tables are either astronomical or meteorological. There are 13 pp. of formulae. Rankine, 1866. T. I. Squares, cubes, reciprocals (to 9 places) and five- iigure logarithms of numbers from 100 to 1000. T. 1 A. Square and cube roots (to 7 places), and reciprocals (to 9 places) of primes from 2 to 97. T. 2. Squares and fifth powers of numbers from 10 to 99. T. 2 A. Prime factors of numbers up to 256. T. 3. Hyperbolic logarithms of numbers to 100, to 5 places. T. 3 A. Ten multiples of the modulus and its reciprocal. T. 4. Multipliers for the conversion of circular lengths and areas, viz. a few multiples of tt and its reciprocal, square roots, &c, T. 5. Circumferences and areas of circles, viz. ird (to 2 places), and 2' (to the nearest integer), from d - 101 to <? = 1000. T. 6. Arcs, sines, and tangents for every degree, to 5 places. Raper, 1846. T. I. Six-figure logarithms of numbers from 1 to 100 and from 1000 to 10,000, with proportional parts at the foot of the page. T. II. Log sines for every second from 0"^ to 1° 30', to five places. T. III. Log sines for every ten seconds from 1° 30' to 4° 31', to 6 places, with proportional parts. T. lY. Log sines, tangents, and secants for every half minute of the qua- drant, to 6 places, with proportional parts. , T. Y. A page of constants. Raper, 1857. T. 21 a. Logarithms for reducinr/ daily variations, viz. log -1440'^^ — log .r, from x = !■" to x = 1440" (= 24'") at intervals of a minute, to 4 places, the arguments being expressed in hour's and minutes. T. 64. Six-figure logarithms of numbers to 100, and from 1000 to 10,000, arranged as is usual in seven-figure tables, except thevt the logarithms are ON MATHEMATICAL TABLES. 123 printed at full length ; the proportional parts are given at the foot of tho page. T. 6-5. Log sines, tangents, and secants to every quarter point, to six places. T. 66. Log sines of small arcs, viz, for each second to 1° 30', thence (T. 67) for every ten seconds to 4° 31', to 6 places, the logarithms being printed at length ; T. 67 has proportional parts. T. 68. Log sines, tangents, and secants (printed at full length) for every half minute of the quadrant, to 6 places, with differences and proportional parts for 1", 2".. ..30" (= half a minute) beyond 3°, scraiquadrantally arranged ; arguments also expressed in time. T. 69. Log siu^ '^ from x = to x = 180° at intervals of 15" (arguments expressed also in time), to 6 places ; all the logarithms printed at full length : no differences. T. 74. Proportional logarit.hms, viz. log 10800" — log .v from .v = 1 to X = 10800" ( = 3° or 3") to every second, the arguments being expressed in degrees (or hours), minutes, and seconds, to 4 places ; the other tables arc nautical &c. Reynaud, 1818.- The trigonometry occupies 182 pages; and after the diagrams are inserted Lalande's logarithms, which are quite disconnected from the work. t [T. I.] Five-figure logarithms to 10,000, arranged in columns, with cha- racteristics and differences ; the number of degrees, minutes, &c. for the first number in each column (viz. for every thirtieth number) is given at the top. [T. XL] Log sines and tangents for every minute of the quadrant, to 5 places, with differences. Riddle, 1824. T. IV, Log sines, tangents, and secants to every point and quarter point of the compass, to 6 places. T. V. Six-figure logarithms of numbers to 100, and from 1000 to 10,000, with differences, arranged as usual. T, VI. Log sines, tangents, and secants to every minute of the quadrant, to 6 places, with differences, semiquadrantally arranged. [The heading of this table in the book is inaccurate.] T. XXVIII. Natural versed and suversed sines, viz. 1 — cos .v and 1 +cos .v, for every minute of the quadrant, to 6 places, with proportional parts for 1 ", 2" . . . 60", so that the tabular results can be taken out very easily to seconds. The extreme left- and right-hand columns serve both for mini\tes in the arguments and for multiples in the proportional parts. The first figure of the versed sine and the first two of the suversed sine are generally omitted throughout. T. XXIX. Proportional logarithms, viz. log 10800" — log .v from x = to .V = lOSOO" (=3^ or S*"), the argiiments expressed in degrees or hours, minutes, and seconds at intervals of 1", to 4 places. The book contains 34 tables, the rest of which are nautical. The navi- gation &c. occupies 299 pages. Rios, 1809. The first edition was published in 1806; and this is the second. The tables are identical with those in the Spanish reprint of 1850 described below, so that the description of the latter will suffice. The numbers both of the tables and the pages are the same in both ; and the only difference is that the headings of the tables &c. in the 1809 edition are in English. A list of errors in this edition is given in the reprint of 1850. '" ._ Although the title of the Spanish reprint is given in the list in § 5, we have 124. REPOiiT— 1873. thought it would bo more convenient to give the work the date of 1809, as this more properly represents the time of appearance than does 1850. T. XIV. Proportional logarithms for every second to 3", to 5 places. This table only differs from T. 74 of Eapek in there being 5 instead of 4 places given. T. XV. Five-figure logarithms of numbers from 10 to 10,200, with the corresponding degrees, minutes, and seconds. T. XVI. (pp. 382-472). Log sines, cosines, secants, cosecants, versed, co- versed, suversed, and sucoversed from 0° to 45° at intervals of 15" (with arguments also in time), to 5 places. The term " versed " (versos) is used for semiversed sine for brevity, and so for the others ; the table thus gives log 2 (1 i eos.r) and log | (1 + sin x). The log sines, cosines, &c. are on the left-hand pages, and the log versed &c. on the right-hand pages. The table, altered in arrangement so as to make it quadrautal, is reproduced in Stansbukt, 1822. There are also given some small tables to convert arc into time, and vice versa, on p. 472. These tables are all included under the heading ' Tablas logaritmicas y tablas para convertir partes de circulo en tiempo y viceversa.' A list of errata in the London edition of 1809 is given at the beginning of the edition of 1850. Roe. T. I. Seven-figure logarithms of numbers from 1 to 100,000, with characteristics unscparated from the mantissas. All the figures of the number are given at the heads of the columns, except the last two, which run down the extreme columns ; 1 ... 50 on the left hand, and 50 . . . 100 on the right-hand side. The first four figures (counting the characteristics) are printed at the top of the columns. There is thus an advance halfway to- wards the modern arrangement, and the final step was made by JohnXewton (1658). This is the first complete seven-figure table that was published. It is formed from Vlacq by leaving out the last three Ggiues, withoiit increasing the seventh when they are greater than 500. T. II. Logarithmic sines and tangents for every hundredth part of a degree (viz. -^^-jj-^y part) of the quadrant, semiquadrantally arranged, to 10 places, with characteristics, which, however, are separated by a comma. The work is very rare : the copy Ave have seen belongs to the lloyal Society. Rumker, 1844. T. I. Six-figure logarithms of numbers from 1000 to 10,000, arranged consecutively in columns and divided into decades, with the proportional parts for each decade by the side of it. T. II. Log sines and tangents for every ten seconds to 2°, and log sines, tangents, and secants for every minute from 0° to 45°, with ditlcrences, to 6 places ; the logarithms written at length. T. III. Xatural versed sines to every minute to 180°, with proportional parts for the seconds, to (5 places. T. IV. Lofjarithmen-Sielc/ezeif, \iz. log versed sines for every minute to 12'', to 6 jjlaces, with diifcrences for one second (corresponding to O"* 0™: the table gives instead of — oo). T. XXIV. Proportional logarithms for every second to 3°, to 4 places ; same as T. 74 of IIapeh. In all cases the logarithms are written at length. The other tables are nautical. ^Salomon, 1827. This Avork avc have not seen ; but as Eogg has given a description of several of the tables, and we see no likelihood of meeting with the book, we here give his account. There are 13 tables, of which the most noteworthy are the following : — • ON MATHEMATICAL TABLES. 125 T, I. Squares, cubes, square and cube roots (to how many places is not stated) of all numbers from 1 to 10,000 conveniently arranged. T. II. Factors (except 2, 3, 5, and 11) of numbers from i to 102,011. T. VII. Six-figure logarithms of numbers to 10,800 (the last 800 to 7 places). T. VIII. Briggian and hyperbolic logarithms of all numbers from 1 to 1000, and of primes from 1009 to 10,333, to 10 places. T. IX. Logarithmic canon for every second of the first two degrees, and then for every ten seconds of the rest of the quadrant (to 6 or 7 places, wo suppose). T. XII. Natural sines and tangents for every minute, with diflPercnces. Kogg adds that the printing and paper are good for Germany, but that he has made no comparison to determine the correctness of the table ; the two pages of errata, however, show (he remarks) that there was not so much care taken as with Sherwin, Gaemnkr, Callet, Hutxon, Taylor, or Vega. Hogg's account is to be found on pp. 254 and 399 of his ' Bibliotheca.' See also Gernorth's tract. "^Schlbmilch [1865 ?]. Five-figure logarithms to 10,909 ; table for the conversion of Briggian into hyperbolic logarithms ; logarithms of constants ; circular measure of degrees, minutes, and seconds ; natural functions for every ten minutes of the quadrant ; log functions for every minute ; reciprocals, square and cube roots, and hyperbolic logarithms of numbers to 100 ; elliptic quadrants ; physical and chemical constants. The above description is taken from an advertisement. Schmidt, 1821. [T. I.J Five-figure logarithms to 100, and from 1000 to 10,000, with proportional parts. [T. II.] Log sines and tangents for every minute of the quadrant (semi- quadrantally arranged), to 5 places, with differences. [T. III.] Natural sines (to 5 places) and tangents (to 5 places when less than unity, above that to 6 figures) for every minute of the quadrant. [T. IV.] Circular arcs, viz. circular measure of 1°, 2° . . . 90° 120° 300°, 360°, of 1', 2' . . . 60', and of 1", 2" . . . 60", to 12 places. [T. v.] Squares and cubes of all numbers from imity to 1000, with two subsidiary tables to extend the table to 10,000 ; the latter are of double entry, and contain :— (i) (2 a + c) c for c= 1, 2 . . . 9 and a=10, 11 ... 99, and b c and 2 be for the same values of c and for 6 = 1, 2 ... 9 ; and (ii) (3 cr + 3ac + c") c for c = 1, 2 ... 9, and a = 10, 11 . . . 99. There are a few other small tables for the solution of triangles, refrac- tions, &c. Schron, 1860. T. I. Seven-figure logarithms to 1000, and from 10,000 to 108,000 (the last 8000 being to 8 places), with proportional parts to' one place of decimals, so that they are in fact multiples. The change in the line is denoted by an asterisk prefixed to the fourtli figure of all the logarithms affected. The degrees, minutes, &c. corresponding to every number (regarded as that number of seconds) in the left-hand column, and also corresponding to these numbers divided by 10, are given. At the bottom of the page also S and T (and also the log sine and tangent) are added for every 10" (§ 3, art. 13, p. 54). When the last figure has been increased there is a bar subscript, wliich, being more obtrusive, is not so good as Babbage's point. The table is followed by the first 100 multiples of the modulus and its reci- procal, to 10 places. T. II. Log sines and tangents for every ten seconds of the quadrant, to 7 places, with very complete proportional-part tables (or more properly mul- tiples of the differences). The increase of the last figure is noted as in T. I, T. III. Interpolation table, viz. the first 100 multiples of all numbers 12G REPORT— 1873. from 40 to 410. The table occupies 75 pages ; and on each double page are given the proportional parts to hundredths of 1, 2, 3, 4, and 5 (viz. the first 100 multiples divided by 100 and contracted to ouo decimal place). The last page of the book is devoted to a table for the calculation of logarithms, and contains common and hyperbolic logarithms of n, 1-On, I'OOn, &c., n being any single digit (or in other words, of 1 + — ^^ from .v = 1 to .r = 9 and n = 1 to n = 10), to 16 places. The figures arc beautifully clear, and the paper very good. The tables are of their land very complete indeed. We have seen errata in this work advei-tised in different numbers of Grunert's ' Archiv der Mathematik und Physik.' See Schron, 1865, below. Schrbn (London edition), 1865. De Morgan remarked that in England, though tliere existed minute- and second-tables of trigonometrical functions, there was no good ten-second table ; and on learning from the publishers that an English edition of kSchkon was contemplated, he offered to write a short preface, as, accuracy being taken for granted, these appeared to him to be the most powerful and best ten-second tables ho had seen : his oft'er, how- ever, was accompanied by the condition that a careful examination should be made by Mr. Farley, sufficient to judge of the accuracy of the work, and that the result should bo satisfactory. Mr. Farley accordingly examined 24 pages selected at hazard, wholly by differences and partly by comparison with Callet ; and the pages were found to be totally free from error ; so that the general accuracy of the tables was assured. They arc printed from the same plates as in the German edition described above ; and the tabular matter in the two seems identical in all respects. Schulze, 1778. [T. I.] Seven-figure logarithms to 1000, and from 10,000 to 101,000, with differences and proportional parts. The proportional parts at the beginning of the table, which are very numerous, arc printed on a folding sheet. . A page at the end of this table contains the first nine multiples of the modulus and its reciprocal, to 48 places ; also c to 27 places, and its square, cube .... to its 25th power, also its 30th and 60th powers, the number of decimals decreasing as the integral portion increases. Log tt (hyperbolic and Briggian) is also given. [T. II.] Wolfram's hyperbolic logarithms of numbers to 48 places. The numbers run from unity to 2200 at intervals of unity, and thence to 10,009, only not for all numbers ; " von 2200 bis 10,000 ist sie hingegen nur f iir die Prim- imd etwas stark componirtc Zahlen berechnet, weil das Uebrige durch leichtes Addiren kaun gefundcn werdcn " (Preface). De Morgan says " for all numbers not divisible by a single digit;"' but this is incorrect, as 2219, 2225, &c. are divisible by single digits, while 9S09 (least factor 17), 9847 (least factor 47) do not occur. In fact, at first a great many composite numbers are tabulated, and near the end very few, if any. All the primes, however, seem to be given ; and by the aid of Wolfram's tables we may regard all hyperbolic logarithms of numbers below 10,000 as known. Space is left for six logarithms, which Wolfram had been prevented from computing by a serious Ulness. These were supplied in the ' Eerliner Jahrbuch,' 1 783, p. 191. Mr. Gray points out an error in Wolfram's table; viz. in log 14()9, .... 1660 ... . should be .... 1 696 .... (' Tables for the formation &c.,' 1 865, p. 38). On Wolfram, see § 8, art. 16. [T. III. J Log sines and tangents for every second from 0° to 2°, to seven places ; the sines are on the left-hand pages, the tangents on the right-hand ; no differences. ON MATHEMATICAL TABLES. 11:^7 [T, IV.] Logistic logarithms to every second to one degree, to four places. The pages in [T. III.] and [T. IV.] are not numbered. [T. v.] is the first table in the second volume. It contains : — natural sines, tangents, and secants to seven places, Avith differences ; log sines and tangents to seven places, with differences (from 0° to 4° the simple difference, and from 4° to 45° one sixth part of the difference, is given) ; and Napisrian (see § 3, art. 17) log sines and tangents to eight places, without differences ; all for every ten seconds for the first four degrees, and thence for every minute to 45°. The Napierian logarithms (see first page of Preface to the second volume) are taken from the ' Canon Mirificus ' of Napier, augmented by Ursinus. The arrangement of the table is not very convenient, but perhaps the best possible. [T. VI.] (pp. 262, 263). First nine multiples of the sines of 1°, 2°, .3° .... 90°. One or two constants are given on p. 264. [T. VII.] Circular measiire of all angles from 1° to 360° at intervals of ]°. This is followed by similar tables for minutes from 1' to 60' at intervals of 1', and for seconds from 1" to 60" at intervals of 1", all to 27 places. [T. VIII.] Powers, as far as the eleventh, of decimal fractions from '0 to J -00 at intervals of -01, to eight places. [T. IX.] Squares of numbexs to 1000, [T. X.] Cubes of numbers to 1000. [T. XI.] Square roots of numbers to 1000, to seven places. [T. XII.] Cube roots of numbers to 1000, to seven places. [T. XIII.] The first six binomial-theorem coefficients, viz. ,r, ^' — '-, .... '—- — -,,L''^ -5 for X = -01 to cc = 1-00, at intervals of -01, to seven places. The other tables connect the height and velocity of falling bodies, and contain specific gravities &c. A table on the last page is for the conversion of minutes and seconds of arc into decimals of an hour. A table headed Jiaiionale Trigonometrie occupies pp. 308-311 , and is very interesting. It gives right-angled triangles whose sides are rational and such that tan |w (w being one of the acute angles of the triangle) is greater than J^. Such triangles (though not so called here) are often known as Pythagorean. Those with sides 3, 4, and 5 ; and 5, 12, and 1.3 are the 1)est-known cases; and 8, 15, and 17, 9, 40, and 41, 20, 21, and 29, &c. are among the next in point of simplicity. This table contains 100 such tri- angles ; but some occur twice. It gives in fact a table of integer values of a, b, c, satisfying (r-\-h'-=c-, subject to the condition mentioned above: tan iio, expressed both as a vulgar fraction and as a decimal, is given, as also are w and 90° — w. For a larger table of the same kind, see Sang, 'Edinburgh Transactions,' t. xxiii. p. 757, 1864. On the whole, this collection of tables is very useful and valuable. [Schi^macher, 1822 ?]. T. V. Five-figure logarithms of numbers for every second to 10,800" (3°), arguments expressed in degrees, minutes, and seconds. T. VI. Log sines for every second to 3°, to five places. There is no name at all on the table ; but it is assigned (and no doubt correctly) to Schumacher in the Royal Society's Librarj- ; and De Morgan, speaking of Waexstoeff's ScnuMACHER (1845), says that the original publication was Altona, 1822; but there was an earlier edition, we believe, at Copenhagen, in 1820. Shanks, 1853. The bulk of this work _([T. I.] jjp. 2-85) consists of the values of the terms in Mr. Shanks's calculation of the value of n by Machin'a 12 REPORT 1873. formula, tt — IG tan ~i g-— 4 tan "i ^i^. The terms in the expansion both of tan ~i i and tan ~i ^iy are given separately to 530 places. The former occupy 60 pp. and extend to ^ ^_r-^, ; and the latter cccujiy 24 pp. and cx- teiid to Qio.osn^ifi ' While the Tvork was passing through the press Mr, Shanks extended his value of tt to 607 decimals ; and to this number of places it is given on pp. 86 and 87 of the book. [T. II.] (pp. 90-!)5) gives every twelfth power of 2 (viz. 2", 2=% &c.) as far as 2^^' (which contains 212 figures). On p. 89 are given the values of <;, log, 2, log^3, log, 5, and log, 10, to 137 places, and the modulus to 130. Values of these quantities were given also by Mr. Shanks to 205 places (I'roc. Roy. Soc. vol. vi. p. 397). The value of e was verified by the reporter to 137 places by calculation from a continued fraction (see Erit. Assoc, lleport, 1871, pp. 16-18, sectional proceedings). The same writer also showed in vol. xix. p. 521 of the ' Proceedings of the lloyal Society,' that Mr. Shanks's values of log 2, 3, 5, and 10 were inaccurate after the 59th place (all owing to one error in a series on which they depended), and deduced the correct values to 100 places. These results were verified by Mr. Shanks, who has recalculated the values of these logarithms, as well as that of the modulus, to 205 places : they are published in vol. xx. p. 27 of the 'Proceedings of the Royal Society' (1871). Mr. Shanks's 607-place value is given in Knight's 'English Cyclojjsedia,' (Art. "Quadrature of the Circle") copied from the work under notice ; and it has been verified by a subsequent calculation of llicliter to 500 places. A list of the calculators of tt, the number of places, &c. to which they have extended their calculations, with references to the places where they aro to be found, is given by Bierens de Haan on a page at the beginning of his " Tables d'lntegrales Definies " in t. iv. of the Amsterdam Transactions. This page, however, does not appear in the separate copies of the tables (the ' Nouvelles Tables,' Leyden, 1867). For an extended and corrected copy of this list, see ' Messenger of Mathematics,' December 1872, and some addi- tional corrections in the same Journal for July 1873 (t. iii. pp. 45, 46). Some years ago Mr. Slianks calculated the reciprocal of the prime number 17389 so as to exhibit the complete circulating period, consisting of 17388 figures, and placed a copy of it in the Archives of the Royal Society. Quite recently he has extended his calculation of w to 707 decimal places (Proc. Roy. Soc. vol. xxi. p. 318). Mr. Shanks has sent us three corrections to this paper : viz. the 459th, 460th, and 461st decimals in n should be 962 instead of 834, and the 513th, 514th, and 515th decimals should be 065 instead of 193; also the 75th decimal of tan "'4- should be 8 instead of 7. The two corrections in tt applj' also to the work under notice. Sharp, 1717. [T. I.] (p. 40). The first hundred multiples of |t, to 21 places. [T. II.] jb-eas of segments of circles. The area of the whole circle is taken as unity ; and the argument is the vei-sed sine (or height of the segment), the diameter being taken as unity. The table then gives areas to 17 places for arguments -0001 to -5000 at intervals of -0001, with difierences. Thus, strictly, the argument is the ratio of the height of the segment to the diameter, and the tabular result the ratio of the area of the segment to that of the whole circle. The table occupies 50 pp., and is the largest of the kind we have seen. [T. III.] Tahle for computing the solidifi/ of the xipright hiipcrhoVic section of a cone, viz. for facilitating the calculation of the volumes of segments of ON MATHEMATICAL TABLES. 129 riglit circular cones, tlie segment being contained by the base of the cone (a segment of a circle), a hyperbolic section perpendicular to the base, and the curved surface. The use of the table (which contains 500 values of the argument and occupies 5 pp.) is explained on pp. 24—26 of the work. [T. lY.] Briggian logarithms of numbers from 1 to 100, and of primes from 100 to 1100, to 61 places; also of numbers from 999,990 to 1,000,010, to 63 places, these last having first, second. . . .tenth differences added. The logarithms in this table were copied into the later editions of Shekwin and other works. The portion of the work which contains the tables is followed by a " Concise treatise of Polyedra, or solid bodies of many bases " (pp. 32). The work is universally attributed to Abraham Sharp, and no doubt exists as to his having been tlie author. [Sheepshanks, 1844.] [T. I.] Four-figure logarithms from 100 to lOOO, arranged as in seven-figure tables, with proportional parts. [T. II.] Log sines and cosines (the arguments being expressed in time) to 24"' at intervals of 1™, to four places, with proportional parts for multiples of 10' (to 60'). Also log sines to l"" for every 10^, with differences for l^ [T. III.] Log sines, cosines, tangents, and secants from 0° to 6° at intervals of 1', thence to 84° at intervals of 10', and then at intervals of 1' to 90°, to four places. In the parts of the table where the intervals are 10', differences for 1' are given. [T. IV.] Natural secants and tangents from 0° to 80"^ at intervals of 10', with differences for 1', and then to 86° at intervals of 1', with difterences for 10", to four places. [T. v.] Mochfied Gaussian logarithms. There are two tables. The first (-.^) gives log I 1 + - 1 as tabular result for argument log .r, the range of log ,^' being from "000 to -909 at intervals of "001, from -90 to 2-00 at intervals of •01, and thence to 4-0 at intervals of -1. The second table gives log I 1 — - | as tabular result, corresponding to the argument log .v, the range being from •000 to 1-000 at intervals of -001, from 1-00 to 3-00 at intervals of -01, and from 3*0 to 6-0 at intervals of •! : both tables to four places, with propor- tional parts. [T. VI.] Log sin" (^ hour angle) from 0'' to 9*^ at intervals of 1'", to four places, with proportional parts for multiples of 10' (from IlirER). [T. VII.] Autilogarithms, for logarithms from -000 to 1*000 at intervals of '001, to four places, with proportional parts. There are also two or three astronomical tables, De Morgan states that the work was issued under the title given in § .5 in 1840, and tAVO years previously without name or titlepage. It is from one of these earlier copies that the above description has been written ; we have seen no copy bearing either author's name or date. Sherwin, 1741. [T. I.] (which follows p. 35 of the introduction) gives Briggian logarithms to 61 places of all numbers to 99, and the logarithms of primes from 100 to 1097, calculated by Abraham Sharp (see Suarp, 1717, [T. IV.]). [T. II.] Briggian logarithms of thirty-five other numbers (viz. 99.9,981 — 1,000,015), to Gl places, with first, second, third, and fourth differences, to 30 places (Sharp [T. IV.]). [T. III.] Seven-figure logarithms of numbers to 1000, and from 10,000 1873. K 130 REPORT— 1873. to 101,000, with proportional parts. The proportional parts near the begin- ning of the table, being too voluminous for insertion on the page, are printed on a fl}'- sheet, and bound up facing the introductory page of the table. [T. IV.] Natural and log sines, tangents, and secants for every minute, to seven places. Differences for the logarithmic functions are added, but not for the natural ones. [T. v.] Natural and log versed sines from 0° to 90° at intervals of a minute, to seven places. Part of a page at the end of [T. V.] is occupied by a small table to convert sexagesimals into decimals, &c., and vice versa. The remaining table (of difference of latitude and departure) is not in- cluded in this llcport (see § 2, art. 12). Sherwin went through five editions ; but as none were stereotyped, some of the later are less accurate than the earher. De Morgan remarks, " Second edition, 1717; third revised by Gardiner, and the best, 1742; fifth and last, 1771, very erroneous — the most inaccurate table Hutton ever met with." In speaking of the third edition we at first thought that De Morgan should probably have written 1741 instead of 1742, as the edition we have described bears the former date, but we have since seen a copy of 1742. Wo possess an edition (1726) which contains a list of " Errata for the second edition of Sherwin's Mathematical Tables " by Gardiner. In this edi- tion, in place of [T. I.] and [T. II.] there are given two pages (pp. 28 and 29) headed " M. Brigg's {sic) Logarithms for all Numbers, from 1 to 100, and for all Ffime Numbers from 100 to 200, calculated by that Ingenious Gentleman and Indefatigable Mathematician, Mr. Abr. Sharp, at Little Horton, near Bradford in Yorkshire." The logarithms are given to from 50 to 60 places (not all to the same extent). We have also before us an edition of 1706 ; and the dedication, which is the same in aU the editions we have seen, is dated July 12, 1705. The table on pp. 27 and 28 is the same as in the edition of 1726 ; but at the end of the introduction is a table of errata, which are corrected in this latter edition. The titlepage of the editions of 1705, 1706, and 1726, and perhaps other dates, runs, " Mathematical Tables. . . .with their Construction and Use by Mr. Briggs, Mr. Wallis, Mr. Halley, Savilian Professors of Geometry in the University of Oxford, Mr. Abr. Sharp" (the names of the authors being placed one under the other); and in the edition of 1700 is added, "The whole being more correct and comjjlete than any Tables extant." Sherwin's name docs not, therefore, occur on the titlepage at aU ; but the preface is signed and the tables were prepared by him, so that the work is universally known as " Sherwin's Tables." In library catalogues, however, it wDl gene- rally be found entered under the name of Briggs, Wallis, Halley, or Sharp. In the edition of 1741, the names of Briggs, Wallis, Halley, and Sharp do not appear on the titlepage, but we have "The third edition, carefully re^^scd and corrected by William Gardiner " instead. It will be seen that there is some confusion in the editions, as, if De Morgan is correct in saying that the second edition was published in 1717, the edition of 1726 would be the third, and that of 1741 the fourth. The Royal Society's Library contains a copy with "1705" on the title- page, while the edition of 1706 (which is in the library of Trinity College, Cambridge) has the date printed in Eoman characters, MDCCVI. We have seen (in the Graves Library) the fourth edition, 1761; and the British Museum contains, besides the editions of 1717 and 1742, the fifth edition, " revised and improved by S. Clark " (1772), while the Cambridge University Library has the same edition with the date 1771. ON MATHEMATICAL TABLES. 131 The editions ^ye have Been are 1705 and 1706, 1717, 1726; the third edition 1741 and 1742, the fourth 1761, and the fifth 1771 and 1772. It thus appears that it was uot at all an uncommon thing (probably as the impression was being made up from time to time) to advance the date by one year. The first four dates we may distribute among the first two editions as we please ; most likely 1705, 1706, and 1717 for the first, and 1726 for the second. Hogg (p. 401) gives the editions as 1706, 1742, 1763, and 1771 ; but else- where (p. 262) he speaks of the fifth as of 1785, which must be incorrect. De Haan (' lets over Logarithmentafels,' p. 57) gives the dates of the editions as 1706, 1717, 1726, second 1742, 1751, 1763, fifth 1771. The subject of the dates of the editions of Sherwin is discussed at some length in the ' Monthly Notices of the Eoyal Astronomical Society ' for March and May 1873 (vol. xxxiii. pp. 344, 454, 455, 457). Mr. Lewis, in his letter to the reporter, printed in the second of these papers, mentions 1717, 1742, 1761, and 1771 as the dates of the editions he had seen, agreeing perfectly with those mentioned by De Morgan, Lalande (' Eibliog. Astron.'), and the results of our own observation. He remarks that Barlow gives 1704 and Callet 1724 as dates of editions, of which the former may bo dismissed at once as an obvious blunder. The editions therefore tliat we have not seen, but which mai/ exist, are those of 1724, 1751, and 1763. About any of these or any others we should be glad to receive information. Eogg mentions that Skerwin has often been confounded with Gardinee, even by Kiistner and Bugge. With regard to the accuracy of the tables, Httttoit writes (we quote from p. 40 of the Introduction to his tables, 3rd edit. 1801) : — " The first edition was in 1706 ; but the third edition, in 1742, which was revised by Gardiner, is esteemed the most correct of any, though containing many thousands of errors in the final figures : as to the last or fifth edition, in 1771, it is so erro- neously printed that no dependence can be placed in it, being the most in- accurate book of tables I ever knew ; I have a list of several thousand errors which I have corrected in it, as well as ia Gardiner's octavo edition." De Haan ('lets' &c., p. 20), speaking of the 1742 edition, says that it contains the logarithms of the numbers from 999,980 to 1,000,020 to 61 places ; but on examination we find that the above descrii^tion of [T, II.] is correct. The advertisement to the book itself is no doubt the source of the error ; for it is there said to contain the logarithms of the 41 numbers from 999,980 to 1,000,020, whereas it rcaUy contains the logarithms of the 35- numbers from 999,981 to 1,000,015. Sherwin's tables are of historical interest as forming part of the main line of descent from Briggs ; and the different editions cover the greater part of the last century. The chief succession (considering only logarithms of num- bers) is Briggs, Yiacq, Eoe, Joitn Newton, Skerwin, Gardiner ; and then: there are two branches, viz. Hutton founded on Sherwin, and Callet on Gardiner, the editions of Vega forming an offshoot. Shortrede (Compendious logarithmic tables), 1844. Small tables of common logarithms with sexagesimal arguments, logarithms to 12,600, anti- logarithms from to -999, log sines and tangents to 5', also from 0° to S\ and from 3° to 5° for every two minutes ; aU to five or six places. The tract only contains 10 pp. Shortrede (Tables), 1844. T. I. Seven-figure logarithms to 10,800 with characteristics, but without difterenccs, and from 10,800 to 120,000, with differences, and their first nine multiples at the bottom of the page : the uum- k2 132 REPORT— 1873. ber of degrees, minutes, and seconds corresponding to the numbers in the number-column multiplied by 10 is given throughout ; and at the top of every page are printed, to seven places, the logarithms of certain constants, viz. of 360°, 180°, 90°, 1°, 2i\ 12^ 3\ 1^, and radius (all expressed in seconds) of arc 1", TT and M the modulus. The change of figure in the line is denoted by a " nokta," the same as that employed subsequently by Mr. Sang (see Sang, § 3, art. 13) ; and its use is open to the same objections here as there. T. II. Antilogarithms, viz, numbers to logarithms from "00000 to 1*00000 at intervals of -00001, to 7 places, with diiferences and multiples at the bottom of the page. The same logarithms of constants are given on the top of the page as in T. I. ; and the change in the line is denoted in the same way. At the end of this table (p. 1 95), under the head " Useful Numbers," the logarithms of some constants are given. T. III. (pp. 59S). Log sines and tangents to every second of the circnni' ference, to 7 places (semiquadrantally arranged), the arguments throughout being also given in time. The use of the word circumference instead of quadrant in this description is justified by the fact that the signs are given for the diiferent quadrants at the top and bottom of the page : thus we have on the first page, at the top, 0° Sin +, 90° Cos—, 180° Sin — , 270° Cos +, and at the bottom 89° Cos +, 179° Sin +, 269° Cos -, 359° Cos -, and the same for the tangent and cotangent, the arguments being also expressed in time. Complete proportional parts are given throughout for tenths of a second of space, and for the first six hundredths of a second of time, both for the sine and tangent ; but near the beginning of the tables coefficients of correction for first and (sometimes) second differences are added instead. The arguments, as before stated, are given also in time ; so that corresponding to 1", 2'', 3", &c. we have -06% -13', -20', &c. This table is the most complete of the kind we know of, and is unique ; the figures are clear ; and the objection to the "nokta" docs not apply here; in one column (p. 142) there are tivo changes on the page. T. V. Seven-place log sines, tangents, and secants to every point and quarter point of the compass. T. XXXVIII. Lengths of circular arcs, viz. circular measure of 1° 2°, 3° .... 180°, of r, 2', . . . . 60', of 1", 2", .... 60", and of 1'", 2'", .... 60'", to 7 places. T. XXXIX. Proportional parts to hundredths of the reciprocal of the modulus, viz. 2-302 . . ., to 8 places. There are thirty-nine tables in the book (T. XLI. is the last ; but XXXV. and XXXVI. are accidentally omitted), the others being astronomical or me- teorological &c. The paging recommences with T. III. and proceeds to p. 634. See Shoet HEDE, 1849 (next below). Shortrede, 1849. This is a second edition of the work of 1844, and is in 2 vols. There is a preface of xxv pages to vol. i. T. I. and II. are the same as T. I. and II. in the 1844 edition; T. III. is a small ten- place table of the lengths of circular arcs. T. IV. and V. are for finding logarithms and antilogarithms to many places ; viz. colog (1 + -Oln) ,. .colog (1 + -01' w), &c. are given for n = 1, 2,.. .100, to 16 places, and colog (1 + -01 n).. .colog (1 + -Ol'^n) for n = 1, 2,. ..10, to 25 places (initial ciphers being omitted). There are added small auxiliary tables for facilitating the resolution of numbers into convenient factors. T. VI. The first hundred multiples of the modulus and its reciprocal to 32 ON MATHEMATICAL TABLES. 133 places. T. YII. (which occupies six closely printed pages). Modified Gaus- sian logarithms. B (=log 1+j) and C ( =log ^rrU ^^^ tabulated for argu- ment A (=log a-), to 5 places, from A=5 to 3 at intervals of -1 ; from A = 3 to 2-7 at intervals of -01; from A = 2-7 to 1-3 at intervals of -001 ; and from A=l-3 to 3-0 at intervals of -01, and thence to A=5 at intervals of -1. T. Yin. Log (1.2.3. .x) from x=l to .r=1000, to 5 and (for the argu- ments ending in 0) to 8 places. Then follow 2 or 3 pp. of barometric &c. tables, and a page of constants (including a small table of log —, and the same for the tangent). The second volume contains T. III. of the 1844 edition, followed by some spherical-trigonometry formulae, and the same page of constants as in vol. i. In the advertisement to the second (1849) edition, Shortrede says "a small edition of this work was published in 1844, before I had an opportu- nity of seeing it complete, which in several respects was such as I did not like. In the present edition many alterations have been made to conform it more to my views ; and for the convenience of purchasers it is now published in two separate volumes." The prices of the two volumes are, Vol. 1. 12s., and Vol. II. 30s. ; it is worth noting this, as we have seen it stated that the price of Shortrede's logarithms (by which some might understand the whole work) is 125. De Morgan says, " They [Shortrede's tables] first appeared in 1844; but some defects and errors having been found, the edition of 1844 was cancelled, and a new edition from corrected plates issued in 1849." This may be true ; but although "we have seen four copies of the 1844 edi- tion in difi'erent libraries, we were not able to obtain a sight of the 1849 edition anywhere till we bought it. Our copy of Vol. i. is dated 1849,- and of Vol. ii. 1858. There are few tables in which, relatively to the number of fgures, the pages are so clear, and the logarithmic canon to seconds is much the most complete we have seen. Every one must agree with De_ Morgan that the work shows extraordinary energy and public spirit. This is the most complete second canon in existence, and is the most accessible. Only two others have been published :— Michael Taylor, 1792, which has several defects attending its use ; and Bagay, 1829, which is scarce. A list of twenty-six errors (nearly all in the antilogarithms) is given by Shortrede himself in the 'Monthly Notices of the Eoyal Astronomical Society' for January 1864; and a supplemental list is added in the same publication for May 1867, where he says that "the unauthorized issue in 1844 contains several others." One erratum is also given in the 'Monthly Notice ' for April 1867. Shortrede adds that the great majority of the errata were communicated to him bj' Mr. Peter Gray. In the ' Insurance Eecord ' Mr. Fxlipowski charged Shortrede with having corrected his table by the aid of his (Filipowski's). That the charge was utterly unfounded is proved by the letter of Mr. Peter Gray (' Insurance Record,' June 9, 1871), who states that the errata in Dodsox were given tv Shortrede by himself (Mr. Gray) ; and we have seen reason to impute un- fairness to Mr. Fihpowski in another matter with regard to Dodson (su FiLiPOWsKi, 1849, § 4). Mr. Gray has kindly placed at our disposal his copious list of errors in Dodson, of which we hope to make use in a sub- sequent Heport. Shortrede did not pay sufficient attention to the examination of the errata- lists of previous works ; and, in consequence, his tables contain a much greater number of the hereditary errors that had descended from Vlacq than do the 134 REroiiT— 1873. best contemporary works. These errors are iusig-nificant in themselves, ex- cept iu so far as they show the acquaintance of the author of a table with the works of his predecessors, Shortredc was absent in India during the publication of the 184-1 edition (which contains seven of these errors) ; but that of 1849 was published under his own superintendence, and still it con- tains six, while Ijabbage, Hulsse's Vega, and other works of earlier date have but one. See ' Monthly Notices of the Eoy. Ast. Soc.,' March 1873, t. xxxiii. p. 33-5; and Gernerth's tract (§3, art. 13, p. 55). Stansbury, 1822. [T. 1.] Small table to convert arc into time. [T. II.] Proportional logarithms for every second to 3°, to 4 places. Same as T. 74 of Eapee. T. D. Log semitangcnts, viz. log — ^-^ from a:=0 to x= 180° at intervals of 15', to 3 places. This table occupies one page. T. G. Proportional logarithms for every minute to 24"', viz. log 1440 —log Xi the arguments being expressed iu houi'S and minutes (and also iu arc), to 4 places. T. H. (pp. 215-304). Log sines and secants, also log versed and sucovcrsed, from 0° to 90° at intervals of 15" (arg-uments also expressed in time), to 5 places. By "versed" and "sucovcrsed" are meant " scmiverscd sine ''and "scraisu- coversed sine " (the terms introduced by De Mcndoza y Eios being used for 1 -I- cos X 1 -l- sm X brevity, see Rios, 1809); so that the table gives log ^ — and log — . This table was copied from T. XYI. of Eios ; but there is a difference of arrangement, as the original table gave log sines, cosines, &c., the arrange- ment being scmiquadrantal, while in the present work it is quadrantal. T. X. Pivc-figure logarithms from lOOU to 10,000 ; no differences. T. Y. Halves of natural sines, viz. | sin x from .r=0° to ar=90° at in- tervals of a minute, to 5 places, with proportional parts for seconds. The other tables are nautical. Stegmann, 1855. T. I. Six-figure logarithms to 119, and five-figure logarithms, with differences, from 1000 to 10,000. T. II. AntHogarithms from -0000 to -9999, to 5 j)laces. A few tables of atomic weights &c. are added. As in Filipoavski's tables, the terminal 5 is replaced by the Eoman V when it lias been increased. The preface to these tables is signed by Stegmann, but his name does not appear on the titlepage. ^Stegijaanu. This work we have not seen. Three errata in it are given by Prof. Wackerbarth in. the 'Monthly Notices of the Eoyal Astronomical Society ' for April 1867 : and this is the only place in which wo have seen the table referred to. It is very possibly a five-figure hyperbolic logarithmic table, similar to the same author's table of common logarithms just de- scribed. Janet Taylor, 1833. T. XVII, Log sines, tangents, and secants to every quarter point, to G places. T. XVIII. Six-figure logarithms of numbers to 10,000. T. XIX. Log sines and tangents for every 10" to 2°, and log sines, tan- gents, and secants for every minute of tho quadrant, to 6 i)laccs, with dif- ferences. T. XX. Natural sines for every mimite of the quadrant, to 6 places, T. XXI. Log versed sines to 8'' at intervals of 5**, to 5 places. T. XXXVI. Proportional logarithms for every second to 3°, to 4 places ; same as T. 74 of Eapee. ■ . ■• . ■ -..-.. ON MATHEMATICAL TABLES. 133 At the cud of tlio preface Mrs. Taylor makes the following curious re- mark : — " Some errors have crept into the calculations from the multiplicity of entries &c. ; these, I trust, will claim the indulgence of the public ; for the system on which I have worked being mathematically correct, and founded on sound principles, any slight oversight in the figures can be of but little moment, and very easily rectified." It is to be presumed that this does not refer to the tables included in this Eeport, as they would not havo been calculated afresh. Mrs. Taylor was also the author of a work on navigation, the tables in which arc described below. Janet Taylor, IS-iS. T. 3. Log sines, tangents, and secants to every quarter point, to 6 places. T. 4. Six-figure logarithms of numbers to 10,000. T. 5. Log sines and tangents for every 10" to 2° ; and log sines, tangents, and secants for eveiy minute of the quadrant, to 6 places, with differences. T. 30. Log versed sines for every 5' to 8^ to 5 places. T. 32. Natural sines for every minute of the quadrant, to 6 places. T, 35. Proportional logarithms for every second to 3°, to 4 places ; same as T. 74 of Rapee. Mrs. Taylor, as we learn from an advertisement, kept a nautical academy in the Minorics. Michael Taylor, 1792. [T. I.] Logarithms of numbers to 1260, to 7 places. [T. II.] Logarithms of numbers from 10,000 to 101,000, to 7 places, with differences and proportional parts. The change in the tliird figure, in the middle of the line is not marked. [T. III.] Table of log sines and tangents to every second of the quadrant, to 7 places (semiquadrantally arranged). The change in the leading figures, when it occurs in the middle of the column, is not marked at all ; and it requires very great care in using the table to prevent errors from this cause. If any one is likely to have to make much use of the table, it will be worth his while to go through the whole of it, and fiU in with ink the first after the change (making it a black circle such as is used to denote full moon in almanacs), and also to make some mark that will catch the eye at the top of every column containing a change. This will be a work of con- siderable labour, but is absolutely necessary to ensure accuracy. It is uo doubt chiefly on account of the absence of any mark at a change that Bagat has so completely superseded this table, though difference of size &c. are also in favour of the former. [T. I.] and [T. II.] present no novelty ; but [T. III.] is an enormous table, containing about 450 pages, with an average number of about 7750 figures to a page, so that it contains nearly three milhons and a half of figures. The left-hand pages contain sines and cosines, the right-hand tangents and cotangents. This is unfortunate, as the sines and cosines (which are used far more frequently than the tangents and cotangents) are thus separated at least a foot from the computer's paper as he works with the table on his left ; and it is well known that the number of errors of transcription is * proportional to the distance the eye has to carry the numbers, [T. III.] was calculated by interpolation from YLAca's ' Trigonometria Ai-tificiahs,' to 10 places, and then contracted to 7 ; so that the last figure should always^ be correct. Taylor was a computer in the Nautical Almanac Office ; he unfor- tunately died almost at the moment of the completion of his work, only five pages remaining unfinished in the press at the time of his death. These 136 KEPOHT— 1873. were cxamiued, and the introduction &c. written, by Maskelync. Some errata, found among Taylor's papers, are given on p. 64 of the work ; and a list of nineteen errata signed by Pond is published iu the ' Nautical Almanac' for 1833. To this list is appended the remark: — "The above errata were detected by collating Taylor's Logarithms with tlic French manuscript tables, now the property of C. Babbage, Esq. The arrangement for this examina- tion was made by the late lamented Dr. Young ; a few days only before his death he gave directions for its completion. — J. Pond." We do not know any thing further with regard to this examination, though the fact that certain errors were found in Taylor by comparison with the French tables is weU known ; but there must be some mistake, as the French tables could not have been even temporarily the property of Babbage. In the preface to his tables Babbage states that while on a visit to Paris he availed himself of the opportunity of consulting the great manuscript tables presei'ved at the Observatory, and'that he " enjoyed every facility for making the comparisons which were requisite for this purpose [the preparation of his seven-figure table], as well as making extracts necessary to me for other calculations." Bagay intimates in his preface that he had found 76 errors in Taylor. Taylor was also the author of the Sexagesimal Table (§ 3, art. 9) ; and we cannot but admire the undaunted perseverance that could enable him to com- plete such monuments of industry in addition to his routine work as computer in a laborious office. Thomson, 1S52. T. I. One-page table to convert arc into time. T. X. Locjarithms for folding the correction of the sun's declination &c., viz. log 1440 — log X, from x=l to .r= 1440, to 4 places. T. XI. Logarithms of the latitude and polar distance, viz. log secants to every minute of the quadrant, to 5 places, without differences; quadrantally arranged. T. XII. Logarithms of the half sum and difference, viz. log sines and cosines to everj- minute of the quadrant, to 5 places, without differences ; qua- drantally arranged. 1? T. XIII. Logarithms of the apparent time or horary angle, viz. 2 log sin ^ from .r=:0'' to a'=9'' at intervals of 10", with proportional parts for seconds, to 5 places. T. XV. Logarithms of the apparent altitudes, viz. log cosec x — •5400, fi'om A"=rO° to .r=89°, at intervals of a minute, to 4 places. T. XVI. Logarithms of the apparent distance, viz. log sines and tangents for every minute, from 18° to 90°, to 4 places. T. XIX. Four-place proportional logarithms for every second to 3° ; same as T. 74 of IIapeh. T. XXIII. Logarithms of the sum and difference, viz. log sin ^, from x — 0° to .^'=180°, at intervals of a minute, to 6 places. T. XXIV. Six-figure logarithms of numbers from 1000 to 10,000, with differences and tables for interpolating at the foot of the page. In this book it is only required to find numbers corresponding to logarithms ; and the tables are constructed with this view. There arc given, therefore, the usual differences (called first differences), and the aj>proximate results of the divi- sion of 1, 2, o, . . . . 10, and ten or more higher numbers by them. By the second difference is meant the difference between the given logarithm and the logarithm next below it iu the table. ON MATHliMATlCAL TABLES. 137 T. XXV. Natural versed sines for every minute to 120°, to 6 places, with proportional parts for seconds. The other tables are nautical &c. Trotter, 1841. [T. I.] Six-figure logarithms of numbers to 10,000, with differences. This is followed by a small table to convert Briggian into hyperbolic logarithms drc. [T. II.] Log sines, tangents, and secants to every quarter point, to 6 places. [T. III.] Log sines and tangents for every fifth minute of tlic quadrant, to 6 places. [T. lY.] Natural sines and tangents for every fifth minute of the quadrant, to 6 places. [T. v.] Areas of circular segments, to 6 places ; same as T. XIII. of Hantschl. [T. VI.] Squares, cubes, square and cube roots (to 6 places) for numbers from 1 to 1000. [T. VII.] Circular measure of 1°, 2°, . . . . 180°, of 1', ... . 60', of 1", ... . 60", and of 1'", 60'", to 7 places. [T. VIII.] lleciprocals of numbers from 1 to 500, to 9 places. [T. IX.] Logarithms of ntimbers from 1000 to 1100, to 7 places. [T. X.] Lengths of sides of inscribed and circumscribed polj'gons (up to a 20-sided figure), the diameter of the circle being unity, to 7 places. [T. XL] Hyperbolic logarithms of numbers from 1 to 100, to 8 places, [T. XII.] For finding the areas of oblong and oblate spheroids. A few constants are given. The other tables are astronomical, meteorological, &c. Some trigonometry &c. is prefixed at the beginning (i^p. 102). Turkish Logarithms &c. [1834]. The book commences on the ]a.st page ; and the first table gives seven-figure logarithms of numbers from unity to 10,080, arranged consecutively in columns, there being three columns of arguments and tabular results to the page. The tables begin at the last page, as before remarked, the extreme right-hand column being the first column of arguments ; to the left of it is the corresponding column of tabular results, then to the left of that the second column of arguments, and so on. The table occupies 84 pp. (up to p. 85). Then " ^Uows " a table of log sines and tano-ents for every minute of the quadrant (semiquadrantally aiTanged), the sines and cosines being side by side, and separated by some " white " from the tangents and cotangents. This table occupies 90 pp., and is followed by a similar table of natural sines and tangents (to 7 places), which also occupies 90 pp. Except that the table runs in the wrong dii-ection, it only differs from an ordinary table in the ten digits being denoted by different marks from those to which we are accustomed. A few minutes' practice, however, is quite sufficient to get used to the new numerals ; and then the table could be used as well as any other. There is no introductory or explanatory matter. The book is in the British Museum ; and the place and date in § 5 are taken from the Catalogue of the Library. Ursinus, 1827. [T. I.] Six-figure logarithms to 1000, and from 10,000 to 100,000, without differences ; the values of S and T for finding log sines and tangents of angles below 2° 46' 40" (see § 3, art. 13) are given at the top of the page. [T. II.] Log sines and tangents for every 10 seconds throughout the quadrant, with differences, to 6 places. [T. III.] Longitudes of circular arcs, viz. circular measure of 1°, 2°, 3°, . . . . 360°, of 1', 2', 60', and of 1", 2", 60", to 7 places. These are followed 138 iiEPOKT~1873. by a page giving the siucs of 3°, 6°, 9°, . . . . 87° accurately (/. e. expressed as radicals). [T. IV.] Longitudes of chords, viz. lengths of chords subtending given angles (the arguments) at the centre. The arguments proceed from 0° to 108°, at intervals often minutes, and thence to 180° at intervals of 1°; and the tabular results are given to 3 i^laces. [T. v.] Abacus irlyonometrlcus, viz. natural sines, tangents, and secants, and log sines and tangents from 0° to 90° (quadrantally arranged), to every ten minutes, to 6 places. Then follow a few formiila; and con- stants. Vega (Thesaurus, fol. 1794). T. I. (Magnus Canon logarithmorum vulgarium). Logarithms of numbers from 1 to 1000, without differences, and from 10,000 to 100,999, with diiferenccs, to 10 places, arranged like an ordinary seven-figure table. Proportional parts are also given, but only for the first two or three figures of the difference. The table can thus be used as an ordinary seven-figure table. A change in the fourth figure in the middle of the line is denoted by an asterisk prefixed to all the logarithms affected. T. I. occupies pp. 1-310. The last page and a half arc devoted to multiples of the modulus, a few constants, and a tabic to convert degrees (1° to 360°) and minutes (1' to 60') into seconds. T. II. (Magnus Canon logarithmorum vulgarium trigonometricus). Log sines, cosines, tangents, and cotangents, from 0° to 2° at intervals of one second, to 10 places, without differences, and for the rest of the quadrant at intervals of ten seconds, also to 10 places, with differences. All this occupies pp. 311-629, and is followed by 3 pp. containing natural sines for angles less than twelve minutes, to every second, to 12 places. The appendix occupies pp. 633-685 : p. 633 contains formulte ; and pp. 634 and 635 are occupied with tables of the longitudes of circular arcs &c. Of these the first gives the circular measure of 1°, 2°, 3°, . . . . 360°, the second of 1', 2', 3', 60', the third of 1", 2", 3", 60", aU to U places ; the fourth is a small table to express minutes and seconds as fractions of a degree. Pp. 636- 640 are occupied with formuhc for the solution of triangles ; and on pp. 641- 684 [T. III.] we have Wolfram's great table of hyperbolic logarithms (see ScHTJLZE, § 4). The six omittcil in ScmjLZE are given ; and it is stated in the preface that several errors have been corrected. The error pointed out by Mr. Gray (see Scuulze [T. II.]) is reproduced. An error in log^ 1099 is pointed out by Prof. Wackerbarth in the ' Monthly Notices of the Iloyal Astronomical Society' for April 1867. Some of the errata found in Ylacq are indicated in the preface. These are, as a rule, corrected in the book ; others, given in a list at the end of the in- troduction, were found after the printing, and must be corrected in manu- script before use. There is a third list at the end of the work (p. 685) ; but it is identical with that at the end of the introduction. In some copies the list at the end of the introduction is much more com- plete than in others, the errors in Vlacq being marked by an asterisk, and the errata being also given in Latin and German. It is probable that additional errata were found before the edition was all made up, and that the original list was suppressed and the new one substituted. In all copies the titlepage is the same. See ' Monthly Notices of the Eoy. Ast. Soc.,' June 1872, and May 1873 (p. 454). There is a great difference in the appearance of different copies of the work. In some the tables are beautifully printed on thick white paper, with wide margin, so that the book forms one of the hand'jomgst collections of tables we ON MATHEMATICAL TABLES. 139 have seen ; while in others the paper is thin and discoloured ; all arc printed from the same type. The arrangement of T. I. (though about half the space that would be required if the logarithms and differenees were written at length is thereby saved) is not nearly so convenient as in Vlacq ; 1628, for there is danger of taking out a wroug difference. Yega took great pains to free his tables of logarithms of num- bers from error ; and he detected all the hereditary errors that had descended from Vlacq which affected the first seven figures of the logarithms. But as several of these errors were corrected in his errata-list and not in the text, his successors, who failed to study these lists sufiiciently, were reaUy less accurate than he was. The last thousand logarithms that appear for the first time in this work were calculated by Lieut. Dorfmund at Vega's instigation. T. II. is not reprinted entirely from Vlacq's ' Trigonometria Artificialis,* as the logarithms for every second of the first two degrees Avere calculated for the work by Lieut. Dorfmund. Vega seems not to have bestowed on the tri- gonometrical canon any thing approaching to the care he devoted to the loga- rithms of numbers, as Gauss estimates the number of last-figure errors at from 31,983 to 47,746 (most of them only amounting to a unit, but some to as much as 3 or even 4). Vega offered a reward of a ducat for every error found in his table ; and it is to be inferred fi'om his preface that he intended to regard inaccuracies of a unit as such, so that it was fortunate that no contemporary of his made an examination similar to Gauss's. The paper of Gauss's in which this estimate occurs is entitled " Einige Bemerkungen zu Vega's Thesaurus Logarithmo- rum," and appeared in the ' Astronomische Nachrichten,' No. 756, for May 2, 1851 (reprinted ' Werke,' t. iii. pp. 257-264). It contains an examination of the relative numbers and magnitudes of the last-figure errors that occur in the sine, cosine, and tangent columns. It is easily shown that the tan- gents were formed by mere subtraction from the sine and cosine columns ; but Gauss was unable to explain the fact that the cosines were more accu- rate than the sines, which appeared as one of the results of the examination. This question is further discussed in the ' Monthly Notices of the Eoy. Ast. Soe. ' for May 1873 ; and it is there shown by the reporter that this result is a direct consequence of the formula by means of which Vlacq calculated the table. So long as all these errors remain uncorrected, the logarithmic trigo- nometrical canon cannot be considered to be in a satisfactory state, as it is certainly desirable that a reliable ten-place table should exist. "We believe no perfect list of errors in Vega has been given : a number of errors in T. I. are given by Lefort (' Annales de I'Observatoire de Paris,' t. iv.) ; but this list could not, from the manner in which it was formed, in- clude any errors that did not also occur in Vlacq. A long list of errors in the trigonometrical tables of Vega is given by Gronau, ' Tafeln fiir die hyperbolischen Sectoren' &e. Dantzig, 1862, p. vi. Copies of Vega are still procurable (but with difficulty and delay) from Germany, through a foreign bookseller, for about =£1 10s. or £1 15s. Vega (Manuale), 1800. T. I. Seven-figure logarithms to 1000, and from 10,000 to 101,000, with proportional parts. The change in the line is denoted by an asterisk prefixed to the fourth figure of all the logarithms affected. A few constants are given on p. 188. T. II. Log sines, tangents, and arcs for the first minute to every tenth of a second. Although there is a triple heading, there is but a single column of tabular results, as for such small angles the sines, tangents, and arcs arc equal to one another. 140 KEPOKT— 1873. Log sines, cosines, tangents, and cotangents, from 0° to 6° 3' at intervals of 10", and thence to 45° at intervals of 1', to 7 places, with diiferences for 1" throughout. An Appendix contains some spherical trigonometry. One page (p. 297) contains longitudes of arcs, viz. circular measure of 1°, 2°, . . . . 90°, and by intervals of 10° to 180° ; also of 360°, of 1', 2', . . . . 60', and of 1", 2", .... 60", to 8 places. At the end some errata are given, and also some in Callex and other -works. The description of this work, according to order of date, should follow the next ; but as it is referred to in the latter it is convenient to place it first, Vega (Tabulae), 1797. Yol. i. — T. I. is identical, page for page, with T. I. of Vega's ' Manualc ' just described, and was most likely printed from the same type. The constants &c. on p. 188 are also identical. T. II. is also identical with T. II. of the ' Manuale,' only with the addition of 40 more pages, containing log sines and tangents from 0° for every second to 1° 30' 0", to 7 places, without differences. Thus the ' Tabulae' and the ' Manuale ' agree to p. 193 ; then the 40 pp. are inserted in the ' Tabulae,' and pp. 233-330 of the ' Tabulas ' are identical with pp. 193-290 of the ' Maniuile,' the coincident portions of the two works being doubtless printed from the same type. T. III. Natural sines and tangents to every minute of the quadrant, to 7 places, with differences for one second throughout. The Appendix contains a table of circular arcs, viz. the circular measure of 1°, 2°, 3°, . . . . 360°, of 1', 2', . . . . 60', and of 1", 2", .... 60" (with the cor- responding number of seconds in these angles), to 8 places, and small tables for the conversion of arc into time, and hours &c. into decimals of a day. On pp. 407-409 are given one or two constants connected with the calcula- tion of TT, the values of a few radicals, and the expression for the sine of every third degree in radicals. Some errata are given at the end of the introduction. Yol. ii. — T. I. Table of all the simple divisors of numbers below 102,000 (2, 3, and 5 excluded) ; a, b, e, d arc printed for 11, 13, 17, 19, to save room. This is followed by primes from 102,000 to 400,000. Ciieenac (§ 3, art. 8) found 39 errors in this table : see his preface. T. II. Hyperbolic logarithms of numbers to 1000, and of primes from 1000 to 10,000, to 8 places. This table is followed by the first 45, 36, and 27 powers of 2, 3, and 5 respectively. T. III. gives e^ and Briggian log c^ (the former to 7 figures, the latter to 7 places), from .r=0-00 to ,i~ 10-00 at intervals of -01, T. IV. The first nine powers of numbers from 1 to 100, squares from 1 to 1000, cubes from 1 to 1000, and square and cube roots of numbers from 1 to 100, to 7 places. T. Y. Logistic logarithms, viz. log 3600 — log (number of seconds in argu- ment), for every second to 1° ( = 3600"), to 4 places. [T. YL] The first six binomial-theorem coefiicients, viz. x, '-^t— s> • • • • x{x—l) (.^■— 5) , , . ^ — 1^ H J from A'=-01 to .r=l-00 at intervals of -01, to 7 places. 1 13 1 13 This is followed by a page of tables, giving j-^, ^^' g , ^-^, 2~i~5' 1-3 . . • • • • 2~Z ^^'j to 10 places, with their logarithms to 7 places. ON MATHEMATICAL TABLES. 141 The rest of the book is devoted to astronomical tables and formiilfe, except two remarkable tables at the end (pp. 364-371). The first of these [T. VII.] is most simply desci'ibed by stating that it gives the number of shot in a py- ramidal pile on a square base, the number n of shot in the side of the base being the argument ; the table extends from n=2 to n = 40. There is also given the number of shot in a pyramidal pile on a rectangular base, tlie ar- guments being n the number of shot in the breadth of the base, and m the number of shot in the top row (so that m+n — 1 is the number in the length of the base). The ranges are, for ?i, 2 to 40, andfor jji, 2 to 44, the table being of double entry. [T. VIII.] gives the number of shot in a pyramidal pile on a triangular base, the number of shot in a side of the base being the argument, which extends from 2 to 40. The other portion of the table is headed " Tabula pro acervis globorum oblongis, ab utraque extremitate ad pyramides quadri- latei'as appositis;" and the explanation is as follows: — Suppose we have two pyramidal piles of shot on square bases (n shot on each side) placed facing one another, at a distance equal to the sum of the diameters of m shot apart ; and suppose it is required to fill this interval up, so as to make a pyra- midal pile on a rectangular base, then this table gives the number for n (latus) to n=40, and for m (longitude baseos) to mi =44, the table being of double entry. Some errata are given after the introduction. "We have seen the third edition (Leipzig, 1812) ; aiid though wo have not compared it side by side with the second (here described), we feel no doubt the contents are identical ; at all events the number of pages in each volume 8 the same, and the preface is dated 1797 in both. Vlacq (Arithmetica Logarithmica), Gouda, 1628, and London, 1631. \T. I.] Ten-figure logarithms of numbers from 1 to 100,000, with differ- ences. This table occupies 667 pages. [T. II.] Log sines, tangents, and secants for every minute of the quadrant, to 10 places, with interscript differences ; semiquadrantally arranged. This table occupies 90 pp. In the English copies, by George Miller, there is an English introduction of 54 pp., and then follows a table of latitudes (8 pp.). The original edition of 1G28 has 79 pp. of introduction ; and a list of errata is given, which does not occur in Miller's copies (but see ' Monthly Notices of the Eoy. Ast. Soc' t. xxxiii. pp. 452, 456, May, 1873). There were also copies with a French titlepage ; and in these there is an Introduction in the same language of 84 pp. We suspect that a Dutch edition was contemplated, but that the copies of the table intended for this purpose afterwards formed Miller's English edition : no Dutch edition is known to exist (see Phil. Mag., May 1873). The titles of the three editions are given in full in § 5 ; in all, the tabular portion is from the same type. The bibli- ograjjhy of this work forms an essential part of the history of logarithms ; and a good many of the references occurring in the introductory remarks to § 3, art. 13, have reference to it. The table of logarithms of numbers contains about 300 errors, exclusive of those affecting the last figure by a unit ; but a good many of these have reference to the portion below 10,000, which need never be used. This is still the most convenient ten-figure table there is (Vega, fol. 1794, is the only othei') ; but before use the known errata should be corrected. References to all the places where the requisite errata-lists are to be found are given in the ' Monthly Notices of the Eoy. Ast. Soc.' for May and June, 1872. We intend, 142 KEPORT— 1873, however, in the next Eeport to give a complete list of errors in the portion of the table from 10,000 to 100,000, We succeeded in obtaining a copy of this work after some difficulty ; Mr. Merrifield informs us that copies have always been procurable from abroad for about £2. Vlacq (Trigonometria Artiflcialis), 1633. [T. I.] Log sines and tan- gents to every ton seconds of the quadrant, to 10 placet?, with characteristics and differences (not interscript) ; semiquadrantaUy arranged. The table occupies 270 pp. [T. II.] Ten-figure logarithms of niimbers to 20,000, with differences, printed from the same type as that used in the ' Arithraetica '(1628 and 1631) (excei)t the last 500). A list of errata is given on the last page. The trigo- nometry «S:c. at the beginning occupies 52 pp. See § 3, art. 15 (introductory remarks), and also Vega (foh), 1794. Vlacq, 1081. This is one of the numerous small editions called after Ylacq, on the Gellibrand model. The contents, shape of type, &c. are exactly the same as in Hentscuen (Vlacq), 1757, § 4, except that in the latter the " whites " arc rather wider. The printed portion of the page of tables is 3| in. l)y 5:j in. There are 48 pp. of trigonometry &c. in Latin. No namo except Vlacq's appears in connexion with the work. [T. I.] ISTatural sines, tangents, and secants, and log sines and tangents for every minute, to 7 places. [T. li.] Logarithms of numbers from 1 to 10,000, arranged consecutively in columns, to 7 places ; no differences. In one of tlie copies we have seen there are several errors corrected in manuscript. This edition must be rather common in England, as wo have seen several copies. Wackerbarth, 1807. T. I. Pivo-figure logarithms (arranged as in seven-figure tables) to 100, and from 1000 tc^ 10,000, with proportional parts to tenths (/. e. multiples of the differences). The degrees, minutes, &c. corresponding to eight numbers on the page are given at the bottom of each. At the end of this table tbere are added seven-figure logarithms of numbers from 10 to 100, and also from 10,000 to 11,000, the latter with proportional parts to tenths. T. IL Log (1.2.3. ....t) for .r=l, 2,. . . .100 ; log (1 .3. 5. . . ..r) for .r=l, 3, 5 65 ; log (2 . 4 . 6 .^■) for w=2, 4, 6 06 : all to 5 places. T. III. Log sines and tangents for every second from 0' to 10' ; log sines and tangents for every ten seconds from 0° to 5° ; log sines and tangents for every minute of the quadrant : all to 5 places. Differences are added throughout, and also proportional parts to tenths (i. e. multiples of the diflferences) for every second to 5°, aud for every 10 seconds in the other portion of the table. T. IV. Circular measure of 1°,2°,. . . .180°, of l',2',. . . .60', and of 1", 2", .... 00", to 5 places. Some constants, such as the unit arc, its logarithm &c., are added. T. V. Hyperbolic logarithms of numbers from 1 to 1010, to 5 places, with proportional parts to tenths, arranged as in seven-figure tables of Briggian logarithms ; followed by the first hundred multiples of the modulus and its reciprocal, to 5 places. A few constants, tt, e, &c., are given, to 30 places, T. VI. Squares of numbers from 1 to 1000. T. VII. Square roots (to 7 places) of numbers from 1 to 1000. T. VIII. Natural sines, cosines, tangents, and cotangents for every 10' to 5°, thence for every 20' to 15°, aud thence to 45° at intervals of 30', to 3 places. ON MATHEMATICAL TABLES. 143 T. TX. Tlcciprocals (to 7 places) of numbers from 1 to 1010. T. XVII. List of primes to 10B3. T. XXI. gives some constants. The other tables are chemical &c. This is one of the most complete five-figure tables we have seen. Tlio change in the leading figures, -where it occurs in the midrlle of a line, is throughout denoted by an asterisk prefixed to the third figure of all the logarithms afl'ected. It may be remarked that though the introduction &c. is in Swedish, the headings of the tables are in Latin. . A list of four errata in the tables is given by Prof. Wackerbarth himself in the ' Monthly Notices of the Iloyal Astronomical Society,' t. xxxi. No. 9 (Supplementary Number, 1871). Wallace, 1815. [T. I.] Six-figure logarithms to 100, and from 1000 to 10,000, "with differences. [T. II.] Log sines, tangents, and secants to every minute of the quadrant, to places, with differences. [T. III.J Natural sines to every minute of the quadrant, to 5 places. This is followed by a traverse table. The tables are preceded by 148 pp. of trigonometry &c. ^VaI•nstorff*s Schumacher, 1845. Out of 221 pages, only 21 (pp. 116-120 and 206-221) come within the scope of this Report. [T. I.] For the conversion of arc into time, and vice versa. [T. II.] The circular measure of ]°, 2=, 90°, 95° 120°, 130° 360°, of 1', 2' ... . 60', and of 1", 2", .... 60", to 7 places, [T. III.] Four-figure logarithms to 1009. [T. IV.] Log sines, cosines, tangents, and cotangents at intervals of 4' to 10°, and thence to 45° at intervals of 10', to 4 places. [T. v.] Gaussian logarithms; B and C are given for argument A from A = •00 to 1-80 at intervals of -01, and thence to 4-0 at intervals of -1, to 4 places, with differences. The other tables are astronomical. "Willich, 1853. T. XX. Seven-figure logarithms to 1200, followed by a few constants, &c, T. XXI. Squares, cubes, square and cube roots (to 7 places), and reci- procals (to 9 places) of numbers to 343, followed by some constants. T. A. Hyperbolic logarithms of numbers from 1 to 1200, to 7 places. T. B. Natural and log sines, tangents, secants, and versed sines, for every half degree, to 7 places. T. C. Circumferences and areas of circles for a given diameter, viz. -nd (to 5 places) and -j- (to 2 jjlaces) for d^l, 2, . . . .9, and from d=l to 100 at intervals of -25. T. D. Circular measure of 1°, 2°, 1 80°, to 7 places. The other tables in the work are of a very varied character. We have also seen the second edition (1852), which does not contain the tables A to D ; and we nave seen a review of the seventh edition, edited by M. Marriott, 1871. § 5. List ofworhs containing Tables that are described in this Report, with refer^ cnees to the section and article in which the description of their contents is to he found. [Thoso works to which an asterisk is prefixed have not come under tho inspection of the reporter ; and the description of their contents is therefore 144 REPORT — 1873. derived from some secondhand source. The author's name is enclosed within square brackets when it docs not occur on the titlepaiye of the work. For other explanations see § 2, arts. 4-14, and § 6 (Postscript), arts. 2-4, 8, 10-12.1 AcADEMiE RoYALE . . . DE Prttsse, Public SOUS la direction de 1'. Recueil de Tables Astronomiques. Berlin, 1776. 3 vols. 8vo. § 4. Adams, Johx. The Mathematician's Companion, or a Table of Logarithms from 1 to 10,800 . . . London, 1796, 8vo. § 4. Airy, G. B., Computed under the direction of; Appendix to the Green-wich Observations, 1837. Loudon, 1838. 4to. § 3, art. 15. Alsteditts, J. H. Scientiarum omnium eneyclopsediae tomus primus . . . Lugduni, 1649 (2 vols. fol.). § 3, art. 4. Andeeay, James. Astronomical and Nautical Tables, with Precepts , . . London, 1805. 8vo (pp. 263). § 4. Anonymous. MultiplicationstabeUe, enthaltend die Producte aller ganzen Pactoren von 1 bis 1000, mit 1 bis 100. Kopcnhagen, 1793. 4to (pp. 247 ; and introduction, pp. 8). § 3, art. 1. Anonymous. Tables de Midtiplication . . . Paris, 1812. § 3, art. 1. Anonymous. Tafel logistischer Logarithmen. Zugabe zu den Vcga-Hiils- se'schen und anderen Logarithmen- Tafeln. Aus Callet's " Tables de Loga- rithmes." Niirnberg. Verlag von Riegel & Wiessuer. 1843 (table, 7 pp.). § 3, art. 18. Anonymous (1844). See Sheepshanks. . Anonymous. Logarithmen. Antilogarithmcn. Berlin. [On a card, 1860 ?] §4. • Auxiliary Tables. See [Schumacher .] Babbaoe, Charles. Table of the Logarithms of the Natural Numbers from 1 to 108000. . . Stereotyped. Pourth impression. London, 1841 (202 pp. and explanations &c. xx). § 3, art. 13. [The 1838 edition (or rather tirar/c) has the following notice of errata contained in it, on the back of the titlepagc : " In the logarithms of 10354, G0670 to 9, 70634 to 9, and 106611 to 9, the fourth figures ought to be small instead of large. In the list of constants the last figure of the value of e should be 8 instead of 9." The tables were stereotyped from their first publication in 1827. Mr. W. Barrett Davis has called our attention to the number of last- figure unit errors iu the portion of the table beyond 100,000 ; thus on p. 192 there are no less than fifteen such errors which are corrected in more recent works, such as Schron and Ivohler. This portion of the table Babbage copied from Callet.] Barrage Catalogue. Mathematical and Scientific Library of the late Charles Babbage of No. 1 Dorset Street, Manchester Square. To be sold by Private Contract. . . . Printed by C. P. Hodgson and Son, Gough Square, Fleet Street [London], 1872. [The catalogue was drawn up by Mr. Robert Tucker, M.A., Honorary Secretary of the London Mathematical Society; and the library was purchased by Lord Lindsay.] Bagay, V. Nouvelles Tables Astronomiques et Hydrographiques .... Edition stcre'otype. . . Paris, Firmin Didot, 1829. Small 4to. § 4. Barlow, Peter. New Mathematical Tables containing the factors, squares, cubes, square roots, cube roots, reciprocals, and hyperbolic logaritlims of all numbers from 1 to 10,000, . . . London, 1814. 8vo (pp. 336, and intro- duction Ixi). § 4. Barlow's Tables of Squares, Cubes, Square roots. Cube roots. Reciprocals of all integer numbers up to 10,000. Stereotype edition, examined and cor- rected. (Under the Superintendence of the Society for the Diffusion of Usefu ON MATHEMATICAL TABLES, 145 Knowledge.) London, 1851, from the stereotyped plates of 1840. Svo (pp. 20.)). § 3, arts. 4 and 7. Eates, David. Logarithmic Tables, containing the logarithms of all num- bers from 1 to 10 000, together with . . . Dublin, 1781. (63 pp. of tables, introduction ccxi pp., and appendix GO pp.) § 4. Bkarbmore, Nathaniel. Manual of Hydrology : containing . . . London, 1862. Svo (pp. 384). § 4. Beexoulli, John. A Sexcentenary Table . . . Published by order of the Commissioners of Longitude. London, 1779. 4to (pp. 165 ; and intro- duction, viii). § 3, art. 9. Berthoud, r. Les Longitudes par la mesure du temps . . . Paris, 1775. Small 4to (34 pp. of tables). § 3, art. 15. Bessel. See [Schumacher.] Beverley, Thomas. The Mariner's Latitude and Longitude Ready-com- puter . . . Cirencester (no date ; but Appendix dated 1833). 4to (pp. 290). §4. Blanchaed. See Gardiner (Avignon edition, 1770). Bonntcastle, John. An Introduction to Mensuration .... The fifteenth edition . . . London, 1831. Small Svo. § 3, art. 22. BoRDA, Ch. Tables trigonometriques decimales ou Tables des logarithmes . . . revues, augmente'cs et publiees, par J. B. J. Delambre. Paris, An ix. [1800 or 1801]. Small 4to. § 4. BowDiTCH, N. The improved Practical Navigator ; ... to which is added a number of new Tables .... Revised, recalculated and newly arranged by Thomas Kirbt. London, 1802. Svo. § 4. Bremiker, C. Tafel der Proportionaltheile zum Gebrauche bei logarith- mischen Rechnungen mit besouderer Berlicksichtigung der Logarithmentafeln von Callct und Vega. . . BerUn, 1843. Svo (pp. 127). § 3, art. 2. Bremiker, C. Logarithmorum VI decimalium nova tabula Berolinensis . . . Beroliui, 1852. Svo. § 4. Bremiker's Vega. See Vega (1857). Bremiker. See Crelle (1864). Bretschneider, C. a. Produktentafel enthaltend die 2, 3 .... 9 fachen aller Zahlen von 1 bis 100 000. Hamburg und Gotha, 1841. Svo (pp. 110). § 3, art. 1. Brigge, H. Tables des Logarithmes . . . 1626. See under de Decker, 1626, § 4. [Briggs, Henrt.] Logarithmorum Chilias Prima. [London, 1617.] Small Svo (pp. 16). § 3, art. 13. BiUGGS, Henrt. Arithmetica logarithmica sive logarithmorum chiliades triginta, pro numeris naturali serie crescentibus ab unitate ad 20,000 : et a 90,000 ad 100,000. Quorum ope multa perficiuntur Arithmetica problemata et Geomctrica. Hos numeros primus invenit clai'issimus vir lohannes JN'epe- rus Baro Merchistonij ; eos autera ex eiusdem sententia mutavit, eorumque ortum et usum illustravit HenricusBriggius, iu celebcrrima AcademiaOxoniensi Geometrite professor Savilianus. Dens nobis usuram vitse dedit et ingenii, tanquam pecunife, nulla prsestituta die. [Royal arms, I. R.] Londini, Ex- cudebat Gulielmus lones, 1624. folio (preface &c. 6j)p., trigonometry 88 pp. ; tables unpaged). § 3, art. 13. (Some copies of this work were also published in 1631, with the same title- page as Vlacq's Logarithmicall AritJimetike. See § 3, art. 13.) Briggs, Henry. Trigonometria Britannica : sive de doctriaa triangulorum llbri duo. Quorum prior continet Constructionem Canonis Siuuum Tangen- tium ifc Secantium, una cum Logarithrais Sinuum &• Tangentium ad Gradus 1873. ' 1 146 KEPORT— 1873. & Graduum Centesimas & ad llinuta & Secunda Centesimis respondentia : A Clarissimo Doctissimo Integerrimoque Viro Domino Henrico Briggio Geomc- trice in Celeberrima Academia Oxoniensi Professore Saviliano Dignissimo, paulo ante inopinatam Ipsius e terris emigrationem compositus. Posterior vero usum sive Applicationem Canonis in Eesolutione Triangiilorum tam Planorum quam Sphajricornm e Geometricis fundamentis petita, calculo facil- lirao, eximiisque compeudiis exhibet : Ab Henrico Gellibrand Astronomia) in Collegio Greshamensi apud Londinenses Professore constructus. [Then foUo-w a quotation of three lines from Yieta and a diagram sbo'vring the trigonome- trical functions.] Goudoj, Excudebat Petrus Rammasenius. m.dc.xxxiii. Cum Privilegio. foho. (Dedication to the Electors to the SavUian Chairs, Gellibrand's preface, and 110 pp. of trigonometry &c., followed by one page containing errata to the page signature /. 3 of the tables ; the tables arc unpaged.) § 3, art. 15. Bhiggs. See Sherwin. Bko^yn, See Wallace. Bro-\vne, Robert. A new improvement of the Theory of the Moon .... Loudon, 1731. Small 4to (pp. 14). § 3, art. 2.5. Bettiins, Dr. A new Manual of Logarithms to seven places of Decimals .... Stereotype edition. Bernhard Tauchnitz. Leipzig, 1870. 8vo(pp. GIO, and introduction xxiii). § 4. Brttno, Faa de. Traite elementaire du Calcul des Erreurs avec des Tables stcreotypees , . . Paris, 1869. Svo (41 pp. of tables). § 3, art. 4. Btteckhakut, J. Cn. Tables des Diviseurs pour tons les norabrcs du deuxiome million . . . Paris, 1814. 4to (pp. 112 and viii). § 3, art. 8. Btteckhahdt, J. Cn. Table des Diviseurs pour tous les nombres du troisieme million . , . Paris, 1816. 4to (pp. 112). § 3, art. 8. BtnRCKHAEDT, .J. Cn. Table des Diviseurs pour tous les nombres du premier million . . . Paris, 1817. 4to (pp. 114, and preface &c. 4 pp.). § 3, art. 8. *BtJEGEE, J. A. P. Tafel zur Erleichterung in Rechnungen &c. 1817. Sec under Centxeeschweh, 1825, § 3, art. 3. Btene, Oliver. Practical, short, and direct Method of calculating the Logarithm of any given N'umber, and the Number corresponding to any given Logarithm, discovered by Oliver Byrne . . . London, 1849, 8vo (pp. 82, and introduction xxiii). § 4. Btene, Oliver. Tables of Dual Logarithms, Dual ITumbers, and corre- sponding Natural Numbers; with proportional parts of differences for. single digits and eight places of decimals . . . London, 1867. Large 8vo (pp. 202, and introduction pp. 40). § 3, art. 23. Bteite, Oliver. Other works. See § 3, art. 23. Callet, Pran^ois. Tables portatives de Logarithmes, contcnant Edition sto're'otype, gravc'e, fondue et imprime'e par Eii-min Didot. Paris : Firmin Didot, 1795 (Tirage, 1853). Svo (pp. 680, and introduction pp. 118). § 4. Callet, F, Table of the logarithms of sines and tangents .... Paris, 1795 (Tirage, 1827). Stereotyped and printed by Firmin Didot Svo. §3, art. 15. Callet (1843). See Aifox-mous. Cextxeeschwer, J. J. Neu erfundene Multiplikations- und (iuadrat-Tafeln . . . mit einer Yorrede von . . . J. P. Griisou und L. Ideler. Berlin, 1825. Svo (45 pp. of tables, and introduction Iv). § 3, art. 3. Chernac, Ladislatts. Cribrum Arithmetieum ; sive tabula contincns nu- meros primes ., . Daventrije, 1811. 4to (pp. 1020). ^3, art. 8. ON MATHEMATICAL TABLES. 147 *CLOTrrn, F, M. Tables pour le Calcul cles Coordonne'es goniometriques. Maycn (clicz Tauteur). 8vo. § 3, art. 10. Coleman, Geouge. Luuar and Nautical Tables .... Stereotype edition. London, 1846. Svo (317 pp. of tables). § 4. Ceelle, a. L. Erleichterungs-Tafel fiir jeden, der zu recbnou hat ; cnthal- tend die 2, 3, 4, 5, 6, 7, 8, und 9 fachen aller Zahleu von 1 bis 10 Millionen . . . Berlin, 1836, (pp. 1000 and explanation xvi.) § 3, art. 1. Crelle, a. L. Eechentafeln wclche alles Multiplicireu und Dividircn mit Zahlen uuter Tausend ganz ersparen . . . Zweite Stereotj'p-Ausgabe . . . von Dr. C. Beemiker. Berlin : Georg Reimer, 1864. Folio (pp. 4.50). [There is also a French titlepage.] Also edition of 1820, in two vols. 8vo. § 3, art. 1. Croswele, William. Tables for readily computing the Longitude .... Boston, 1791. 8vo. § 4. Dase, Zacharias. Tafel der natiirlichen Logarithmeu der Zahlen. In der Form und Ausdehnung wie die der gewohnlichen oder Brigg'schen Logarithmeu... Wien, 1850. 4to (pp. 195). § 3, art. 16. Dase, Zacharias. Factoren Tafeln fiir alle Zahlen der Siebenten Million ... Hamburg, 1862. 4to (pp. 112). § 3, art. 8. Dase, Zacharias, Factoren Tafeln fiir alle Zahlen der Achten Million.. . Hamburg, 1863, 4to (pp. 112), § 3, art. 8. Dase, Zacharias. Factoren-tafeln fiir Zahlen der Neunten Million . . erganzt von Dr. H. Eosenbeeg, Hamburg, 1865, 4to (pp, 110). § 3, art. 8, Dechales (Cursus Mathematicus). § 2, art. 3. De Decker. Nieuwe Telkonst, inhoudende de Logarithm! voor de Ghetallen beginnende van 1 tot 10000. . . Door Ezecuiel de Decree, Rekcnmeester, ende Lantmeter residerente ter Goude . . , Ter Goude, By Pieter Eammaseyn , .. 1626, Svo (260 pp. of tables, and introduction pp. 50 + , (copy imper- fect)). [De Haan gives 51 as the number of pp. in the introduction, ' Phil. Mag.' May, 1873]. § 4. Degen, C. F. Tabularum ad faciliorem et breviorem Probabilltatis com- putationem utilium Enneas .... Havnia), 1824, Svo (pp, 44, and intro- duction xxii). § 4. De Haan (lets over Logarithmentafcls). § 3, art, 13 (p, 55), De JoNcotTET. See Joncourt, De la Lande, See Lalande, Delambre. Sec Borda. De Mendoza. See Rios, De Monteerriek. See Montperrier, [De Morgan, A.], Tables of Logarithms (Under the superintendence of the Society for the Diffusion of Useful Knowledge), London, 1854, From the stereotyped plates of 1839, Small Svo (pp. 215). § 4, _ De Moegan, a, Encyelopa;dia Metropolitana, Pure Sciences, vol, ii. {Theory of Prohahilities). London, 1843, § 3, art. 25, De Moegan (Article on tables in the Penny and English Cyclopa;di;i3 and • Arithmetical Books '), § 2, art. 3. De Morgan. See Scheon (1865), De Prasse, Tables logarithmiques, pour les nombres, lea sinus ct les tangentes, dispose'es dans un uouvel ordre . . , Accompaguce do notes et d'un. avertissement par M. Halma. Paris, 1814. 12mo (pp, 80), § 4, Dessiott, See J. H, Moore. Dilling, J. M. Probeschrift eines leichtfasslichen logarithmischen Sys- tems. . . .fiir Burger und Landschulen .... Leipzig, 1826. 12mo (pp. 53). § 3, art. 1, L 2 148 REPORT — 1873. DoBSOx, Jambs. The Antilogarithmic Canon .. . London, 1742. folio. §3, art. 14. DoDs w, Jame^. The Calculator : being correct and necessary tables for computation. Adapted to Science, Business, and Pleasure .... London, 1747. Large 8vo (pp. 174). § 4. DoMKE, F. Nautische astronomische und logarithmische Tafeln . . . f iir die Koniglich Preussischen Navigations-Schulen . . . Berlin, 1852. Svo (353 pp. of tables). § 4. _ DoNxV, Benjamix. Mathematical Tables, or Tables of Logarithms . . . Third edition, with large additions. London, 1789. Svo (pp. 351). § 4. Douglas, George. Mathematical Tables, containing the Logarithms of Numbers ; Tables of Sines, Tangents, and Secants .... and Supplementary Tables. Edinburgh, 1809. Svo (pp. 106). § 4. DouwES. See under Bowditch, § 4. DucoM, P. Cours d'Observations nautiqucs, conteuant . . .suivi d'une col- lection des meilleurcs Tables . . . Bordeaux, 1820. Svo (296 pp. of tables). § 4, Dumas. See Gardiner (Avignon edition, 1770). Dunn, Samuex. Tables of correct and concise logarithms for numbers, sines, tangents, secants .. . London, 1784. Svo (pp. 144). §4. Dupuis, J. Tables de Logarithmes a sept decimales d'apros Brcmiker, CaUet, Vega, etc. par J. Dupuis. Edition stereotype .... troisieme tirage. Paris, 1868. Svo (pp. 578). § 4. Dupuis. See imder Callet, 1853. § 4. [Encke J. F.] Logarithmen von vier Decimal-SteUen. Berlin, 1828. Small Svo (pp. 22). § 4. Ersch (Litteratur der Mathematik). § 2, art. 3. Everett, J. D. Universal Proportional Table .... William Mackenzie. London [no date, 1866]. § 4. Farley, Richard. Tables of six-figure logarithms , . . Stereotyped edition. London, 1840. Svo. § 4. [Farley, E.] Natural versed sines from 0° to 125°, and Logarithmic versed sines from 0° to 135°, or O*" to 9*", used in computing Lunar Distances for the Nautical Almanac. London : Eyre and Spottiswoode, 1856. folio (pp.90). §4. Faulhaber, Johann. Ingenieurs-Schul, Erster Theyl : Darinnen durch den Canonem Logarithmicum alle Planische Triaugel zur fortification . . . . zu solviren . . . Auss Adriano Vlacq, Henrico Briggio, Nepero, Pitisco, Berneck- Lero . . . gezogen . . . Gedruckt zu Frauckfurt am Mayn . . . 1630. Small Svo (pp. 170) (with an Appendix of 14 pp.). Followed by an engraved titlepage. § 4. [Faulhabek, J.] Zehentausend Logarithmi der Absolut oder ledigen Zahlen, von 1. biss auff 10000. nach Herrn Johannis Neperi Baronis Merchistenii Arth und Inuention, welche Heinricus Briggius illustriert, und Adrianus Vlacq augiert, gerichtet. Gedruckt zu Augspurg, durch Andream Aperger, auff unser lieben Frawen Thor. Anno m.dc.xxxi. Small Svo (pp. 104). § 4. [Faulhaber, J.]. Canon Triangulorum logarithmicus, das ist : Kiinstliche Logarithmische Tafeln der Siuuum, Tangentium imd Seeantium, nach Adrian! Vlacqs Calculation Rechnung und Manier gestelt. Gedruckt zu Augspurg, durch Andream Aperger, auff unser lieben Frawen Thor. Anno m.dc.xxxi. Small Svo (pp. 190). § 4. Felkel, Anton. Tafel aller Einfachen Factoren der durch 2, 3, 5 nicht theilbaren Zahlen von 1 bis 10 000 000. I. Theil. Enthaltend die Factoren von 1 bis 144000 "Wien, mit von Ehelenscheu Schriften gedruckt, 1776. Large folio (pp. 26, and preface, &c. 4 pp.). § 3, art. 8. ON MATHEMATICAL TABLES. 14(> Felkel. Sec Lambeet. FiLipowsEi, Herschell E. a table of Anti-logaritlims, containing to seven places of decimals, natural numbers, answering to all logarithms from -00001 to -99999, and an improved table of Gauss's logarithms. . . . London, 1849, 8vo (pp. 220, and introduction xvi). § 4. FiLiPowsKi, H. The -wonderful canon of logarithms ... by John Napier . . . .retranslated from the Latin text, and enlarged, with a table of hyper- bolic logarithms to aU numbers from 1 to 1201. By Herschell Filipowski Edinburgh, 1857. 16mo. § 3, art. 16. FiNCK. Tlioma; Fiiikii Flcnspurgensis Gcomctria) rotundi Libri xiv. ad Fridericum Secundum, Serenissimum Daniae, & NorvegijB regem &c. Cum Gratia & PrivHeg. Ctes. Majest. Basilea) per Sebastianum Henricpetri [1583J. 4to. § 3, art. 10. Fischeb's Vega. See Vega. French Manuscript Tables. See Tables dtj Cadastre. Galbraith, D. The Piece-Goods Calculator, consisting of a series of tables . . . Glasgow, 1838. 8vo (pp. 53). § 3, art. 25. Galbraith, J, A., and S. Haughton. Manual of Mathematical tables . . . London, 1860. Small 8vo (pp. 252). § 4. Galbraith, "William. Mathematical and Astronomical Tables . . . Edin- burgh, 1827. 8vo (112 pp. of tables). § 4. Gardiner, William. Tables of Logarithms for all numbers from 1 to 102100, and for the Sines and Tangents .. . London, 1742. 4to. §4. Gardiner, W. Tables de Logarithmes, contenant les Logarithmes des nombres . . . des sinus & des tangentes . . . Nouvelle edition, Augmentee des Logarithmes des sinus & tangentes pour chaque seconde des quatre premiers dcgres. Avignon, 1770. 4to. (This reprint was edited by Pezenas, Dumas, and Blanchard.) § 4. *Gardiner. Paris edition, 1773. § 4. Garrard, William. Copious trigonometrical tables .... intended to com- plete the requisite tables to the Nautical Almanack .... London, 1789. 8vo. § 4. Gauss, C. F. Tafel zur bequemern Berechnung des Logarithmen der Summe oder Differenz zweyer Griissen, welche selbst nur durch ihre Loga- rithmen gegeben sind. Zach's ' Monatliche Correspondenz,' t. xsvi. (pp. 498- 528). Gotha, 1812. § 3, art. 19. Gauss. Carl Friedrich Gauss Werke .... herausgegeben von der koniglicheu Gesellschaft der Wissenschaften zu Gottingen. Still in course of publication : 4to, t. i. (1863, and 'zweitcr Abdruck,' 1870) ; t. ii. (1863) § 3, arts. 6 and 7 (introductory remarks); t. iii. (1866) § 3, art. 19 (introductory remarks) ; and under De Prasse, Hulsse's Vega, Pasquich, Vega (1794) in § 4 &c. (t. iii. includes the reprints from the ' Astronomische Nachrichten ' and the ' Gottingische gelehrte Anzeigeu,' on logarithmic tables.) Gellibrand. See Briggs (1633). Gellibband. See John Newton (1658). Gernerth (Tract on the accuracy of logarithmic tables). Under Eheticus (§ 3, art. 10), and § 3, art. 13 (introductory remarks, p. 55). Glaisher, J. W. L. ' Monthly Notices of the Eoyal Astronomical Society : ' May, 1872 (On errors in Vlacq's (often called Briggs' or Neper's) table of ten-figure logarithms of numbers) ; June, 1872 (Addition to a paper on errors in Vlacq's ten-figure logarithms, published in the last Number of the ' Monthly Notices ') ; March, 1873 (On the progress to accuracy of logarithmic tables) ; May, 1873 (On logarithmic tables). ' Philosophical Magazine : ' October, .J50 UEPOKT — ]873. 1872 (Notice respecting some new facts in the early historj' of logaritliraie tables) ; December (Supplementary Number), 1872 (Supplementary remarks on some early logarithmic tables) ; May, 1873 (On early logarithmic tables and their calculators). ' Messenger of Mathematics ' (new series) : (July, 1872 (Pineto's table of ten-figure logarithms of numbers) ; May, 1873 (Eo- marks on logarithmic and factor tables, with special reference to Mr. Drach's suggestions). § 3. art. 13 (introductory remarks ; Bbiggs, 1617 ; Pineto), art. 15 (Gunxer), art. 17 (Napieb, 1614), § 4, Borda and Delambeb, db Decker, Hulsse's Vega, Shortrede, Yega, 1794, YiAca, 1633, &c. [GoDWARD, William, Jun.] Interpolation tables used in the Nautical Almanac Office. Loudon : Eyre and Spottiswoode, 1857. 8vo (pp. 30). § 3, art. 21. GooDWTN, Henry. The first centenary of a series of concise and useful tables of all the complete decimal quotients, which can arise from dividing a unit or any whole number less than each divisor, by all integers from 1 to 1024. [London, Preface dated 1816]. Small 4to (pp. 18 and introduction xiv). § 3, art. 6. GooDAYXN, Henry. The first centenary of a series of concise and useful tables of all decimal quotients, which can arise from dividing a unit, or any whole number less than each divisor, by aU integers from 1 to 1024. To which is now added a tabular series of complete decimal quotients, for all the proper vulgar fractions, of which, when in their lowest terms neither the numerator, nor the denominator is greater than 100 : with the equivalent vulgar fractions prefixed. London, 1818. Small 4to (pp. 18 and 30, and introductions xiv and vii). § 3, art. 6. [GooDWYN, Henry.] A tabular series of decimal quotients for all the proper vulgar fractions, of which, when in their lowest terms, neither the numerator nor the denominator is greater than 1000. London, 1823. 8yo (pp. 153 and introduction v). § 3, art. 6. [GooDWYN, Henry.] A table of the circles arising from the division of a unit or any other whole number by all the integers from 1 to 1024 ; bsing all the pure decimal quotients that can arise from this source. London, 1823. 8vo (pp. 118 and introduction v). § 3, art. 6. Gordon, Jaues. Lunar and Time Tables .... for finding the Longitude London, 1849. 8vo (92 pp. of tables). § 4. Graesse (Trcsor de livres rares). § 2, art. 3. Gray, Peter. Tables and formulas for the computation of life contin- gencies . . . London, 1849. 8vo (68 pp. of tables). § 3, art. 19. Gray, Peter. Addendum to tables and formute for the computation of life contingencies .... Second issue, comprising a large extension of the prin- cipal table. . . . London, 1870, 8vo (26 pp. of tables) (noticed under the pre- ceding work, § 3, art. 19). This title is copied from the wrapper of the "Addendum," the titlepage of which is intended to apply to the whole work when the " Addendum " is included, and runs, " Tables and formula) for the computation of life contingencies .... Second issue, with an addendum, com- prising a large extension of the principal table .... London, 1870." Gray, Peter. Tables for the formation of Logarithms and Anti-logarithms to twelve places ; with explanatory introduction .... London, 1865. 8vo (55 pp. of introduction &c. and xi pp. of tables). § 3, art. 13. Gregory, Olinthus. Tables for the use of nautical men, astronomers, and others ; by Olinthus Gregory, W. S. B. Woolhouse and James Hann. London, 1843. 8vo (pp. 168 and introduction sxiv). § 4. Gregory, Olinthus. See Huiton (1858). ox MATIIEMATKJAL TABLKS. 151 GuiEiTBEEGEK. Eloiucuta trigoiiomctrica, id est simis tangentes, secautes In Purtibus Sinus totius 100000. Christophori Grienbergeri E Societate lesu. Eerum Mathematicarum Opusculum Secundum. [Device — globe with IHS.] Eomte, Per Hajred. Barthol. Zan. 1630. Superiorum permissu. 12nio (pre- face and tables unpaged, trigonometry 88 pp., and 4 pp. of corrections). § 3, art. 10. GiuFi'iN, James, A complete Epitome of Practical Navigation .... to wliich is added an extensive set of lleqnisito tables . . . London, 1843. 8vo (325 pp. of tables). § 4. Gruenbekger, GruexperctEr, or Griembergek. See Gkienbergek. Grxtson, J. P. Pinaeothe'que, ou collection do Tables d'une utilite ge'nerale pour multiplier ct divisor inventees par J. P. Gruson. Avec une table do tons les facteurs simples de 1 a 10500. Berlin, 1798. 8vo (pp. 418 and introduction xxiv). § 3, art. 1. Grusox, J. P. Grosses Einmaleins von Eins bis Hunderttausend. Erstes Heft vons Eins bis Zehntausend . . . Berlin, 1799. Large folio (pp. 42). § 3, art. 1. Gruson, J. P. Bequeme logaritbrnische, trigouometrische und andero niitzliche Tafeln zur Gebrauch auf Schulen . . . Dritte verbesserte Auflage. Berlin, 1832. 8vo. § 4. Gruson. See Cenxnerschwer. GuNTER, Edmund. Canon Triangulorum sive Tabulte Sinuum et Tangen- tiuni artificialium ad Eadium 10000,0000 & ad scrupula prima quadrautis. Per Edm. Gunter, Professorem Astronomiae in CoUegio Greshamensi. Londini, excudebat Gidiclmus Jones, sidcxx. Small Svo (p. 94). § 3, art. 15. Gunter, Edmund. The works of ; . . . with a canon of artificial sines and tangents . . . The fifth edition, diligently corrected ... By William Ley- bourn, Philomath. London, 1073. Small 4to. § 3, art. 15. Halley. See [Sherwin.] Halma. See De Prasse. Hann. See Olinthus Gregory (1843). Hantsohl, Joseph. Logarithmisch-trigonometrisches Handbuch . . . Wien, 1827. Large 8vo. § 4. Hartig, G. L. Kubik-Tabellen fiir geschnittene, beschlagene und rund& Hblzer . . . und Potenz-Tabellen, zur Erleichterung der Zins-Bereehnung . . . Dritte Auflage . . . Berlin und Stettin, 1829. Svo. (pp. 488 and introduc- tion xviii). § 4. Hassler, r. R. Tabulae logarithmicee et trigonometricae, notis septem decimalibus expresste, in forma minima . . . Novi-Eboraci, 1830. 12mo [stereotyped]. § 4. Hasslee, E. R. Logarithmic and trigonometric tables, to seven places of decimals, in a pocket form . . . New York, 1830. 12mo [stereotyped]. § 4. Hassler, F. 11. Tables logarithmiques et trigonometriques a, sept de'ci- males, en petit format . . . Nouvelle-York, 1830. 12mo [stereotyped]. _ § 4. Hassler, E. R. Logarithmische und trigouometrische Tafeln, zu sieben Dczimal-Stellen ; in Taschen-Eormat . . . Neu-York, 1830. 12mo [stereo- typed]. § 4. Hassler, F. R. Tablas logaritmicas y trigonometricas para las siete deci- males, corregidas . . . Nueva-York, 1830. 12mo [stereotyped]. § 4. Haughton. See J. A. Galbkaith. Heilbeonner, C. Historia Matheseos Universae . . . Lipsiae, 1742. 4to. § 3, art. 25 ; and see § 2, art. 3. Henbion, Denis. Traictc des logarithmes. Par D. Henriou, Professour 152 REPOKT — 1873. cs Mathcmatiqucs. [Typographical ornament]. A Paris, chcz rAutheiir, demeurant en I'lsle du Palais, a I'lmage S. Michel, m.dc.xxyi. Aucc priuilcgc du Eoy. 8vo (paging begins at 341, and proceeds to 708). § 4. Hensel, See Hulsse's Yega, § 4. Hentschen. Adrian Ylaeq Tabellen der sinuum, tangentium . . . ISTeue und verbessertc Auflage von Johann Jacob Hentschen. Franckfurt nnd Leipzig, 1757. SmaU Svo (280 pp. of tables, 48 pp. of trigonometry, &c.). § 4. Heekmann. ' Vienna Sitzuugsberichte ' (Verbesserung der II. Callct'schen Tafel). See under Callet, 1853, § 4. Heewaet ab Hohejtbueg. Tabulae arithmetics TIpondcKpcupeaetiis Tni- versales, quarum subsidio numerus quilibet, ex multiplicatione producendus, per solam additionem : et qnotiens quilibet, e divisione eliciendus, per solam subtractionem, sine tcediosa & lubricfi Multiplicationis, atque Divisionis ope- ratione etiam ab co, qui Arithmetices non admodum sit gnarus, exacte, celeriter & nullo negotio invenitur. E museo loannis Georgii Herwart ab Hohenburg, V. I. doctoris, ex assessore summi tribunalis Imperatorii, et ex Cancellario 'supremo serenissimi ntriusque BavariiB Ducis, suae serenissimte Celsitudinis Consiliarii ex intimis, Pra;sidis proviutiai Schuabaj, & inclytorum iitriusque Bavarife Statuum Cancellarii. Monachii Bavariarum, ex officina Nicolai Henrici. Anno Christi m.bc.x. obi. folio (pp. 999 and introduction 7 pp.). § 3, art. 1. ... Hill, John. Decimal and logarithmical Arithmetic explained . . . with a table of logarithms from 1 to 10,000 . . . Edinburgh, 1799. Svo (pp. 4G). § 3, art. 13. Hind, J. E. See [Fakley] (Versed Sines, 1856). Hobeet, Jean Philippe and Louis Ideler. Nouvellcs Tables trigonomc- triques calculees pour la division decimale du quart de cerclc . . . Berlin, 1799. 8vo (pp. 351, and introduction Ixxii). § 4. HOHENBUEG. ScO HeEWAET. HotTEL, J. Tables de Logarithmes a cinq de'cimales . . . Paris, 1858. Svo (116 pp. of tables, 32 of introduction). § 4. HotTEL, J. Tables pour la reduction du temps en parties decimales du jour . . . Publication der astronomischen Gescllschaft, iv. Leipzig, 186G. 4to (pp. 27). § 3, art. 12. HuLssE, J. A. See Vega (Sammlung, 1840). Hulsse's Vega. See Vega (Sammlung, 1840.) HuTTON (Tracts). § 2, art. 3. HuTTON, Chaeles. Tables of the Products and Powers of Numbers . . . Published by the Commissioners of Longitude. London, 1781. folio (pp. 103). § 4. HuTTON, Chaeles. Mathematical Tables : containing common, hyperbolic, and logistic logarithms. Also sines, tangents, secants, and versed sines . . . to which is prefixed a large and original history of the discoveries and writings relating to those subjects . . . London, 1785. Svo (pp. 343 of tables and 176 of introduction). § 4 (under Hutton, 1858). HuTTON, Chaeles. A Philosophical and Mathematical Dictionary ... (in 2 vols.), vol. ii. London, 1815. 4to. § 3, art. S. HuTTON, Chaeles. Mathematical Tables, . . . with seven additional tables o trigonometrical formulae by Olinthus Geecoey . . . Ncav edition. London, 1858. Svo (368 pp. of tables). § 4. Idelee. See Cenineeschwee. Idelee. See Hobeet, ON MATHEMATICAL TABLES. 153 Inman, J. jN'aiidcal Tables, designed for the use of British Seamen. New edition, rcyised by the Rev. J. "W. Inma^^ London, Oxford and Cambridge, 1871 . 8vo (445 pp. of tables). § 4. InsENGAETn, H. F. Gemcinniitziges Compendium von Quadrat-Plachen- TabcUen , . . Small 8vo. Hannover, 1810 (pp. 148 and xxxvi). § 4, JjiGEit. See under Keuger, § 3, art. 8. Jahn, Gustav AcoLi-n. Tafcln der sechsstelligcn Logarithmen fiir die Zahlen 1 bis 100 000, fiir die Sinus und Tangenten . . . Leipzig. 2 vols, vol. i. 1837; vol. ii. 1838. 4to (vol. i. pp. 79, and introduction, &c., xvi ; vol. ii. pp. 4G3, and introduction, &c., viii). There is also a Latin title on the same titlepage. § 4. JoxcoruT, E. DE. De natui-a et ptECclaro usu simplicissimse speciei nume- rorum trigonalium . . . Hagaj Comitum, 17G2. Very small 4to (pp. 267). § 3, art. 25. JxjNGE, August. Tafel der wirldichen Liinge der Sinus und Cosinus fiir den lladius 1 000 000 und fiir alle Winkel des erstcn Quadranten von 10 zu 10 Secundon .... insbesondcre fiir diejenigen, welche bei trigonometrischea Berechnungen die Thomas'sche llcchenmaschine benutzen. Leipzig, 1864. Small folio (pp. 90). § 3, art. 10. Kasxner (Gescbichte der Mathematik). § 2, art. 3. Keith. See [Mayxaed.] Keplee, J. Joannis Kepleri . . . Chilias logarithmorum ad totidem nu- meros rotundos . . . quibus nova traditur Aritbmetica . . . Marpurgi, 1624. Small 4to (55 pp. of introduction and table unpaged). § 3, art. 18. Keeigan, Thomas. The young Navigator's Guide to . . . Nautical Astro- nomy . . . London, 1821. 8vo (204 pages of tables). § 4. KiRBY. See BOWDITCH. KoHLEE, H. G. Jerome de La Lande's logarithmische-trigonometrische Tafelu durch die Tafel der Gausschen Logarithmen und andere Tafeln iind FormeLn vermehrt . . . Stereotypen-Ausgabe. Dritter Plattenabdruck . . . Leipzig, 1832. 32mo (pp. 254, and introduction xlv). There is also a French titlepage. § 4. KoHLEE, H. G. Logarithmisch-trigonometrisches Handbuch . . . Zweite Stereotypausgabe. Leipzig, 1848. 8vo (pp. 388, and introduction xxxvi). §4. Keugee, J. G. Gcdanckcn von der Algebra nebst den Primzahlen von 1 bis 1 000 000 . . . Halle im Magdeburgischen, 1746. 12mo (Algebra pp. 124, and the list of primes pp. 47). § 3, art. 8. KuxiK, Jakoi! PniLipp. Tafeln der Quadrat= und Ivubik-Zahlen aller natiirlichen Zahlen bis Hundert Tausend . . . nach eiuer neuen Methode be- rechnet . . . Leipzig, 1848. 8vo (pp. 460, and preface vii). § 3, art. 4. Laeande, ^Jeeome de. Tables de logarithmes pour les nombres et pour les sinus . . . Edition stereotype . . . grave'e, fondue et imprimee, par Eirmin Didot . . . Paris, 1805 (tirage de 1816). 16mo. § 4. Lalakde, Jerome de. Tables de logarithmes par Jerome de Lalande eten- dues a sept decimales par E. C. M. Maeie . . . precedees d'une instruction . . . par le Baron Eeynaud. Edition stereotypee . . , Paris, 1829. 12mo (pp. 204 and introduction xlii). § 4. Lalande (Bibliographic Astronomique). § 2, art. 3. Lalande. Sec Kohler (1832). Lalande. See Eeynaud. Lambeet, J. H. Supplementa tabularum logarithmicarum et trigonome- tricarum .... cum vcrsione introdutionis (sic), Gcrmanica) in Latiuxim ser- 154 UEi'oiiT — 1873. moncm, secuudum ultima auctoris consilia aiuplilicata. Curaute Antokio FjiLKEL, Olisipoiio, 1798. 8vo (pp. 198 and introduction Ixxv). § 4. Lambert, J. H. Zusiitse zu den logarithmisclicn und trigouometrisclicn Tabcllen, 1770. tSco the Supplementa ^'t. of the same author next above, §4. Lattndy, Samuel Linn. Table of Quarter-squares of all integer numbers up to 100,000, by which the product of two factors may be found by the aid of Addition and Subtraction alone. . . London, 185G. 8vo (pp. 214 and intro- duction xxviii). § 3, art, 3. Laundi', S. L. a Table of Products, by the factors 1 to 9 of all numbers from 1 to 100,000 . . . London, 18G5. 4to (10 pp. of tables and introduc- tion vi). § 3, art. 1. Lax, Eev. "VV. Tables to be used with the Nautical Almanac for finding the latitude and longitude at sea . . . London, 1821. 8vo. § 4. Lefoet, F. Description des grandes Tables logarithmiques et trigonome- triques calculoes au Bureau du Cadastre, &c. Aimales de I'Observatoii'o Imperial do Paris, t. iv. (1858) pp. [123]-[150]. § 3, art. 13, under Tables DU Cadasxke. Leonelli. Leonelli's logarithmische Supplcmento . . . aus dem Franzo- eischen nebst einigen Zusiitzen von G. W. Leokhardi . . . Dresden, 1806. SmaU 8vo (pp. 88). § 3, art. 19. Leonhaedi. Sec Leonelli. Leslie, John. The Philosophy of Arithmetic .... with tables for the multiplication of numbers as far as one thousand . . . Second edition, im- proved and enlarged. Edinburgh, 1820. 8vo (pp. 258). § 3, art. 3. LiTXEOw, C. L. VON. Hiilfs-Tafeln fiir die Wiener Univorsitats-Stcrnwarte. ZusammengesteUt im Jahre 1837 . . . 8vo (pp. 88). § 3, art. 12. LuDOLF. Tetragonometria tabularia, qua per tabulas quadratorum a Radiee quadrata 1. usque ad 100 000 . . . Autore L. Jobo Ludolffo, P. P. Math. iii Universitate Hierana ibidemque Senatore. Amstelodami, 1690. Small 4to (introduction, 150 pp., and tables about 420 pp.). § 3, art. 4. Lynn, Thomas. Horary tables, for finding the time by inspection . . . London, 1827. 4to (300 pp. of tables). § 4. Mackay, Andeeav. The Theory and Practice of finding the Longitude . . . with new tables. In 2 vols., the third edition, improved and enlarged . . . London, 1810. 8vo (vol. ii. contains about 340 pp. of tables). § 4. Magini, J. A. Tabula tetragonica sen quadratorum numcrorum cum suis radicibus ex qua cujuscunque numeri perquam magni minoris tamen triginta tribus notis, quadrata radix facile, minimaque industria colligitur. Venetiis, 1592. § 3, art. 4. Maginus, J. a. . . . De Planis triangulis liber unicus. De dimetiendi ratione . . . libri quinque. Venetiis, 1592. SmaU 4to (contains the Tabula Tetragonica, see Magini above). § 3, art. 4. Maeie. See Lalaitde (1829). Maeeiott. See under Willich, § 4. Maetin, C. F. Les tables de Martin, ou le regulateur univorsel .... troisieme edition. Paris, 1801. 8vo. § 3, art. 1. Maseees, Feancis. The Doctrine of Permutations and Combinations . . . together with some other useful tracts . . . London, 1795. 8vo. § 4. [Masiceltne, Nevil.] Tables requisite to be used with the Nautical Ephe- meris . . . Published by order of the Commissioners of Longitude. The third edition, corrected and improved. London, 1802. 8vo (206 pp. of tables, and appendix (see next below) 106 pp. of tables). § 4. 0.\ MATilEJiAXlCAL TABLKS. 155 [Maseelyne, Netil.] Ai^pendis to the tliird oclitiou of the llccjuisite Tubica . . . [London, 1802]. 8vo (pp. 106). § 4. Maskelyne. See Michael Tatlok (1792). Massaloup, J. V. Logaritlimisch-trigononietrische Hiilfstafeln . . . Hand- buck flir Geometer, Markscheider . . . Leipzig, 1847 (pp. G67 and intro- duction xii). § 3, art. 10. [Matxhiessen, E. a.] Tafel zur bequemern Berechnung des Logaritbmcn der Summe oder Differcnz zweyer Grossen welcbe selbst nur durch ihre Logarithmen gegobcn sind. Altona, 1818. Large 8vo (pp. 212 and intro- duction 53). There is also a Latin titlepage. § 3, art. 19. [Matnakd, Samuel.] A table containing useful numbers often required in calculations, together with their logarithms. 8vo (pp. 12, numbered 169- 180). From Templeton's 'Millwright and Engineer's Pocket Companion' [see title under Temi'Letoh]. It is stated on tho first page that a portion of the table had appeared in other publications, and in particular in Keite's ' Measurer,' 24th edit. 1846, by the same editor (Maynard). § 3, art. 24. Mendoza. See Rios. Mebpaut, J. M. Tables Arithmonomiques fondees sur le rapport du rect- angle au carre, on le calcul rcduit h son dernier degre de simplification . . . Vannes, 1832. 16mo (500 pp. of tables, introduction 40 pp.). § 3, art, 3. MiCHAELis. See under Hulsse's Yega, § 4. MiNsiNGEE, Prof. Die gemeinen oder Briggischen Logarithmen der Zahlen . . . Augsburg, 1845. 8vo (31 pp. of tables and introduction &c. vi). § 4. MoNTPERKiEB, A. S. HE. Dictionnaire des sciences mathematiques pures et appliquees . . . Tome troisieme (Supplement). Paris, 1840. folio. § 3, art. 13. MoNTUOLA (Histoire des Mathematiques). § 2, art. 3. [Moore, Sib, Jonas.] A canon of the squares and cubes of all numbers under 1000. Of the squared squares under 300. And of the square cubes and cubed cubes under 200 . . . [London, 1650 ?] § 3, art. 4. Moore, Sib Jonas. Excellent Table for the finding the Periferies or Cir- cumferences of aU EUeipses or Ovals . . . (no place or date. ? London, 1660). 1 page folio. § 3, art. 22. Moore, Sir Jonas. A new Systemo of the Mathematicks ... In 2 vols. Vol. ii. (Tables). London, 1681. 4to (351 pp. of tables). § 4. [Moore, Sib Jonas.] A Table of Yersed sines both natural and artificial. 4to. [London, 1681] (pp. 90). § 4. Moore, J. H. The new Practical Navigator ; being a complete epitome of navigation, to which are added all the Tables requisite . . . The nineteenth edition, enlarged and carefully improved by Joseph Dessiou. London, 1814. 8vo. § 4. Mouxon's sines &c. to every second. See Gardiner (Avignon reprint, 1770). MtJLLER, J. H. T. Yierstellige Logarithmen der natiirlichen Zahlen und Winkel Eunctionem . . . (Preface dated from Gothai 1844.) 8vo (25 pp. of i tables). § 4. *MuLiiPLicATiON, Tables de . . . Paris, 1812. § 3, art. 1 (Introductory remarks). MuBHARD (Bibliotheca Mathematica). § 2, art. 3. Napier. Mirifici Logarithmorum Canonis descriptio, Ejusque usus, in utraquc Trigonometria ; ut ctiam in omni Logistica Mathematica, Amplissimi, EaciUimi, & expeditissimi explicatio. Authore ac Inventore, Ioanne Nepebo, Barone Merchistonii, &c. Scoto. Edinburgi, Ex ofiicina, Andrea) Hart Bib- liopoloe, cio.DC.xiv. [On an ornamented titlepage.] 4to (dedication, preface &c. 6 pp., text 57 pp., tables 90 pp.). § 3, art. 17. 156 iiEPORT— 1873. Napier. Mirifid logarithmorum canonis construetio ; Et eorum ad natu- ralcs ipsorum numeros habitudines ; una cum Appcndice, de alia eaque prtestaniiore Logarithmorum specie condenda. Quibus accessere Proposi- tiones ad triangula sphaerica faciliore calculo resolvcnda : Una cum Anno- tationibus aliquot doctissimi D. Henrici Enggii, in eas & memoratam appen- dicem. Authorc & Inventore loanne Nepcro, I3arone Merchistonii, &c. Scoto. [Tj'pograijhical ornament, a thistle.] Edinburgi, Excudebat Andreas Hart. Anno Domini 1619. 4to (prefoce 2 pp. and text 67 pp.). § 3, art. 17. [The above is a transcript of the titlepage of the ' Construetio;' but in the only copy of this work that we have seen it is immediately preceded by an ornamental titlepage, which, as far as the ornamentation is concerned, is a facsimile ol that of the ' Descriptio,' 1614. The letterpress, however, is very difi'erent, and runs, " Mirifici logarithmorum canonis descriptio, Ejusque usus, in utraque Tri- gonometria ; ut etiam in omni Logistica Mathematica, amplissimi, facillimi, (it expeditissimi explicatio. Accesserunt opera posthnma : Primo, Mirifici ipsius canonis construetio, &■ Logarithmorum ad naturales ipsorum numeros habitudines. Secundo, Aj^pendix de alia, eaque praistantioro Logarithmorum specie construenda. Tcrtio, Propositiones quajdam eminentissima), ad Tri- angula sphterica mirii facilitate resolvenda. Autore ac Inventore loanne Nepcro, Baroue Merchistonii, &c. Scoto. Edinburgi, Excudebat Andreas Hart. Anno 1619." This would imply that the ' Descriptio' and ' Construetio' were issiied together in 1619 ; and whether this was so or not, it shows that such was intended. Some writers speak of a rei>rint of tlie ' Descriptio ' in 1619 ; but this title may be all their authority, as few of those who have written on the subject seem to have looked beyond the titlepages of the works they were noticing. On the other hand, of course, the 'Descriptio' may have been torn out from the copy before us. The ' Construetio' is a much rarer work than the ' Descriptio ;' we have seen half a dozen copies of the latter and but one of the former (Camb. Univ. Lib.). In any case, as the leading words of the title of the ' Construetio' (on the first titlepage) are " Mirifici logarithmorum canonis descriptio," it could only be distinguished from the ' Descriptio' in most library catalogues by the date 1619. We have thought it worth while, since the description in § 3, art. 17 (p. 73), was printed, to add the first title of the work containing the ' Construetio,' and to point out the uncertainty relating to the reprint of the ' Descriptio,' in hopes that some one may settle the matter. The 1619 edition of the 'Descriptio' (supposing there to have been one of this date) is the only book of importance relating to the early spread of logarithms of which we have seen no copy ; and the question of its publication is almost the only point of bibliography, in reference to the tables of this time, that we are obliged to leave iiudeeided for the present.] Neper, Nepair, or Nepper. See Napier. Newton, John. Trigonometria Britanica (sic) : or, the doctrine of tri- angles. In Two Books. . . . The one Composed, the other Translated, from the Latine Copie written by Henry Gellibrand, ... A table of logarithms to 1 00.000, thereto annexed, AVith the Artificial Sines and Tangents, to the hundred part of every Degree ; and the three first Degrees to a thovisand parts. By John Newton . . . London : MDCLVIII. fol. (Dedication and preface 6 pp., trigonometry 96 pp. ; tables unpaged.) § 4. NoEiE, J. AV. A complete set of Nautical Tables containing all that are requisite . . . Eighth (stereotype) edition. Loudon, 1836. 8vo (360 pp. of tables). § 4. NoraE, J. AV. A complete epitome of Practical Navigation , . . Thirteenth ON MATHEMATICAL TABLES. 157 (stereotype) edition, considerably augmented and improved. London, 1844. 8vo (360 pp. of tables). § 4. [We have also seen the " fourteenth (stereotype) edition .... by George Coleman," 1848, the " twelfth (stereotype) edition," 1839, the " eleventh edition," 1835, all containing 360 pp. of tables — and, besides, an edition of 1805 containing 252 pp. of tables, in which it is stated that the tables were published two years previously under the title " Nautical Tables." j Norwood, Richard. Trigonometric, or the Doctrine of Triangles . . . per- formed by that late and excellent invention of logarithms . . . London, 1631, Small 4to. § 4. Oakes, Lieut. -Col. W. H. Table of the reciprocals of numbers from 1 to 100,000, with their differences, by which the reciprocals of numbers may be obtained up to 10,000,000. . . London, 1865. 8vo (205 pp. of tables and xii of introduction). § 3, art. 7. Oakes. Machine table for determining primes and the least factors of composite numbers up to 100,000. Dedicated, by permission, to Professor De Morgan. By Lieut.-Col. W. H. Oakes. Printed and published by Charles and Edwin Layton. . . . London, 1865. § 3, art. 8. Oppolzer, Theodor. Vierstellige logarithmisch-trigonometrischc Tafeln. . . . Wien, 1866 (pp. 16). § 4. QpTTS Palatinum. See Rheticus, Otho. See Rheticus (Opus Palatinum). OuGHTRED, "William. Trigonometrie, or. The manner of calculating the Sides and Angles of Triangles, by the Mathematical Canon, demonstrated . . . published by Richard Stokes and Arthur Haughton .... London, 1657. Small 4to. (Trigonometry 36 pp., tables 240 pp.). § 4. OzA>rAM, M. Tables des sinus tangentes et secantes et des logarithmes des sinus et des tangentes . . . Paris, 1685. Small 8vo. § 4. Parkhurst. Astronomical Tables, comprising logarithms from 3 to 100 decimal places, and other useful Tables. Ry Henry M. Parkhurst. Revised edition. Printed and publislied by Henry M. Parkhurst (Short Hand Writer and Law Reporter), No. 121 Nassau Street, New York City. 1871. 12mo (176 pp. of tables, 66 pp. of formulte, explanations, &c.). § 4. PAsauiCH, loANNES. Tabulffi logarithmico-trigonometricce contractse cum novis accessionibus . . . Lipsia;, 1817. 8vo (pp.228 and introduction xxxviii). There is also a German titlepage. § 4. Peacock (Arithmetic). § 2, art. 3. Pearson, W. An introduction to Practical Astronomy containing Tables .... London, 1824. 2 vols. Large 4to. § 4. [Pell, J.] Tabula Numerorum Quadratorum dccies millium, una cum ip- sorum lateribus ab unitate incipientibus & ordine naturali us(iue ad 10 000 progredientibus . . . London, 1672. 4to (pp. 32). § 3, art. 4. Peters, C. E. W. Astronomische Tafeln und Eormeln. . . Hamburg, 1871. 8vo (pp. 217). § 4. Pezenas. See Gardiner (Avignon edition, 1770). Phillips, Sir Thomas, Bart. An improved Numeration Table to facilitate and extend Astronomical Calculations .. . [London?], 1829. 12mo (pp. 18). § 3, art. 25. Picarte, R. La Division rcduite h, uue Addition, ouvrage approuve par I'Academie des Sciences dc Paris . . . augmente dune Table de Logarithmes . . . Paris [1861]. 4to (pp. 104 and introduction &c. xvi). § 3, art. 7. PiGRi, Giuseppe. Nuove Tavole degli Elementi dei Numeri dall' 1 al 10 000 .. . Pisa, 1758. 8vo (pp. 195). § 3, art. 8. 158 REPORT~1873. PiKETO, S. Tables de Logarithmcs vulgaires a dix flecimalcs construitcs d'apros iin iiouveaii mode . . . S.-Petersbourg, 1871. 8vo (pp. 50 and intro- duction xxiv). § 3, art. 13. PiTiscirs. Thesaurus mathematicus Sive canon sinuura ad radium 1.00000.00000.00000. et ad dena qureque scrupula secunda Quadrantis : una cum sinibus primi ct postremi gradus, ad cundem radium, ct ad singula scrupula secunda Quadrantis : Adjuuctis ubiquc difFerentiis primis et secun- dis ; atq, ubi res tulit, etiam tertijs. jam olim quidcm incredibili laboro & Bumptu a Georgio Joachimo Ehetico supputatus : at nunc primura in lucom cditus & cum viris doctis communicatus a Bartholomffio Pitisco Grunbcrgensi Silesio. eujus etiam accesserunt: I. Princ-ipia Sinuum, ad radium, 1.00000. 00000.00000.00000.00000. quam accuratissime supputata. II. Sinus deei- morum, tricesimorum & quinquagesimorum qirorumq ; scnrpulorum secundo- rumperprima &postrema35. scrupula prinm, ad radium, 1.00000.00000.00000, 00000.00. [Tj'pographical ornament.] Prancofurti Excudebat Nicolaus HofFmannus, sumptibus JonoB Posfc Anno cio. ij. xiii. folio [part of the title is printed in red] (preface 5 pp., tables pp. 2-271, pp. 2-Gl, pp. 3-15). Tliei-e are four titlepages altogether, including that to the whole work (copied above) ; on the first two the date should be cio. loc. xiii. and not as printed. § 3, art. 10. PoGOENDOEFF (Haudworterbuch). § 2, art. 3. Prasse. See 1)e Prasse. Prony, Sec Tables du Cadastre. Sec also § 3, art. 13 (introductory remarks, p. 54), and § 3, art. 16 (introductory remarks, p. 09). Rahn, J. H. Teutsche Algebra, oder Algebraischc Eechcnkuust ... Zurich, 1659. Very smaU quarto (pp. about 200). § 3, art. 8. Rankine, W. J. li. Useful Eules and Tables relating to Mensuration, Engineering, Structures, and Machines . . . London, 1866. 8vo. § 4. IIaper, Henry, Lieut. P.N". Tables of logarithms to six places . . , London^ 1846. 8vo (pp. 122 and introduction xi). § 4. Pi,APER, Henry, Lieut. R.S'. The Practise of Navigation and Nautical Astronomy . . . Sixth Edition. London, 1857. 8vo (454 pp. of tables), § 4. Eees, Aeradam. The CyclopaBdia, or Universal Dictionary of Arts, Sciences, and Literature ... In 39 vols. London, 1819. 4to. Vol. xviii. Hyperholic logarithms, § 3, art. 16. Vol. xxi. Logarithms. § 3, art. 13, Vol. xxviii. Prime numbers. § 3, art. 8. Eeishammer, Pi:Lix. Manuel general pour les Arbitrages de Changes . . . par Nomhrcs fixes ou par Logarithmes . . . suivi d'une Table de Logarithmes depuis Ijusqu'a 10400 (et, a I'aide de la Tables des Differences, jusqu'a 104000) . . . Paris, An viii (1800). 8vo (pp. 326 and 131 pp. of tables). § 3, art, 13, Eeqtjisite Tables. See [Maskeltne.] Eetjss (Repertorium). § 2, art. 3. Eeynaud, a, a. L. Trigonometric , . . troisiemc edition ; suivic dcs tables de logarithmes . . . de Jerome do Lalandc. Paris, 12mo, 1818 (203 pp. of tables). § 4. ■ Reynaud. See Lalande (1829). RnETicus. Opus Palatinum de triangulis a Georgio loachimo Ehetico coeptum : L. Valentinus Otho Principis Palatini Frideriei IV. Eloctoris mathematicus consummavit. An. sal. hum. cio. lo. xcvi. Plin. lib. xxxvi. cap, ix. Rerum naturaj iuterpretationem iEgyptiorum opera philosophia) continent. Cum privUegio ctes, majes. folio, 2 vols, [on an ornamented title- page]. § 3, art. 10, ON MATHEMATICAL TABLES. 159 IlnETICTTfl. ScO PiTISCTJS. lliDDLE, Edavakd. Trcatisc ou Navigation and Nautical Astronomy . . . ■with all the Tables requisite in nautical computations . . . London, 1824. Svo (239 pp. of tables). § 4. Riley's Arithmetical Tables for miiltiplying and dividing sums to the utmost extent of numbers . . . London, 1775. Svo (pp. 176 and intro- duction xii). § 3, art. 1. Eios, Joseph de Mendoza. A complete collection of Tables for Navigation and Nautical Astronomy . . . Second edition, improved. Loudon, 1809. 4to (604 pp. of tables). § 4. Kios, Jose de Mendoza t. Coleccion complcta de Tablas para los uses de la Navegacion y Astronomia Nautica . . . Primera Tirada. Madrid, 1850. 4to. § 4. Eoe, N. Tabulte Logarithmicae, or two tables of logarithmes ... by Na- thaniel Roe, Pastor of Benacre in Suffolke . . . Unto which is annexed Iheir admirable use ... by Edm. "Wingate, Gent. London, 1633. Svo (preface and tables unpaged, the Use &c. pp. 70, and 10 addit. pp. of tables). § 4. RooG (Bibliotheca Mathcmatica). § 2, art. 3. Rosenberg, See Dase (ninth million). Rouse, William. The Doctrine of Chances, or the Theory of Gaming made easy . . . with Tables on Chance, never before published . . . London [no date]. Svo (pp. 350, preface &e. Ivi). § 3, art. 25. RirjtKER, C. Handbuch der Scliifffahrtskunde mit einer Sammlung von Scemanns-Tafeln . . . Vierte Auflage. Hamburg, 1844. Svo (531 pp. of tables). § 4. Saigey. See under Callet, 1853, § 4. *Salomon, Jos. M. Logarithraische Tafeln, enthaltend die Logarithmen der Zahlen 1-10800, die Logarithmen der Sinusse und Tangenten von Sekunde zu Sekunde, etc. "VVien, 1827. 4to (pp. 466 and introduction xxxviii). Also with French text. § 4. Sang, Edward. Five-place logarithms . . . Edinburgh and London, 1859. 32mo (pp. 32). § 3, art. 13. Sang, Edward. A new table of seven-place logarithms of all numbers from 20 000 to 200 000 .. . London, 1871. Large Svo (pp. 365). § 3, art. 13. Sang, Edward, 'Edinburgh Transactions,' vol. xxvi. 1871. (Account of the new table of logarithms to 200 000). See under Sang, § 3, art. 13. ScHEijftEL (Mathematical Bibliography). § 2, art. 3. [ScHEiTTz, G. and E.] Specimens of Tables ; calculated, stercomoulded, and printed by Machinery. London, 1857. Svo (pp. 50). § 3, art. 13. *ScnL03iiLcn, 0. FiinfstelLige logarithmische und trigonometrische Tafeln. Braunschweig. Svo, § 4, Schmidt, G. G. Logarithmische, trigonometrische und anderc Tafeln . . . Giessen, 1821. 12mo (pp. 217 and introduction xxii). § 4. ScHEoN, LuDwiG. Tafeln der drei= und fiinfsteUigen Logarithmen . . . Jena, 1S38. (Small quarto tract, without cover, 20 pp.) § 3, art. 13. ScHRON, LuDwiG. Sicbcnstellige gemeine Logarithmen der Zahlen von 1 bis 108000 und der Sinus, Cosinus, Tangenten und Cotangenten . . . nebst eiuer Interpolationstafel zur Berechnung der Proportion altlieilc . . . Sterco- typ-Ausgabe. Gesammt-Ausgabe in drei Tafeln. Braunschweig, 1860, Largo Svo (pp. 550). § 4, ScuRoN, LuDwio. Seven-figure logarithms . . . Fifth edition, corrected and stereotyped. With a description of the tables added by A. de Morgan . . « London and Brunswick, 1865. Svo. § 4. 160 REPORT— 1873. ScHULZE, JoHANN Carl. Nguc und er^reitcrte Sammlung logarithmischer, trigonometrisclier unci andercr. . . .Tafcln. Berlin, 1778. 2 vols. 8to (each about 300 pp.)- There is also a French titlepage. § 4. ScHTjLZE. See Acad£mie Rotale de Prusse, § 4. Schumacher, H. G. Sammlung von Hiilfstat'eln hcrausgegeben im Jahre 1822 von H. G. Schumacher. Neu herausgegeben \ind vermehrt von G. H. L. Warnstoeff. Altona, 1845. 8vo (pp. 221, and 31 pp. of explanation in French). § 4. [Schumacher.] Auxiliary Tables for Mr. Bessel's method of clearing the Distances. 8vo (pp. 91). [No editor's name, date, or place.] § 4. Schweigger-Seidel (Litteratur dcr Mathematik). § 2, art. 3. S£guin, M. Manuel d' Architecture ou Principes des Operations primi- tives de cet Art .... Cct ouvrage est termine par une table des quarrcs et des cubes, dont les racines commencent jjar Tunlte, et vont jusqu'a dix mille .... Paris, 1786. 8vo (the table occupies 100 pp.). § 3, art. 4. Shanks, William. Contributions to Mathematics, comprising chiefly the Rectification of the Circle to 607 places of decimals . . . London, 1853. Printed for the Author. 8vo (pp. 95). § 4. [Sharp, Abraham.] Geometry Improv'd. 1. By a lai'ge and and accurate table of segments of circles .... with compendious tables for finding a true proportional part . . . exemplify'd in making out Logarithms or natural numbers from them, true to sixty figures, there being a table of them for all primes to 1100, true to 61 figures. 2. A concise treatise of Polyedra. ... By A. S. Philomath London, 1717. Small 4to (pp. 136). § 4. Sharp. See Sherwin. Sheepshanks, R. Tables for facilitating Astronomical Reductions. London, 1846. 4to. § 4. (Also Anonymous, 1844). § 4. [Sherwin, Henry.] Sherwin's Mathematical Tables, contriv'd after a most comprehensive method. . . . The third edition. Carefully revised and corrected by "William Gardiner. London, 1741. 8vo. § 4. Shortrede, Robert. Compendious Logarithmic Tables .... Edinburgh, 1844. 8vo (pp. 10). § 4. Shortrede, Robert. Logarithmic Tables to seven places of decimals containing.... Edinburgh, 1844. Largo 8vo (pp. 829, and introduction, pp. 39). § 4. Also 1849 (2 vols.). See next title, Shortrede, Robert. Logarithmic Tables : containing logarithms to num- bers from 1 to 120,000, numbers to logarithms from -0 to TOOOOO, to seven places of decimals ; . . . . Edinburgh, 1849. 8vo (pp. 208 and preface xxv). This is the title of the first volume; that of the second is, "Logarithmic Tables to seven places of decimals, containing logarithmic sines and tan- gents to every second of the circle, VFith arguments in space and time ..." Edinburgh, 1858 (pp. 602 and preface pp. 2), 8vo. The two volumes seem to have been regarded as separate works, as the book is not stated to be in 2 vols ; nor are they called vol. i. and vol. ii. § 4, under Shortrede, 1849. SoHNEE (Bibliotheca Mathematica). § 2, art. 3. Speidell, J. New logarithmes. the First iuuention whereof, was, by the Honourable Lo ; John Nepair Baron of Marchiston, and Printed at Edinburg in Scotland, Anno : 1614. In whose vse was and is required the knowledge of Algebraicall Addition and Subtraction, according to + and — These being Extracted from and out of them (they being first ouer scene, corrected, and amended) require not at all any skill in Algebra, or Cossike nnmbers. But may be vsed by cuery one that can onely adde and Subtract, in whole numbers, according to the Common or vulgar Arithmeticke, without any consideration ON MATHEMATICAL TABLES. 161 or respect of + and — [Typographical ornament] By lohn Speidell, pro- fessor of the Mathematickes ; and are to bee soldo at his dwelling house in the Fields, on the backe side of Drury Lane, betweene Princes streete and the new Playhouse. [Erasure in ink.] 1619 (unpaged, pp. 90 and titlepage). § 3, art. 16. Stansburt, Daniel. Tables to facilitate the necessary Calculations in Nautical Astronomy New York, 1822. 4to (337 pp. of tables). § 4. [Stegmann, F.] Tafel der fiinfstelligen Logarithmen und Antilogarithmen. Marbitrg, 1855. § 4. *Stegmann. Tafel der natiirlicher Logarithmen. Marburg, 1856. § 4. Steinberger, a. Tafel der gcmeinen oder Brigg'schen Logarithmen aUer Zahlen von 1 — 1 000 000 mit fiinf und beliebig sieben DecimalsteUen .... Eegensburg, 1840. 8vo (pp. 65). § 3, art. 13. Tables du Cadastre, calculated under the dii-ection of Prony (manu- script). § 3, art. 13. Taylor, Janet. Lunisolar and Horary Tables, with their appKcation in Nautical Astronomy ... . London, 1833. 8 vo (pp. 232). §4. Taylor, Janet. An Epitome of Navigation and Nautical Astronomy, with the imjiroved Lunar Tables .... London, 1843. 8vo (320 pp. of tables). § 4. Taylor, Michael. A Sexagesimal Table .... and the Sexagesimal Table turned into seconds as far as the 1000th column .... Published by order of the Commissioners of Longitude. London, 1780. 4to (pp. 316 and intro- duction xlv) § 3, art. 9. Taylor, Michael. Tables of logarithms of All numbers, from 1 to 101000, and of the sines and tangents to every second of the quadrant .... With a preface. . . .by Nevil Maskelyne. . . . London, 1792. Large 4to (about 600 pp.). § 4. Templeton, W. The Millwright and Engineer's pocket Companion . . . corrected by Samuel Maynard : London, 1871. 8vo. (Noticed under [May- nard], § 3, art. 24). Thomson, David. Lunar and Horary Tables .... Forty-fourth edition. London, 1852, 8vo (218 pp. of tables). § 4. Todd, Charles. A series of Tables of the Area and Circumference of Circles ; the Solidity and Superficies of Spheres ; the Area and Length of the Diagonal of Squares.... Second edition. London, 1853. 8vo (pp. 114). § 3, art. 22. Trotter, James. A Manual of Logarithms and Practical Mathematics .... Edinburgh, 1841. 8vo (82 pp. of tables). § 4. Turkish Table of Logarithms &c. [Bulak] 1250 [1834]. 8vo (pp. 270). §4. Ursin. See G. F. Ursinus. Uksinus, B. Beni. Ursini Mathematici Electoralis Brandenburgici Trigo- nometria cum magno logarithmor. Can one Cum Privilegio Coloniae Sumptib. M. Guttij. tipijs G. Rungij descripta CDDCXXV (sic). (This is the title of the volume, and is printed on an ornamented titlepage.) The trigonometria occupies 272 pp. ; and then follows the Ckinon, unpaged, with a fresh title- page. "Benjaminis Ursini Spottavi Silesi .... Magnus Canon triangulorum logarithmicus ; ex vote & consilio lUustr. Nepcri, p. m. novissimo, Et sinu toto 100000000. ad scrupulor. sccundor. dccadas usq; vigili studio & perti- naci industria diductus . . . ColoniiE. Typis Georgij liungij . . . M.DC.XXIY"; but the colophon (at the end of the canon and of the whole work) is " Berolini, Excudebat Georgius Rungius Typographus, impensis & sumtibus 1873. M 162 REPORT— 1873. Martini Guttij. Bibliopolas Colouiensis. Anno CIq IqC XXIV." 4to. § 3, art. 17. Ursinus, G. F. Logaritlinii VI Decimalium scilicet numerorum ab 1 ad 100 000 et Siiiuum. et Tangeutium ad 10" . . . (Impensis autoris.) Hafniie, 1827. 8vo. § 4. Vega, G. Thesaurus logarithniorum completus, ex arithmetica logarithmica, et ex trigouometria artificial Adriani Vlacci coUectus, jjluriniis erroribus purgatus, in novum ordinem redactus, .... "VVolframii denique tabula logarith- morum naturalium locujJetatus a Georgio Vega .... Lipsiaj, 1794. folio (pp. 685 and introduction xxx). There is also a German titlepage. § 4. Vega, G. Georgii Vega .... tabulae logarithmico-trigonometricae cum diversis aliis in Matheseos usum constructis Tabiilis et Formulis. . . . Editio secunda, emendata, aucta penitusque reformata. Lipsia;, 1797. 2 vols. 8vo (pp. 409 and 371 ; vol. i. has also Ixxxiv pp. introduction). There is also a German titlepage. § 4. Vega, G. Georgii Vega . . . . manuale logarithmico-trigonometricum. . . . Editio secuuda, aucta et emendata. Lipsitc, 1800. 8vo (pj). 304 and intro- duction Ixiv). There is also a German titlepage. § 4. Vega, G. Saramlung mathematischer Tafeln. . . . Herausgegeben von Dr. J, A. HtEssE. Stereotyp-Ausgabe. Erstur Abdruck. Leipzig, 1840. 8vo (pp. 681 and introduction xxiv). § 4 (described as Hulsse's Vega). Vega, G. Logarithmisch-trigonometrischcs Haudbuch (einuud^^erzig8te Auflage) .... bearbeitet von Dr. C. Bremiker. Berlin, 1857. 8vo (pp. 575 and introduction xxxii). § 4 (described as Bremiejse's Vega). Vega, G. Logarithmic Tables. . . .by Baron von Vega, translated from the fortieth edition of Dr. Bremiker's by "\V. L. F. Fischer. . . . Thoroughly revised and enlarged edition . . . . Stereotj-ped . . . . Berlin, 1857. (pp.575 and introduction xxvii) § 4 (under Bremiker's Vega). Versed Sines, A Table of. See [Sir Jonas Moore.] Versed Sines, Natural . . . and Logarithmic . . . See [Farlet]. Vlacq, Adrian. Arithmetica logarithmica, sive logarithniorum chiliades centum, pro Numeris naturali serie crescentibus ab Unitate ad 100000. una cum canoue triangulorum sen tabula artificialium Sinuum, Tangentium, & Secantium, Ad Radium 10,00000,00000. & ad singula Scrupula Prima Qua- drantis. Quibus novum traditur compendium, quo nullum uec admirabHius, nee uinius solvendi pleraque Problemata Arithmetica & Geometrica. Hos numeros primus inveuit Clarissimus Vir Johannes Neperus Baro Merchis- tonij : eos autem ex ejusdem sententia mutavit, eorumque ortum & usum iLLustravit Henricus Briggius, in celeberrima Academia Oxoniensi Geometriae Professor Savilianus. Editio Secunda aucta per Adrianum Vlacq Goudanum. Deus nobis usuram vitse dedit et ingenii, tanquam pecuuiaj, nulla prasstituta die. [Typographical ornament.] Goudae, Excudebat Petrus Eammasenius. M.DC.XXVIII. Cum Privilegio Illust. Ord. Generalium. fol. (preface and errata 5 pp., trigonometry &c. 79 pp.; tables unpaged). Part of the title is printed in red. § 4. Vlacq, Adrian. Arithmetique logarithmique ou la constmction et usage d'une table contenantlesLogarithmesde tons lesNombresdepuisl' Unite jnsques a 100000. et d'une autre table en laquelle sont comprins les Logarithmes dcs Sinus, Tangentes & Secantes, de tons les Degrez & Minutes du quart du Cercle, selon le Raid de 10,00000,00000. parties. Par le moyen desquelles on resoult tres-facilement les Problemes Arithmetiques & Geometriquea. Ces nombres premierement sont inventez par lean Neper Baron de Mar- chiston : mais Henry Brigs Professeur de la Geometric en TUniversite ON MATHEMATICAL TABLES. 163 d'Oxford, les a change, & leur Nature, Origine, & Usage illustre selon rinten- tion du dit Neper, La description est traduite du Latin en Frangois, la premiere Table augmentee, & la seconde compose'e par Adriaen Vlacq. Dieu nous a donne I'usage de la vie et d'entendement, plus qu'il n'a fait par le temps passe. [Small typographical ornamentl. A Goude, Chez Pierre llammasein. M.DC.XXVIII. Avee Privilege des Estats Generaux. fol. (preface 3 pp., errata 1 p., trigonometry &c. 84 pp. ; tables unpaged). Part of the title is printed in red. § 4. [The radius is erroneously describedin the above two titles as 10,00000,00000 ; it is really 1,00000,00000, viz. the logarithms are given to ten decimal places.] Vlacq, Adeian. Logarithmieall arithmetike. or tables of iogarithmes for absolute numbers from an unite to 100000 ; as also for Sines, Taugentes and Secantes for everj^ Minute of a Quadrant : with a plaine description of their use in Arithmetike, Geometrie, Geographic, Astronomic, Navigation, &c. These Numbers were first invented by the most excellent lohn Neper Baron of Marcliiston, and the same were transformed, and the foundation and use of them illustrated with his approbation by Henry Briggs Sir Henry Savils Professor of Geometrie in the Universitie of Oxford. The uses whereof were written in Latin by the Author himselfe, and since his death published in English by diverse of his friends according to his mind, for the benefit of such as understand not the Latin tongue. Deus nobis usuram vitoe dedit, et ingenii, tanquam pecuniae, nuUa pi-aestituta die. [Printer's device and motto, Anchora spei.] London, Printed by George Miller. 1G31. fol. (54 pp. of trigonometry &c. followed by " a Table of Latitudes" (8 pp.), and then the logarithmic tables, unpaged). § 4. Vlacq, Adrian. Trigouometria artificialis : sive magnus canon triangu- lorum logarithmicus, Ad Radium 100000,00000, & ad dena Scrupula Secunda, ab Adrian© Vlacco Goudano Consfcructus. Cui Accedunt Henrici Briggii Geometriae Professoris in Academia Oxoniensi p.m. Chiliades logarithmorum Viginti pro numeris naturali serie crescentibus ab Unitate ad 20000. Quorum ope triaugula jjlana & sphaeriea, inter alia Nova eximiaque compendia e Geometricis fundamentis petita, sola Additione, Subtractione, & Bipartitione, exquisitissime dimetiuntur. [Here follows a quotation of seven lines from Kepler. Harm. lib. iv. cap. vii. p. 168,] GoudiB, Excudebat Petrus llam- masenius. Anno M.DC.XXXIII. Cum Previlegio. folio. (Dedication and preface 4 pp., trigonometry &c. 52 pp. ; tables unpaged). § 4. Vlacq, Adrian. Tabulae sinuum, tangentium et logarithmi sinuum tangen- tium & numerorum ab unitate in 10,000 .... Editio ultima emendata & aucta. Amsteltedami : Apud Henricum & Viduam Theodori Boom. 1(581. Small 8vo. § 4. Vlacq's works (Chinese reprint). § 3, art. 13 (introductory remarks, p. 54). Vlacq. See Hentschen. *VoisiN, Antoine. Tables de Multiplications on Logarithmes des Nombres Entiers depuis 1 jusqu'a 20,000 Paris, 1817. § 3, art. 3. Wackeebarth, a. F. D. Fem-staUiga Logarithm -TabeUer, jemte en Samling TabeUer. . . . Upsala, 1867. Small 8vo (pp. 224 and introduction xviii). § 4. Wallace, John. Mathematical Tables containing the logarithms of num- bers, logarithmic sines, tangents, and secants .... By J, Broavn. The third edition, improved, enlarged with many useful additions, by J. Wallace. Edinburgh, 1815. 8vo. § 4. Wallis. See Sheewin. Warnstobff. See Schumacher, ji2 164 REPORT — 1873. 07 4-1 Weidenbach. Tafel um den Logarithmen von ' zu finden wenn der x—1 Logarithme von x gegeben ist . . . . Mit einem Vorworte von Herrn Hofrath Gauss. Copenhagen, 1829. 16mo (pp. 24). § 3, art. 19. Wells, I. Sciographia. London, 1635. See under De Decker, 1626. WiLLicH, C. M. Popular Tables arranged in a new form .... Third edition. London, 1853. 8vo (pp. 166). § 4. WiNGATE. See EOE. WiTTSTEiTS', Theodoe. Logarithmes de Gauss a sept decimales .... Han- nover, 1866. 8vo (pp. 127 and introduction xvi). § 3, art. 19. "Wolfram. 48-place hyperbolic logarithms: these first appeared in Schulze's Sammlung. See Schijlze (1778). "WooLHousE, W. S. B. On Interpolation, Summation, and the Adjustment of Numerical Tables London, 1865. 8vo (pp. 100). § 3, art. 21. "Woolhotjse. See Olinthtjs Gregory (1843). WucHERER, W. F. Beytrage zum allgemeinern Gebrauch der Decimal- Briiche. . . . Carlsruhe, 1796. 8vo (152 pp. of tables aud 48 pp. of intro- duction). § 3, art. 6. Zech, J. Tafeln der Additions- und Subtractionslogarithmen fiir sieben Stellen .... Aus der Vega-Hiilsse'schen Sammlung besonders abgedruckt. Leipzig, 1849. 8vo (pp. 201). Also " Zweiter Auflage," 1863. § 3, art. 19. § 6. Postscript. Art. 1. The foregoing Report is that which was presented to the Brighton Meeting in 1872, considerably enlarged. After the Meeting it seemed de- sirable to extend some of the articles in § 3, and to add descriptions of several works to § 4 ; and it then appeared that the Report was so lengthy that it was thought better to delay its publication till the ensuing volume, so as to afford time for its passage through the press without undue haste. The printing therefore was commenced in February or March, and is now (September 30, 1873) all but finished. It was arranged, as the completion of the Report by a supplement depended in great measure on the coopera- tion of others possessing information on the subject of tables, that a certain number of separate copies should be placed in the hands of the Committee, as soon as the printing was effected, for circulation amongst those interested in the matter, so as to avoid the delay of a year that would otherwise take place before the work undertaken by the Committee became known to those who could render assistance. Art. 2. While the Report has been passing through the press a good many alterations have been made which were necessitated by increased informa- tion on the subjects treated of, and by repetitions &c. which were detected for the first time when the whole appeared in print. But no attempt has been made to increase the extent of the Report by introducing descriptions of fresh works ; in fact only about a dozen have been added since the Brighton Meeting, and but four or five since the MS. was placed in the printer's hands. The tendency of the Report has been fi-om the first to become more and more bibliographical. Originally it was intended to introduce nothing of a bibliographical nature ; but experience showed that this was impossible, and attention to such matters has been continually forced upon us. A report on tables differs from a report on any other scientific subject in this — that whereas in a progressive science the earlier works become superseded by ON MATHEMATICAL TABLES. 165 their successors, and are only of historical interest, a table forms a piece of work done, and, if done correctly, is done for all time. Thus Briggs, 1624, or Vlacq, 1628, when procured, are as useful now as if the tables had been calculated and published recently, subject to the one drawback, that it needs a bibliographical research to determine how far their accuracy is to be relied upon. A table is calculated for a special purpose, which purpose in process of time ceases to be an object of practical interest, and the table is forgotten ; but, for all that, it is the expression of a certain amount of abstract truth, and as such is always of value, and is liable at any moment to be utilized again for some other purpose. Thus one of the most useful objects of the Report is to give in an accessible form accounts of old tables that have passed out of notice, as even the most special table is never so obsolete that some fresh use may not be found for it in the future ; and it is of little value to describe an old and unimportant work without such additional explanation as may lead to its easy identification, with references to the works that contain information of importance to its user. Art. 3. But, apart from the necessity of giving bibliographical information with regard to some works in order to render the descriptions useful, it is to be noticed that mathematical history is practically nothing but mathematical bibliography, as the number of letters and other manuscript documents bear- ing upon the subject is very small. This being so, it seemed a pity when the examination of any work showed it to possess some interest, even though of a purely historical kind, to ignore it entirely merely because the table it contained was clearly destitute of practical value*. The whole additional space thus devoted to bibliography does not altogether amount to more than a very few pages ; and the chief concession that has been made to it is in the list of titles in § 5, where in several cases the full titlepage has been tran- scribed. This, with one or two exceptions, has only been done in the case of the tables of logarithms immediately following their invention in 1614. An examination of a great number of works of reference in regard to this matter has shown us how inaccurate, not only in details but even in pro- minent facts, are the accounts usually given. With the exception of Delambre, Lalande (in his ' Bibliographic Astronomique '), and De Morgan, it is not too mvich to say that not a single writer on the subject is to be trusted. Those only who have had occasion to investigate any historical point, like that of the invention of logarithms, can appreciate the slight value that was set on accuracy previously to the dawning of a more careful age at the beginning of the present century. It is necessary to give this caution, as any one who took the trouble to compare certain statements made in this Heport with those given in such works as Thomson's ' History of the Eoyal Society,' or even Hallam's ' Literature of Europe ' (founded on earlier works), might imagine that our account involved matters of opinion and was liable to be disputed ; whereas we cannot find that any previous writer ever did (or perhaps could in the then state of Kbraries) examine or even see all the works relating to this period. It is also worthy of remark that the early logarithmic tables form a most remarkable bibliographical tangle. For some years it was customary to always place the name of J!fapier on the titlepages * " It would be something towards a complete collection of mathematical bibliography, if those who have occasion to examine old works, and take a pleasure in doing it, would add each his quotum, in the shape of description of such works as he has actually seen, without any attempt to appear more learned than his opportunities hare made him." — De Morgan, 'Arithmetical Books,' p. x. See also 'Companion to the Almanac,' 1851, p. 5. 166 REPORT — 1873. of works on logarithms, as being their inTentor, and, if the logarithms were decimal, that of Briggs (and perhaps also that of Ylacq) in addition. Thus the ' Arithmetica ' of 1628 will be found in bibliographies and library cata- logues usually under the name of Napier or Briggs, and very rarely under that of its author Vlacq. If to this confusion be added the additional com- plication produced by the varieties of ways in which the names of the three leading logarithmic calculators were spelt, it may easily be inferred how incorrect and confused is all the infonnation to be obtained from bibliogra- phical sources, whether general or mathematical*. It is on this account that we have thought it desirable to give the titles of these works in full in § 5. Perhaps it would not have been possible to sec so many of them in any one other country except this ; and the value of a number of such titles collectively in the same list is much greater than the sum of their separate values when scattered in different works. Art. 4. While on the subject of bibliography, it is proper to remark that, in the cases where the full titles have been given in § 5, there is a certain slight want of uniformity in the way in which they have been transcribed, viz. in the use of capitals, the writing at fuU length of words abbreviated, and the modernizing the language by the substitution of u for v or i for j, and vice versa. Titlepages are printed partly in capital and partly in Roman and italic characters ; and when they are transcribed whoDy in Eoman letters, there arise several uncertainties. Thus it is usual in the portion printed in capitals to replace U by V and J by I, and very often not to use a larger letter after a full stop or for a proper name ; and in copying the whole in Roman letters it is doubtful whether to write these as they are, or to recon- vert them. We are inclined to think that the best plan (except when capitals are reprinted as capitals &c., in which case no difficulty occurs) is to make an exact copy, and not even introduce a capital letter after a full stop, although the author would no doubt have done so himself had he printed his title- page in Roman characters throughout. Exception must, however, be made in the case of proper names. These rules have not been followed out com- pletely in one or two of the earliest titles that we copied, before experience hsd taught us that in bibliographical matters the greatest attainable accu- racy should be invariably striven after ; also one or two abbreviations have been replaced by the words at length (such as e.g. " serenis™'-' by " sere- nissimi " or " atq ;" by " atque"). Whejiever, of course, any difference from ordinary spelling is observed, it may be taken for granted that the title is so printed in the book ; the utmost change that has been made being that some words in a few of the titles are modernized. The foregoing remarks apply to the titles that are transcribed at length ; but a few words must also be said with regard to those in which only enough is given to identify the books described without possibility of mis- take. Wherever words are left out from the title, the omission is marked * Even Babbage makes a bibliographical error on the first page of the preface to his tables, where he says that " the first 20,000 were read with those in the Trigonoinetria Artificialis of Briggs." The 'Trigonometria Artificialis' was calculated by Vlacq, and published by him two years after Briggs's death, though the 20,000 logarithms ap- pended were of course originally computed by Briggs. Any one who will look at the title of the ' Trigonometria Artificialis ' in § 5 will see how easily a mistake of this kind can be made ; and in fact an inspection of the titles of the other works of this period will show that it would be difficult for any one who had not bestowed some attention on the history of logarithms to assign them to their true authors. Part of the confusion that exists is due to Vlacq's excessive modesty, which led him on the titlepages of his works to give quite a subordinate position to his own name compared with those of Napier and Briggs. ON MATHEMATICAL TABLES. 167 by dots, except between place and date, where the publisher's name almost invariably occurs ; so that, this being understood, the separation by a comma was considered sufficient. If the work of the Report had to be performed over again, we should adopt a set of fixed rules with regard to the use of initial capitals in the printing of words in titles, instead of leaving the matter to caprice or the printer ; as it is, the treatment in this respect has been fairly uniform, but might have been better. Such details may seem insignificant ; but it is desirable that nothing should be regarded as arbitrary. With regard to the number of pages assigned to books in § 5, there is also a certain want of uniformitj^: at first we merely looked at the number on the last page, and (having assured ourselves that the pagination was continuous) regarded that as the number of pages, ignoring the few pages at the beginning (usually with a roman pagination) that are devoted to preface &c. ; but afterwards we included these also. Our object merely was to give an idea of the size of the work ; so that (except in the cases where the interest of the book was bibliographical, when we took pains to be quite accurate) it was not thought necessary always to count pages that were not numbered. Sometimes it seemed desirable to give the number of pages occupied by the tables instead of the number in the whole book ; and in a few cases, where the pages were not numbered, it was not considered worth while to count them, or even give an estimate. It may be remarked that very frequently (we think we might say more often than not) the pages on which extensive tabular matter is printed are not numbered. Art. 5. The distinction mentioned in § 2, art. 8, between works that are and works that are not described in the Report, viz. that the names of the authors of the former, when the works are referred to, are printed in small capitals, and of the latter in roman characters, has been adhered to as carefully as possible ; but it has been found to be very troublesome and unsatisfactory. We have generally thought it sufficient to print the name in small capitals only once in a paragraph ; and when there is no risk of mistake (as in the description of the work in question itself) the name has been printed in ordinary roman type : the distinction will not be retained in future Reports. Also, with reference to the meanings to be attached to the words 8vo, 4to, &c., explained in § 2, art. 9, experience has shown that it is more conve- nient to use these terms in their technical significations, viz. as defined by the number of pages to the sheet ; and in future Reports they will be so used. It should be stated that, except in the case of a few books of no bibliographical interest, these have been the meanings actually adopted. Care was taken that this should be so in regard to all works of bibliogra- phical interest ; and in most other cases the size, as estimated by the eye, agrees with the technical signification. Art. 6. In § 1 it is stated that the Committee had determined to print and stereotype certain tables of e-' and e"^, and of hyperbolic sines and cosines which had been commenced by the reporter, and that they were then in the press. Only four pages were set tip when the above statement was written ; and shortly afterwards, when the elliptic functions (referred to further on in art. 16) were in process of calculation, it became clear that they would occupy BO much attention that it was not likely that the tables of e^ &c. could be continued by the reporter till after their completion, and, further, that the publication of the elliptic functions would tax the resources of the Committee to such an extent that it was not probable that they would have the means of printing any thing else, at all events for some time. These tables were therefore withdrawn ; and the reporter contemplates completing 168 REPORT— 1873. them (very little more remains to be done) after the publication of the elliptic functions, when they will probably be communicated to one of the learned societies. The table of powers by the reporter, mentioned in § 3, art. 5, is entirely completed, except for the final verification by differ- ences, which is in progress ; and the printing will be commenced very shortly ; but as it is intended to prefix to it a list of constants, with historical notices of the calculation of each, the publication may be somewhat delayed. Art. 7. Any one who studies the Keport attentively cannot faU to notice differences of modes of description in it. These are only verbal, and will be seen to be unavoidable when it is considered that, as a rule, the account of each book was written by itself on a separate piece of paper, and that not till aU had been arranged, and the Report was in print, was it easy to com- pare the descriptions of the same table occumng in different works, and therefore written under different circumstances. Very few of these " dis- crepancies " have been removed, partly because, as each description was cor- rect, it seemed scarcely worth while to make alterations for the sake of a fictitious uniformity, and partly because we made it a rule that, a descrip- tion having been written in the presence of the book, it ought not to be altered when the book was absent. Slight differences of style and manner are inevitable in a work the performance of which has extended over the space of two years, as experience must always continually modify to some extent both opinions and modes of thought and expression ; of course, if the work could be done over again with the experience already obtained, the descriptions would be more uniform. Art. 8. An objection might be made on the ground that descriptions are given of some very minor works, which have not even the bibliographical interest due to age. In answer to this it is to be noted (1) that it is sometimes as important to know that a book does not contain any thing of value as to know what is in it if it does, and that the reader alone should be left to decide what is and what is not valuable ; and (2) that no book is so insignificant that in the future a correct account of its contents will not be of value. " The most worthless book of a bygone day is a record worthy of preserva- tion. Like a telescopic star, its obscurity may render it unavailable for most purposes ; but it serves, in hands which know how to use it, to deter- mine the places of more important bodies " (De Morgan, ' Arithmetical Books,' page ii). Although the primary object of the Report is utility in the present, still it is not detsirable to entirely forget the wants of the future. The difficulty the historian of science meets with consists not so much in getting a sight of the books the existence of which he knows, as in finding out the names of the second- and third-rate authors of the period he is con- cerned with. Bibliographies grow more valuable as they increase in age ; and it may be predicted with confidence, that long after every vestige of claim to represent the " state of science " has passed away from this Report, the list of names in § 5 wiU be consulted as a useful record of nineteenth- century authors of tables. It might be thought that a less detailed descrip- tion of unimportant books would sufiice ; but it is only necessary to point out in reply, that work, unless done thoroughly, had better be left alone. An account of all the tables in a book is absolute, whereas an account only of those that seem to the writer worth notice is relative. Want of thorough- ness is the thing most to be dreaded in all work of a bibliographical, his- torical, or descriptive nature. It is this want that renders all but valueless the greater part of seventeenth and eighteenth-centuiy writings of this class ; and any one who performs such work in an incomplete or slovenly ON MATHEMATICAL TABLES. 169 manner, merely accumulates obstructions which obscure the truth, and ren- ders more difficult the task of his successors, who will have to be at the pains not only of doing the work again de novo, but also of correcting the errors into which others have fallen through his imperfect accounts. Art. 9. With regard to the future Eeport on the subject of general tables that has been mentioned more than once, and is intended to be supplemen- tary to the foregoing, it may be stated that a number of additional tables have already been described and will be included in it ; but the cooperation of others in the matter is requested. Whether the descriptions in the Sup- plement will resemble those in this Report will of course depend on the ex- tent of the former, as, if the number of works described be large, it may be necessary to practise some curtailment. It is requested also that notices of errors detected in the Eeport may be sent to the reporter (see p. 12). Art. 10. Although, as already stated, this Report has no pretensions to completeness, still any one who notices the non-appearance of names well known in calculation (such as that of Legendre) is asked to read the con- clusion of § 1, the list of articles in § 3, and enough of the introductory matter in § 2 to comprehend clearly the spirit that has directed the selection of works included, before coming to the conclusion that the omission was not intentional. Books such as Legendre's ' Ponctions Elliptiques ' and Jacobi's ' Canon Arithmeticus,' though forming separate publications, yet belong more properly to a later portion of the Committee's work, as they are conclusive, not subsidiary tables ; the former belongs to Division II., and the latter to Division III. (see § 1, p. 4). It is perhaps worth noting explicitly, that the word Report has sometimes been used to denote the whole Report that is contemplated by the Committee, including 'the accounts of the Integral and Theory-of-Number tables, and sometimes only the portion of it that will form one year's instalment ; but the context always shows, without risk of confusion, the meaning to be assigned. Art. 11. It was originally intended that the list in § 5 should merely con- tain the titles of the books described in §§ 3 and 4, with references to the section and article where each description was given. But it has been found convenient to render it in addition more of an index to the whole Report by adding cross references, and also a few titles of papers often referred to, as well as references to the places where certain other works or tracts (besides books of tables) were noticed. One or two remarks that should have appeared in the accounts of the works themselves in §§ 3 and 4 have been added after their titles in § 5 (see Babbage, Noeie, 1844, and Napier, 1619, in A table of contents is given at the conclusion of this postscript. Whether a work of reference ever gets into use or not depends more on the complete- ness with which it is indexed than on any thing else. Art. 12. The following statistics will not be found without interest. The number of separate books of tables described at length in this Report (ex- clusive of diiferent editions and of works only noticed incidentally) is 235, of which only 5 are derived from second-hand sources. The 230 that have thus come under the eye of the reporter are thus distributed among the dif- ferent countries : — Great Britain and Ireland .... 109 France 27 Germany (including Austria &c.) 66 Holland 8 170 REPORT— 1873. Denmark 7 Portugal 1 Italy 3 Sweden 1 United States 3 Eussia 1 Switzerland 2 Egypt 1 Spain 1 Belgium supplying none. These figures afford no comparison between Great Britain and other countries ; but they give a fair idea of the relative table- publication of foreign countries, or, at all events, of the relative proportions in which their tabular works are to be found in English libraries. The numbers of tables published in some of the chief towns are as follows : — London 94, Paris 23, Berlin 18, Leipzig 17, Edinburgh 11, Vienna 5, Copenhagen 4, New York 3. Of the 109 works published in Great Britain and Ireland the following is the distribution : — England 96 (London 94, Boston 1, Ci- rencester 1), Scotland 12 (Edinburgh 11, Glasgow 1), Ireland 1 (Dublin), showing the paramount position of London in the publishing trade in this country. Art. 13. Contents of the Eeport that was intended to be presented to THE Bb.u)ford Meeting, 1873. — Owing to the great amount of space already occupied in the present volume by the foregoing Eeport, it seemed desirable to postpone for a year the Eeport which it was till recently intended should be presented to the Bradford Meeting, and only to give here a brief description of the work performed in 1872-1873. This latter Eeport (which is not lengthy) consists of three parts — (1) Tables of the Legendrian Func- tions ; (2) List of errors in Vlacq's ' Arithmetica Logarithmica,' 1628 or 1631 ; (3) Account of the tabulation of the Elliptic Eunctions. Art. 14. The Tables of the Legendrian Functions (La^tlace's Coefficients). — These give P"(a') to n=7 from x=0 to x=l at intervals of -Ol, viz. the functions arc : — P' = l, P'=.i', P= = i(3.j--1), P3 = l(5.r=-3.r), P' = l(35.v'-.30.r' + 3), P= = ^(63,t''-70.f^ + lo,r), P« = JL(231.r'' - 3U.v' + 1 05.v= - 5), p'' = yV(^-9'^'' - '^^^•^'' + '^i^-^'' - 2^-^') ; and as only powers of 2 appear in the denominators, all the decimals ter- minate, and their accurate values are therefore given. The work was per- formed in duplicate — one calculation having been made by Mr. W. Barrett Davis, and the other under the direction of the reporter, by whom the two were compared, the errors corrected, and the whole differenced. As the accurate values of the functions were tabulated, the verification by differ- ences was absolute. A short introduction on the use of the tables in inter- polation was written by Prof. Cayley, who has also made drawings of the curves y = Y"{x) over the portion calculated. Art. 15. The List of Errors in Vlacq's 'Arithmetica Logarithmica' (1628 or 1631). — It seemed very desirable that a complete list of the errata in Vlacq, 1628 or 1631, should be foi-med for the convenience of those who have occasion to employ ten -figure logarithms. No less than five copies of this work have been continually in use in the calculation of the Elliptic ON MATHEMATICAL TABLES. 171 Functions (see next article) during the last year ; and it is tlie ten-figure table chiefly used. Besides this, the errata in Vlacq are known with more certainty than are those in Vega, 1794. This list had only been partially formed when it was determined to post- pone the Heport ; and it is believed that the year's delay may possibly result in its being made more complete. It is proposed to add a list of errata also in Dodson's ' Antilogarithmic Canon,' 1742 (§ 3, art. 14), and perhaps to consider the subject of errors in tables generally. Art. 16. The account of the Tahidation of the Elliptic Functions. — In Sep- tember 1872 it was resolved to undertake the systematic tabulation of the Elliptic Functions (inverse to the Elliptic Integrals), or, more strictly, of the Jacobian Theta Functions which form their numerators and denomi- nators. The formulae are : — 2K.t' e — -= 1 — 2o cos 2a; + 27* cos 4.r—2o^ cos 6.r+. . ., 7r ^ 2Kx 1 ^2K.r TT Jc^ IT 1 / 1 . a 25 \ = —A 2q'^ sm 0-'— 25"* sin 3x-\-2q ■» sin 5.r — . . . J, ^* COS ^ + 22-^ cos3.r + 22 ■»cos5.r + . . .), =(',)^c%i = k'^ (1 + 22 cos 2^+22* cos 4:X-[-2q^ cos 6a? + , . .) ; 80 that 2X.V ^ 2Kr . ^2Kx Sin am = Q^ —Q , TT TT TT cos am = 0., -i- 6 , TT ' TT It 2Xx' „ 2K.i; . ^2K.r A am =63- — ~" J TT IT IT irK' 2 being, as always, e ^ ; and the tables, when completed, will give 9, Gj, e^, 63 and their logarithms to eight decimals for x=r, 2°, . . . 90°, ^- = sin 1°, sin 2°,... sin 90°. The tables are thus of double entry, and contain eight tabular results for each of 8100 arguments, viz. 64,800 tabular results. The arrangement will be so that over each page h shall be constant ; and at the top of each page certain constants (i. e. quantities independent of x), such as K, K', J, J', E, iK {^f, (^.)^ q, «fec., 172 REPORT— 1873. and their logarithms, which are likely to be wanted in connexion with tho tables, will be added. K and K' (complete elliptic integrals) were, as is well known, tabulated by Legendre, and published by him in 1826. For the performance of the calculation of and G3 (63 being deduced from 9) 8500 forms were printed and bound up into 15 books (550 in each, with a few over). Each book, therefore, contains forms for the calculation of six nineties, viz. from ^•=sin a° (say), x = 0°, to ^ = sin (a°+5°), x = 90°. Similar forms for the calculation of 0^ and 0^ were priated and bound up into 15 other books. The work has been in active progress since the beginning of October 1872 ; and eight computers have been engaged from that time to the present, under the superintendence of Mr. James Glaisher, F.R.S., aud the Reporter. About three quarters of the work is now performed — having been calculated com- pletely, and its accuracy verified by differences, and 03 being nearly finished also, while very considerable progress has been made with 0^ and 0. . It is intended that the tables, which wiU be completed, it is hoped, by February 1874, shall form a separate work, and that they shaU be preceded by an introduction, in which aU the members of the Committee wiU take part, — an account of the application of the functions in mathematics generally being undertaken by Professor Cayley, of their application in the theory of numbers by Professor H. J. S. Smith, and of their use in physics by Sir W. Thomson and Professor Stokes, while the account of the method of calcula- tion &c. will be written by the Reporter. The magnitude of the numerical work performed has not often been ex- ceeded since the original calculation of logarithms by Briggs and Vlacq, 1617-1628 ; and it is believed that the value of the tables wUl be great. After the circular and logarithmic functions there are no transcendants more widely used in analysis than the Elliptic Functions ; and the tables will not only render the subjects in which they occur more complete, but wiU also, to a great extent, render available for practical purposes a vast and fertile region of analysis. Apart from their interest and utility in a mathematical point of view, one of the most valuable uses of numerical tables is that they connect mathematics and physics, and enable the extension of the former to bear fruit practically in aiding the advance of the latter. Art. 17. Note on the Centesimal Division of the Degree. — In the note on p. 64 we have expressed an opinion that Briggs and his followers, by dividing centesimally the old nonagesimal degree, showed a truer appreciation of how far improvement was practicable, or indeed desirable, than did the French mathematicians who divided the quadrant centesimally. On reading Stevinus's ' La Disme,' the celebrated tract in which the invention of decimal fractions was first announced, we found that the centesimal division of the degree was there suggested. The following extract from ' La Disme ' is taken from pp. 156 and 157 of ' La Pratique d'Arithmetique de Simon Stevin de Bruges ' (Leyden, 1585), near the end of which ' La Disme ' appears in French. The first publication of the tract, as far as we can find, was in Dutch, under the title " De Thiende .... Beschreven door Simon Stevin van Brugghe " (Leyden, 1585). " Article V. Des Computations Astronomiques. — Aians les anciens Astro- nomes parti le circle en 360 degrez, ils voioient que les computations Astro- nomiques d'iceUes, auec leurs partitions, estoient trop labourieuses, pourtant ils ont parti chasque degre en certaines parties, & les mesmes autrefois en autant, &c., a fin de pouuoir par ainsi tousiours operer par uombres entiers, en choissisans la soixantiesme progression, parce que 60 est nombre mesurable ON MATHEMATICAL TABLES. 173 par plnsienrs (sic) mesures entieres, a sgauoir 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, mais si Ton peut croire Texperience (ce que nous disons par toute reuerence de la venerable antiquite & esmeu auec I'vlilite commune) certes la soixan- tiesme progression n'estoit pas la plus commode, au moins entre celles qui consistoient potentiellement en la nature, ains la dixiesme qui est telle : Nous nommons les 360 degrez aussi Commencemens Ics denotans ainsi 360(0) * & chascun degre ou 1(0) se diuisera en 10 parties egales, desquelles chascune fera 1(1), puis chasque 1(1) en 10(2), & ainsi des autres, comme lesemblable est faict par plusieurs fois ci deuant " f- At the end of the ' Appendice du Traicte des Triangles,' which concludes the fourth book of the " Cosmographie " in Albert Girard's edition of Stevinus's collected works, Leyden, 1634 (p. 95), there occurs the following note : — " Notez. — J'ay descrit un chapitre contenant la maniere de la fabrique & usage de la dixiesme progression aux parties des arcs avec leurs sinus, & de- clare combien grande facilite en suit, comparee a la vulgaire soixantiesme progression, de 1 deg, en 60(1), & 1(1) en 60(2), &c. laquelle matiere pour- roit ici sembler requerir sa place : Mais veu que les principaux exemples d'icelle se prennent des cours moyens des Planetes & autres comptes communs avec iceux, qui jusques ici ne sont point encores descrits, nous avons applique le susdit chapitre derriere le traicte d'icelles Planetes, a sgavoir en V Appen- dice du cours des Planetes." To which is appended the following note by Girard : — " Ceste promesse ne se trouve pas avoir este effectuee." Steichen, in his ' Memoire sur la vie et les travaux de Simon Stevin ' (Brussels, 1846), p. 52, says that Stevinus promises a chapter on the manner of constructing a table of trigonometrical lines " pour la division de la cir- conference en parties decimales." This is not correct, as the quotation from ' La Disme ' shows that Stevinus's idea was to divide the deyree cen- tesimaUy. Briggs, in the ' Trigonometria Britannica' (p. 1), states that he was led to divide the degree centesimally by the authority of Vieta (" Ego vero adductus authoritate Vietae, pag. 29. Calendarij Gregoriani, & aliorum hortatu, Gradus partior decupla rations in partes primarias 100, & harum quamlibet in partes 10. quarum quselibet secatur eadem ratione. Atque hae partes cal- culum reddunt multo facilorem {sic), & non minus certum "). We have looked through ' Francisci Vietse Fontenaeensis .... Relatio Kalendarii vere Gregoriani. . . .1600 " (Colophon: ' Excudebat Parisiis. . . . ,' 40 leaves, as only the rectos are numbered, 1 to 40) without finding, either on p. 29 or elsewhere, any mention of the division of the degree. Without venturing to say that there is nothing of the kind in the book, it is not unlikely that the wrong work of Vieta's is referred to, as we have found many other seven- teenth-century references inaccurate ; and this is rendered more probable when it is remembered that the ' Trigonometria Britannica ' was published after Briggs's death. But granting, as is likely, that Briggs did derive the idea from Vieta, it is very probable that the latter himself obtained it from Stevinus, and perhaps adopted it without acknowledgment, as unfortunately it is to be feared that * Stevinus encloses tne exponential numbers in complete circles, for which we have throughout substituted parentheses, for convenience of printing. t This refers to the preceding articles of the ' Disme,' where the decimal division is explained. 174 REPORT — 1873. Vieta was bigoted enough to suppress tke name of a heterodox author, such as in all likelihood Sttivinus was. There can therefore be but little doubt that the original suggestion for the centesimal division of the degree is contained in the sentence quoted from ' La Disme ;' but we intend to inves- tigate the question further, and endeavour to decide it conclusively. Contents of Part I. (1872 and 1873) of the Report on Mathematical Tables. Page § 1. General Statement of the Objects of the Co7nmittee 1 § 2. General Introduction to the present Report, and explanation of its Arrangement a7id Use. Art. 1. (Report inoludes general tables ; see also conclusion of § 1) 4 2. (Object of the Report) 5 3. (Previous works on the subject of tables; bibliographies, &c.) 5 4. (Mode of arrangement of the Report ; meaning of a prefixed asterisk) 7 5. (Explanation of the marks, conventions, terminology, &c. adopted) 8 6. (The particular edition of a work described is arbitrary) 10 7. (The tables themselves, and not merely their titlepages, hare been ex- amined) 10 8. (Why certain names are printed in small capitals, or enclosed in square brackets; see also § 6, art. 5) 10 9. (Use of the words 8vo, 4to, &c. ; see also § 6, art. 5) 11 10. (Libraries consulted) 11 11. (The Report is imperfect; information is asked from persons possessing knowledge on the subject of tables) 12 12. (Traverse tables omitted) 12 13. (Errors in tables) ' 13 14. (The works are described from inspection ; care taken in preparation of the Report) 14 § 3. Separate Tables, arranged according to the nature of their contents ; with Introductory Ecmarks on each of the several kinds of Tables included in the present Beport. Art. 1. Multiplication tables 15 2. Tables of proportional parts 20 3. Tables of quarter squares 21 4. Tables of squares, cubes, square roots, and cube roots 25 5. Tables of powers higher than cubes 29 6. Tables for the expression of vulgar fractions as decimals 30 7. Tables of reciprocals 33 8. Tables of divisors (factor tables), and tables of primes 34 9. Sexagesimal and sexcentenary tables 40 10. Tables of natural trigonometrical functions 41 11. Lengths (or longitudes) of circular arcs 47 12. Tables for the expression of hours, minutes, &c. as decimals of a day, and for the conversion of time into space, and vice versa 48 13. Tables of (Briggian) logarithms of numbers 49 14. Tables of antilogarithms 62 1 5. Tables of (Briggian) logarithmic trigonometrical functions 63 16. Tables of hyperbolic logarithms (viz. logarithms to base 2-71828 ) 68 17. Napierian logarithms (not to base 2-71828 ) 70 18. Logistic and proportional logarithms 73 19. Tables of Gaussian logarithms 75 20. Tables to convert Briggian into hyperbohc logarithms, and wee wrsa 78 21. Interpolation tables ... 79 22. Mensuration tables 79 23. Dual logarithms 80 ON COAL-CUTTING MACHINERY. 175 Page 24. Mathematical constants 81 25. Miscellaneous tables, iigurate numbers, &c •. 83 §4. Works containing Collections of Tables, arra^igedin alphabetical order 85 § 5. List of Works containing 7'ables that are described in this Report, with references to the section and article in which the description of their cotitents is to be found 143 § 6. Postscript. Art. 1. (Eeport is that presented to the Brighton Meeting enlai-ged) 164 2. (Alterations since the Brighton Meeting; Report has been made more bibliographical) 164 3. (Reasons for introducing bibliography ; inaccuracy of previous writers) ... 165 4. (Explanations with regard to the list of books in § 5) 166 5. (Supplementary explanations referring to § 2, arts. 8 and 9) 167 6. (The tables of hyperbolic antilogarithms and powers calculated by the Re- porter; § 1, and § 3, art. 5) 167 7. (Slight differences in mode of description observable in the Report) 168 8. (Why some unimportant works are included) 168 9. (The Supplementary Report on general tables) 169 10. (Some books omitted intentionally, as. belonging more properiyto subse- quent Reports) 169 11. (§ 5 has been made an index as well as a list of titles of books) 169 12. (Statistics with regard to books described in the Report from inspection) ... 169 13. (Contents of the Report that was intended to be presented to the Bradford Meeting) 170 14. (The tables of the Legendrian functions) 170 15. (Thelistof errors in Vlacq, 1628 or 1631) 170 16. (The account of the tabulation of the elliptic functions) 171 17. (Note on the centesimal division of the degree) 172 Page 6, line 8 from bottom, for Poggendoff read Poggendorlf. Page 15, line 25 from top, for multiplication read multiplication t; ERRATA. Poggendorlf. table. Observations on the Application of Machinery to the Cutting of Coal in Mines. By William Firth, of Birley Wood, Leeds. [A Communication ordered by the General Committee to be printed in extenso.'] The object of this paper is to submit for consideration some matters toucli- ing the history of the now more than ever absorbing subject of cutting coal in mines by mechanical means. It is intended to avoid all technical and scientific symbols, and to convey, in the most simple manner, whatever information is at my command, and to give, from practical experience, spread over long periods, the results derived therefrom, and to show that machinery can be, and is now, applied to the purpose equally to the advantage of the masters and of the men. I am aware that there are noiv several distinct modes of doing the work, and doing it well ; but it is not in my power to give any reliable information upon the competitive status which the successfid machines hold towards each other. I shall tlierefore have in this paper to confine myself more particularly to the introduction of coal-cutting machinery driven by com- pressed air, and the results obtained from the invention now known as " Firth's Machine," which was unquestionably the first that ever succeeded in reducing to actual practice the cutting of coal in mines. 176 REPORT— 1873. When the severe nature of the employment of manual labour for the " hewing of coal " and the great dangers which beset that occupation are taken into thoughtful consideration, it is not surprising that much sympathy should have been always excited in favour of the coal- working class. All men who have thought upon the subject have felt a strong desire that some mechanical invention might be made to amehorate the severe conditions of that occupation. The statistics of the comparative longevity of the working classes show that the duration of the lives of colliers (apart from special accidents) is lamentably low ; and as respects the " hewers" or "pickmen," whose work is the most exhausting, they must especially, and in a large degree, contribute to, and account for, much of that average shortness of hfe. The really hard work of a colliery faUs upon the " hewers ; " and the effect is very often to stamp the men with the mark of their trade, and (through the constrained position of their daUj' toil) to alter and distort many of the more delicately formed persons ; and it is due to these men as a class, that their weaknesses should be mUdly judged, having regard to the scanty oppor- tunities hitherto afforded to them for intellectual culture, and the unequal sacrifices which press so heavily upon them in the most valuable and im- portant branch of all our indusbries. In 1862 some experiments were commenced at West Ardsley, by the em- ployment of compressed air, to devise a cutting-instrument in the form of a pick. It was to be moved on the face of the coal, strildug in a line and with such force as would cut a groove deep enough to admit of its being easily taken out. In the early stages there were many and serious discouraging symptoms discovered, but on the whole we were well satisfied that they could be overcome by perseverance. We set about to improve the defects, and battle with the difficulties as they presented themselves ; and after some years we were in possession of a coal- getting machine, in combination with air-power, more suitable for the performance of the work which we had undertaken than we ever anticipated. Much surprise has been expressed at our slow progress during the ten years which have elapsed since the time when we believed that we had reached success ; but when the peculiar circumstances which surround the work, and the nature of the work to be done, are taken fairly into account, the delay need not excite any astonishment. It was in many respects a new field to be broken up, and accompanied by numerous uncertainties. It has been more or less so with most of the important inventions which have gone before it ; indeed the steam-engine, whose origin cannot be traced back, was known as a prime mover nearly two centuries before it was sufficiently developed to be recognized as a valuable machine. We found, however, that we had to contend against much jirejudice and resistance. Those who were the most likely to be benefited by it were either openly hostile or manifested an unfriendly disposition towards the machine ; and, added to these embarrassments, we failed to obtain any general encouragement from those who exerted the greatest influence over the coal-mining interests of the country ; but through the recent deai-ness of coal, the attention of the country has been drawn to the subject, the public mind has been powerfully impressed with the necessity for some improved means of working the mines, and coal-cutting machinery is now universally looked to as the principal source from which rehef is to come. Erom the altered fecHngs of the miners as to the number of hours which they consider to be sufficient for their labour, and with the new restrictions ON COAL-CUTTING MACHINERY. 177 imposed by the Legislature, there is found already at every colliery in the country a deficiency of hands to fully man the works now existing ; and coal has in consequence been scarce and exorbitantly dear. The consumption goes on increasing ; the continual enlargement of the old iron works, and the establishment of new ones in new districts, indicate a progressive en- largement in the demand for coal, unless a general collapse in our foreign commerce should, through high prices of production, come upon the country. New coal-fields, too, are sought after ; and new pits are being opened in every direction, at enormously increased cost ; and the question naturally arises, where are the colliers to come from to work them, or how is the in- creased demand to be reasonably met ? Labourers from the agricultural districts, and other unskilled workmen, may, through the influence of high wages, be drawn off to the mines ; but it is only in " dead work " where they can be immediately made use of, and only a small proportion make eificient " pickmen." By the figures laid before Mr. Ayrton's Committee of the House of Com- mons (1873), it appears that whilst in 1871 the average production of coal per man was 313 tons, it had declined to 296 tons per man in 1872. There had been an increase in the number of persons employed at and about the mines of 42,184. The disturbance which has been felt in nearly every other occupation seems to me to be traceable to the heavy drafts which have been made upon them to supply the increased demand for the coal and iron trades during the last two years ; and until stagnation and distress in those trades shall throw back the suffering masses again upon their former employment, that disturbance must continue, with all its inconveniences. A. continuance of the present high price of coal may, and I think will, make itself felt upon the foreign commerce of this country. I believe, how- ever, that a decided modification of these evils may be found in the speedy adoption of coal-cutting machinery. Other countries are now turning their attention vigorously to the employ- ment of coal-getting machinery ; and it is not improbable that foreigners will in this matter take the lead in the employment of an invention purely English. In the earlier stages of machine-working, it was contended that the " creep " of the floors, and the natural disturbances of the strata, would so dislocate or break the joints of the air-pipes, that continuous working could not be carried on, the out-put would be intermittent and uncertain, and the cost of compressing the air would be enormous and overwhelming to the enterprise. The coal-owners during many years had had an unprofitable trade, and they were unwilling to encounter a considerable outlay of new capital in the work incident to the new system, which, indeed, had not then met with the approbation of the engineers and mining agents, whUst tlie mining inspectors, with very few exceptions, were decidedly mistrustful of the success of the invention. There were others who believed that the heavy work which they saw done would knock the machines to pieces, and that they could not stand the test of long-continued service. Five or six years, however, of regular and daily working of the machines at Ardsley and elsewhere have effectually negatived these fears. In the collier class there is a good deal of professional pride or esprit de corps, especially amongst the older men. There was, and still is, an unwill- ingness to give up the social dignity which they consider belongs to the expert wielder of the time-honoured pick ; and some of them have been heard to declare that they " would adhere to the ancient implement to the end of 1873. N 178 REPORT— 1873. their days," and that they would not come down to the humiliating condition, as they considered it, of " following the machine." This feeling on the part of the colliers has hindered the progress of machine- work more than any other difficulty ; and although it yet prevails to some extent, the more intelligent and tlie younger men evince a contrary disposi- tion towards it. The leaders of the miners of Yorkshire and other districts have seen the machines at work, and, whilst they express -without hesitation their un- qualified approbation of them*, state frankly that their object wiU be to gain as fuU and fair a share of the advantage of the machines as possible for their own class. Wow nobody will object to that claim ; and when we come to con- sider the figures of cost, as we presently shall do, it will be seen that that claim has not been neglected. InteUigence is what is required to manage these machines, rather than muscular development ; and any youth of ordinary capacity can in a few days acquii'e sufficient knowledge to do so. In 1761, Michael Menzies, of Newcastle, obtained a patent for cutting coal in mines ; and that is the earliest evidence which we have of any attempt having been made to produce a mechanical coal-cutter ; and his plans, having regard to the time at which they were produced, were remarkable for their ingenuity. Menzies's specification is also remarkable in other respects, as showing that it was his intention to make use of the " Fire Engine " as his motor, which engine had about two years previously, tkrough the improvements of Watt and of Smeaton, attained only to so much perfection as to become a doubtful rival to the " Water Miln," the " Wind Miln," and the " Horse Gin." By the power of one or other of these agents, he proposed to give motion to a heavy iron pick, made to reciprocate by means of spears and chains, carried down the pit, and with wheels and horizontal spears, on rollers, extended to the working places, and there to " shear " the coal exactly as it is now performed. In the same patent Menzies included a " Saw" to cut the coal ; and although nothing came from his labours, he displayed so much mechanical knowledge as to have deserved success ; and I am satisfied that his failure was due to the absence of an eligible power, and not to his defi- ciencies as a mechanic. During the hundred years that followed these events, more than a hundred other patents were applied for and granted ; but I cannot find, amongst them all, that there was one machine that approached nearer to success than the invention of Michael Menzies. This fact is not referred to in disparagement of the patentees ; for there were many curious devices, ingeniously arranged ; but I name the matter to show * Extract from Letter received by the West Ardsley Company, dated 22nd February, 1872, from Mr. Philip Casey, of Barnsley, Secretary to the South Yorksliire Miners' Union. " Will you allow me to express the gratitude which I feel for the pleasure I derived in visiting your works yesterday ? " For many years the name of IVIr. Firth has been known to me in connexion with his efi'orts to lighten the heavy labour incidental to mining operations ; and the coal-bearing machine that I saw in operation at the West Ardsley Works altogether exceeded my expectations. " I cannot see how the coal could possibly pay to be got by hand ; its extreme hardness, coupled with the thinness of the seam, would make it utterly impossible. This machine is the best friend the collier ever had ; but it will Ije our business to obtain a full and fair shave of its benefits to our people.'' ON COAL-CUTTING MACHINERY. 179 that the object excited much contiauous interest, and that amongst so many miscarriages our mechanics were still hopeful. Amongst these devices may be enumerated the " Saw," " Catapult," " Battering-Earn," " Plough," " Rotary Wheel," " Endless Chain," " Planing Machine," and many others. There had been no suitable power made known for diivrng the machines ; and it was to that cause, in my opinion, that so many failures and disap- pointments were attributable. The steam engine, even since it attained to its most perfect form, is in itself insufficient for the purpose, because steam cannot be produced near to the place where the work has to be done, nor can it be carried long distances in effective condition, by reason of its rapid con- densation. Moreover an escape of exhaust-steam could not be permitted in the coal-mine, because of its tendency to soften and bring down the roof, the difficulty of maintaining which is already the most serious and troublesome part of the coal-mining operations. Hydraulic power might in certain cases be, and has been recently tried ; but its unfavourable conditions exceed its advantages for the purpose of cutting coal in mines, and may be put aside from present consideration. But in compressed air, so far as the moving power is concerned, every requirement is found, and from the date of the experiments at West Ardsley, in 1862, the question was undoubtedly settled. The elastic property of air under compression is an old and well-known power ; but until these experiments had been completed, its value was but im- perfectly understood, and its future beneiicial influence, being dormant, was imappreciated. The engine for compressing the air is generally placed on the surface, near to the top of the shaft ; a receiver is fixed in close proximity thereto ; and the air is taken from the compressor to the receiver, which is 30 feet in length and 4 feet in diameter. The pressure is generally of about three atmospheres. Iron pipes of sufficient area are laid on from the receiver to the bottom of the shaft, and there, being spht into smaller sizes, are led in every needed direction through the roads and passages .of the mine, exactly as the gas and water services are laid on in our towns. At the entrance into the working places, a screw joint and stopcock are fixed to the iron air-pipe, at which point an india-rubber hose, fifty or sixty yards in length (as the length of the " bank " may require), is screwed on ; the other end of the hose is attached to the cutting machine ; and when aU is in readiness, the tap at the receiver is turned on, and the air rushes down, and throughout the whole service of pipes. The air does not require to be forced from the receiver, for by its own elasticity it is carried forward at a velocity depending on its own pressure. Apparently it loses none of its power by distance, excepting the frictional retardation ; and machines are working nearly two miles distant from the air-engine without any material loss of force. I have no doubt that if the compressor were stationed in Bradford the air would travel, and the machines work by it at Ardsley (ten miles; as satisfactorily as they now do by the engines on the spot. In calculating the cost of compressed air, I am satisfied that, although it is admittedlj' not a cheap power relatively to steam, yet there is no other available power so cheap or so good for the purpose of cutting coal in mines ; and I invite attention to the figures on this head which follow, viz. :— With well-constructed machineiy, 45 to .50 per cent, of the steam power N 2 180 REPORT— 1873. exerted will be given off iu compressed air at a pressur* of three atmospheres iuto the receiver ; and this pressure is sufficient for effectually working our machiner)''. Some makers of air-engines offer to guarantee a much larger product ; but I base my calculations upon the smaller yield. If the pressure be much higher than three atmospheres, there is a material increase in the frictional heat disengaged by the act of compression. The engines do not work wdth the same ease ; and the result of our experience is, that at 45 to 50 lbs. the maximum point of economy is attained. Calculating its cost and taking a 40-horse-power boiler to consume 10 lbs. of coal per hour per horse-power, or 2 tons of engine-coal per day of 11 hours at 8s. per ton at the pit, we have a cost of 16s. per day. It is safe to calculate that this boiler will drive an engine of sufficient power to supply four coal-cutting machines, being 4s. per day for each machine ; and each machine will cut more coal in any given time, and do it in a better manner, in an ordinaiy seam, than twelve men ; it follows, therefore, that the equivalent of a man's power exerted for a whole day in cutting coal, can be obtained, out of compressed air, at a cost in fuel of but 3|cL Assuming, then, that this comparison is an accurate one, it may be taken for granted that the objection to its use, on the score of cost, has no founda- tion in fact. And considering its many and remarkable properties for employment in coal-mines, it may be useful to dwell briefly on some of those peculiarities. It is a power from which, and under no circumstances, can an explosion happen ; and when an escape from the pipes takes place, it is more or less beneficial, and not in any wise injurious. At every stroke of the piston the air is discharged from the cylinder of the coal-cutting machine at a temperature of about freezing-point, compressed into one third of its natural bulk ; and it has been found that the working of only one machine has had the effect of reducing the temperature at the working face of the coal to the extent of two degrees Fahrenheit. Occasionally ice is formed at the escape valves of the machine, but with- out producing any inconvenience to their working. Now any thing that will reduce the temperature of a mine is an inesti- mable advantage. It diminishes the risk of explosion ; and by increasing the velocity of the ventilating current, it renders the occupation of a miner more tolerable and more healthy. In very deep mines the internal heat will probably be found to be so great, that manual labour of an exhausting character will be unendurable ; but the discharge of so large a volume of pure air at a pressure of three atmospheres, and at freezing-point, must exert a powerful and highly favourable influence under the peculiar circumstances. It is well known that the lives which are lost through explosions of gas are far more numerous from the effect of the damp wliich folloivs the fire, than from the fire itself ; and in many cases nearly, if not all, the sufferers have died from this cause. There has been no case of fatal explosion within the experience of our machine w^orkings ; and therefore we have no facts upon which absolute reliance can be placed ; but we draw the inference, that where coal-cutting machinery may be in general use in any mine where an explosion of gas does take place, those who escape from the first effect of the fire will most pro- bably be saved from death. At a lamentable accident in this neighbourhood about two years ago, ■when thirty-one lives were lost, twenty- five or twenty-seven of those unfor- ON COAL-CUTTING MACHINERY. 181 tunate persons died from the effect of the " afterdamp ; " two of the mea were fortunately saved by a very smaU current of air which was turned upon them by a brattice cloth, and which supported life until they were released*. If the compressed air-pipes had been in those workings at that time, it is not unreasonable to believe that very few, if anj^, of those twenty-live men would have succumbed. There is another useful purpose incidental to the use of coal-cutting machinery in mines, which it is worth while to notice ; and that is in the event of a jnt being on fire. At West Ardsley a " blown-out shot " ignited the gas and set fire to the goaf. It extended to the face of the coal, and had taken strong hold of it, and the whole pit was in the greatest danger. There is a large water-tank at the surface for suppljdng the boilers and coke-ovens ; and the manager promptly connected the air-pipes to the water-tank and turned the water into the fire. In less than an hour the fire was completely extinguished without any serious damage. On a previous occasion the same colliery was on fire, and had to be closed up. That fire cost us many thousands of pounds. It hap- pened before the introduction of the coal-cutting machinery. Compressed air is also becomiug extensively used for " hauling," and with very great advantage. Small engines can be set up wherever convenience or necessity may require ; they are portable and removable at a trifling expense, and are available where no other mechanical power for traction can be obtained. It is also valuable for pumping water, and " drilling " the holes where the coal has to be " blasted," or broken down by the hydraulic press. Enough has been said respecting this remarkable and diversified power to justify the expectation that it is the key to vast and important improvements upon the present system of working coal ; and bearing in mind that the wealth, the power, and the greatness of this nation depend primarily upon an abundant supply of coal, it is hardly possible to overrate the importance or overvalue the advantage which this power places at our disposal. I now turn to the consideration of the machine for cutting the coal, which has for several years been employed at West Ardsley without any interrup- tion. [A model and photograph were exhibited to show its form and con- struction.] The weight is about 15 cwt. for a machine of ordinary size, its length 4 feet, its height 2 feet 2 inches, and the gauge 1 foot 6 inches to 2 feet ; it is very portable and easily transferred from one benk to another. The front and hind wheels of the machine are coupled together in a similar manner to the coupled locomotive engines. The " pick," or cutter, is double- headed, whereby the penetrating power is considerably increased. The groove is now cut to a depth of 3 feet to 3 feet 6 inches at one course, whereas by the old form of a single blade we had to pass the machine twice over the face of the coal to accomplish the same depth. The points are loose and cottered into the boss ; so that when one is blunted or broken, it can be replaced in a few moments. This dispenses with the necessity of sending the heavy tools out of the pit to be sharpened, and is an immense improvement upon the old pick. When all is in readiness for work, the air is admitted and the reciprocating * I am informed that at the accident at the Oaks Colliery, near Bai*n8ley, in 1866, forty-five persons were found dead in one place, and seventy in another, vfIio were lost for want of a little air ; and it is bolieved tliat many more al that time died from the same cause. 182 REPORT— 1873. action commences. It works at a speed of sixty to ninety strokes per minnte, varying according to the pressure of the condensed air, the hardness of the strata to be cut, or the expertness of the attendant. As to the quantity of work in " long wall," a machine can, under favoui- ahle circumstances, cut 20 yards in an hour to a depth of 3 feet ; but we consider 10 yards per hour very good work, or say 60 yards in a shift. This is about equal to the day's work of twelve average men ; and the per- sons employed to work the machine are one man, one youth, and one boy, who remove and lay down the road and clear away the debris. The machines are built so strong that they rarely get out of working con- dition. Some of those now working at West Ardsley (and other places) have been in constant use for three or four years. At that colliery there are about eight machines in use. One of the seams is so hard and difficult to manage that it could not be done " by hand," and the proprietors had to abandon, and did abandon it ; but now, by the employment of the machines, it is worked with perfect ease. It is a thin cannel seam with layers of ironstone ; and the machines now " hole " for about 1200 tons per week. The groove made by the machine is only 2 to 3 inches wide at the face, and 1| inch at the back; whereas by bandit is 12 to 18 inches on the face, and 2 to 3 inches at the back. Thus, in thick seams worked by hand, the holing is often done to a depth of 4 feet 6 inches to 5 feet, and the getter is quite within the hole that he has made ; and where the coal does not stick well up to the roof, or where there is a natural parting, there is great difficulty and danger from " falls of coal." Referring to a section, it was observed that the angle of tlie cut is such that, when the upper portion falls off, there is nothing for it but to pitch forward into the road ; but by machine "holing" with a perfectly horizontal groove, when the coal falls it simply settles upon its own bed, and has no tendency to fall forward. The cost of applying coal-cutting machinery is an important part of the question ; but it frequently happens that at old-established collieries there may be surplus power, which can be utilized; but supposing that everything has to be provided new, then the following may be taken as an approximate estimate of the necessary outlay : — 2 Boilers at .£500 each .£1000^ 1 Steam-engine 1250 | 10 Machines at ^150 each 1500 > say £5000 Pipes, receiver, fixing and sundry other 1 -. ^rn outgoings J ■ ■ J This outlay would provide aU necessary power and plant for the regular working of eight machines, with two in reserve ; and estimating that each machine will cut 60 yards per day, the product in a 4-feet seam would be 85 tons per day, or per week say 500 tons per machine ; and 8 by 500 is 4000 tons. Now at this rate of expenditure and work done, an allowance of 2c?. per ton would in three years liquidate the entire outlay. But there is no reason why the machines should be restricted to a single shift daily ; indeed it is far more economical to work double shifts : there is no additional outlay of capital ; and so far as depends upon the machinery, the output might be easily increased to 8000 tons per week. We now come to the relative costs of cutting the coal by hand and by ON COAL-CUTTING xMACHINKRY. 183 machine ; and the following figures may be taken as representing a somewhat favourable state of things for the latter. The seam is the " Middleton Main " or " Silkstone bed." The depth of the mine is 160 yards, and the coal 4 feet thick ; there are two bands of shale, with a thin layer of coal between them. The bottom portion is not always wholly merchantable ; but when it is so, it yields one ton and a third of a ton per running yard. For the purpose, however, of this comparison, I take 60 tons only per day (which would come out of 45 yards of machine working). The Cost by Hand. 30 men cutting, filling, timbering, drilling, road- All cut on the end. laying, blasting, and all other needful wor'i ready in the corves for the" hurricr" at 4s. 5|rf. £ s. per ton 13 8 9 By Machine. machine man at 8s. 6d youth at 5s. Gd. ] , i . i x boy at 3s. 6^^. | (^'i^'^^ *« ^ "^«'^) men cleaning and packing at 8s. 4c7. 6 men filling 10 tons each man, at 8^d. per ton 3 men timbering at 6s. lOd. ........ 3 men drilling and blowing down at 6s. lOd g portion of cost of steam and air ex- penses Maintenance at Id. per ton Eedemption of capital at 2d. per ton .... s. 8 5 3 5 1 d. 6 6 6 3 6 6 1 14 .5 10 8 13 9 Difference, in money, in favour of the machine, or Is. 7d. per ton 4 15 £13 8 9 The two boys, it wUl be noticed, are taken as equal to one man ; and for the purpose of another comparison, I wiU assume that by hand labour thirty men will produce 60 tons per day, or two tons each, and that by machine seventeen men will produce the same tonnage. The saving in number, therefore, would be twelve men to every 60 tons, or upon a colliery getting 4000 tons per week, the saving would be 132 men. I do not wish to press this point further than to say that the cost of dwellings properly to domicile one half of this number would exceed the first outlay of capital in furnishing a first-class colliery with first-class machinery for cutting the coal ; and it must not be forgotten that the equipment of the hand-cutters in tools forms a considerable item in the first cost of fitting up a colliery. It has been generally supposed that our machines are not adapted for *' pillar and stall work." That their locomotion " is not so easy as that of men," must of course be 1H4! RKPORT — 1873. admitted ; but they are removed from place to place with little more trouble than a full corve ; and we have recently made some careful experiments, which prove that there is in " pillar and staU " about equal advantage as in " long wall;" and we can confidently assert that the opinions upon the difficulty of moving them which have been recently enunciated from high quarters are quite erroneous. The items of cost in working contained in the previous account, are con- fined to the actual working of the two systems, up to the coal being put into the corves, and ready for being sent out of the pit, all the other work, whether for hand or machine, being exactly alike. But there are some advantages in the machine over the hand-working, which pertain to the general mine account, viz. the larger size of the coal brought out, and an increased average price, on sale, with a saving in timber and other stores. I may say in conclusion, that, putting aside entirely all reduction in the cost of getting out the coal, there are other and collateral considera- tions which are, in my opinion, sufficiently important and worthy of your attention, I now recapitulate the most prominent points upon which I rely, viz. : — 1. Greater safety for the workmen from falls of coal and roof. 2. Less danger of explosion, and greater security against the effect of choke damp. 3. Less strain upon the physical powers of the labourers, and great amelioration in the hard conditions of their employment, conse- quently adding to the comfort and length of their lives. 4. Saving from destruction much of the most valuable of all our com- modities. 5. Saving of timber and other materials employed in mining. 6. Increased control over production, enabling sudden demands to be suddenly met. 7. Preparing for other important improvements in mining, without any addition to the first outlay, such as drilling, hauling, and pumping. 8. The peculiar adaptability of the means set forth for working the very deep seams of coal, without which it is very doubtful whether they can ever be profitably worked. 9. Greater saving of time in opening new pits, and quickening the means of such becoming remunerative. Considering the vast extent of the trade in coal and the stupendous con- sequences of a short and insufficient supply, and believing that the speediest adoption of coal-getting machinery is desirable, I have myself made some efforts to stimulate that object by an ofi'er of a premium of £500 for the best machine that could be produced ; but those efforts have failed, and I now submit that the question, being of national importance, is one specially entitled to the support and encouragement of the Government, and that the British Association is preeminently the channel through which that object could be obtained in the best manner. ON MALTESE FOSSIL ELEPHANTS. 185 Concluding Report on the Maltese Fossil Elephants. By A. Leith Adams, M.B., F.R.S., F.G.S. It is with mucli pleasure I have to announce to the members of the Asso- ciation that my labours in connexion with the fossil elephants of Malta havo been completed. It is now thirteen years since these researches were begun ; and although frequently interrupted by other engagements, the importance of the subject has all along stimulated me to make every sacrifice within my power in order to accomplish a work of so much scientific interest. The monograph descriptive of the elephantine remains discovered by me was read at the con- cluding meeting of the Zoological Society of London in June last, and will appear in due course in the Transactions of the Society. It is illustrated by a mapand 21 Quarto plates. In mySecondReportinl866, drawn up immediately after the termination of my explorations, I was dis- posed towards an opinion that the exuviae I had brought together represented only one form of Elephant, distinct from any known member of the genus, and somewhat under the ordinary dimensions of the living species. Subse- quent examinations, however, showed, in addition, that there were good indications of the presence of the two dwarf elephants previously determined by Dr. Falconer and Mr. Busk, from the collection made by Capt. Spratt in the Zebbug Cave in Malta in the year 1859. Ist. With reference to the largest species. This is represented in my col- lections by nearly the entire dentition and many bones of an elephant which varied in height between 6| and 7 feet. The last figure, however, represents the maximum proportions as far as I have been enabled to determine from my own specimens and from all other remains hitherto discovered in the island. It is apparent, therefore, that the largest Maltese fossil elephant was, comparatively speaking, a small animal. The dental specimens I have assigned to this species are very numerous, and for the most part perfect. They represent every stage of growth, from the first to the last, showing what appears to me an unbroken series of molars which display the progressive succession of ridges characteristic of the subgenus Loxodon, and are therefore allied to the existing African elephant, from which, however, they differ not only in relative dimensions, but also in well-marked specific characters. The ridge-formulae of the deciduous and true molars of this species seem to me to stand thus * : — Milk-Molars. True Molars. X 3 x: X 6 X : X 8-9 X : : x 8-9 x : x 10 x : x 12-13 x. From these figures it will be apparent that the nearest alliance as regards the ridge-formula would be to the gigantic Loxodon meridionalis, whilst the crown sculpturing of the molars resemble the same in Elephas antiquus ; but they do not agree in further particulars with other species excepting the Elephas melitensis, to which I will refer presently. With reference to the skeleton generally, the majority of the characters of the long bones are more in keeping with the African than the Asiatic elephant. The presence of this larger species of elephant, in conjunction with the dwarf forms, was pointed out by Dr. Falconer, and subsequently by Mr. Busk ; but their specimens were much too fragmentary to allow of specific determination, a want, however, which is amply supplied by the materials collected by me. * X stands for talons. 186 HEPORT— 1873. In the choice of a name for this proboscidian I have been prompted by considerations purely incidental, inasmuch as the gap or rock-fissure from which I obtained the most perfect specimens of its teeth and bones is situ- ated in the immediate vicinity of a remarkable megalithic structure supposed to have been built during the Phoenician occupation of the Maltese Islands. I have accordingly named this new species the Eleplias mnaidriensis. 2nd. The dwarf species named Elej;>has melitensis by Falconer and Busk is well shown in my collection by many important bones, besides what appears to me to be the entire dental series. This species seems to have varied con- siderably in size ; indeed it would appear to link the two extremes represented by the Elephas mnaidriensis and the smallest form, Elephas Falconer i. The majority of the bones indicate, however, that its average height may have been nearly 5 feet, as previously estimated by Dr. Falconer and Mr. Busk, from the Zebbug collection. The dentition of Elejjhas melitensis, as determined by Falconer, receives ample confirmation from the data furnished by my collections, the ridge formula being : — Milk Molars. True Molars. X 3 X : X 5 X : X 8 X. x 8-9 x : x 9-10 x : x 12 x. The only discrepancy between our estimates is an additional ridge in the penultimate true molar of my specimens, which it may be observed is not a rare occurrence in the equivalent tooth of the African elephant. It is clear therefore that, like the larger form, the above belonged to the Loxodon group, with a ridge-formula almost identical to that of E. mnaidriensis, ex- cepting in the penultimate milk-molar, which in the former holds 5 instead of 6 plates, besides talons — a distinction maintained in various specimens in my collection. The crown-patterns of worn molars in the two elephants are also very much alike ; but the relative dimensions of teeth of equivalent stages of growth differ a great deal, indeed more so than perhaps in large and small indivi- duals of any known species. Again, we find thick- and thin-plated varieties among the last true molars of both forms, just as obtains in other species ; so that, taken in conjunction with the bones, it seems to me that they cannot be reconciled with sexual or individual peculiarities of one species of elephant. 3rd. The smallest adult bones in my collection represent a very diminu- tive elephant. In some instances, as compared with other species, there are evidences of individuals even under 3 feet in height. With reference to dental materials, there is some variety in dimensions of molars ascribable to the Elephas melitensis ; but, allowing a fair margin in this respect, and taking into consideration their absolute similarity in every other particular, it seems to me impossible to make out a third species from the teeth alone. There are, however, vertebrae and other bones which fairly establish the pigmy proportions of the Elephas Falconeri of Busk ; at the same time there is no difficulty in arranging a graduated series of specimens, from the smallest up to the largest bones ascribable to the Elephas melitensis. But whilst the differences in size between the two dwarf forms are not so great as usually obtains between large and small individuals of living species, there is a remarkable dissimilarity in this respect between the lai'gest specimens representing the Elephas mnaidriensis and the smallest of Elephas Falconeri ; indeed the estimated height of the former shows an elephant nearly three times as tall as the latter, thus displaying a range much exceeding any known instances of individual variation among recent and extinct species. ON MALTESE FOSSIL ELEPHANTS. 187 I am thus particular to record these facts in order to show what appears to me evidence that the dwarf forms were not females or small individuals of Elejjhas mnaidriensis, although the latter was, comparatively speaking, a small species, and agreed, at all events, with Elephas melitensis in many important particulars. Unless, therefore, a far greater variability of species existed in those times than at present, after making every allowance for size and other characters, I see no avoiding the inference the materials force on us, viz. that there lived in the Maltese area two, if not three, distinct species of elephants diiferent from any known forms. It is necessary to say a few words with reference to their associated fossil fauna. In the first place, all the elephantine forms have been found in the same deposits, and usually in- termingled. Along with them we find bones and teeth referable to the Hi])- popotamus Pentlandi and H. minutus. The former has been met with in great abundance in the island, whilst only a few teeth and other portions of the skeleton of the latter have turned iip. Here again we observe a great varia- bility in dimensions ; indeed in this respect these two riverhorses resemble the large and pigmy forms of the elephants ; and although the former have been found in a fossil state in Sicily and Crete in conjunction with other mammals, this is not the case with the giant dormice and large extinct swan, which have hitherto tvirned up nowhere out of Malta. I may state that the Reptilian remains found by Admiral Spratt and myself in union with these quadrupeds and birds have not, as a whole, been critically examined ; but, in consideration of the importance of the subject, 1 am in hopes of seeing this accomplished soon. The moUusca found in connexion with foregoing represent several recent species, which have been already noticed in my first Eeport for 1865. It must be apparent, therefore, that this (for the most part) unique fossil fauna, restricted to a small mid-ocean island, presents several interesting contrasts with reference to the Mammalia in general, and elephants in par- ticular, which frequented Europe during late geological epochs. Eor example, between Eome and Sicily we find remains of the Elephas jjrimigenius, Elephas antiqims, and Elephas meridionalis. In the caves of Sicily traces of the African elephant have been discovered, and also molars, barely distinguish- able from those of the Asiatic species, and which, under the name of Elephas armeniacus, are traceable eastward into Asia Minor, in the direction of the present habitat of the living species. It looks, indeed, as if the eastern basin of the Mediterranean had been at one time a common ground where aU these extinct and living elephants met, and whence, with other animals, they have disappeared or been repelled to distant regions. In fine the importance of late discoveries in this area, and the circumstance that the explorations have been hitherto restricted to isolated points along the shores and islands of the great inland sea, promise weU for future re- searches ; indeed I might be permitted to say that if one quarter of the super- fluous zeal and energy of the rising generation of English geologists were directed towards the ossiferous deposits of Southern Europe and JSTorthern Africa, we should not have long to wait for novelties equally interesting with any yet produced. In conclusion, I beg once more to express my deep obligations to the British Association for the valued assistance extended to me not only during the prosecution of the explorations, biit also with reference to the illustration of the various and interesting materials I have described at length in my memoir, of which this is but a brief abstract. 188 REPORT — 1873. Report of the Committee, consisting of Professor Ramsay, Professor Geikie, Professor J. Young, Professor Nicol, Dr. Bryce, Dr. Arthur Mitchell, Professor Hull, Sir B,. Griffith, Bart., Dr. King, Professor Harkness, Mr. Prestwich, Mr. Hughes, Rev. H. W. Crosskey, Mr. W. Jolly, Mr. D. Milne-Holme, and Mr. Pengelly, appointed for the purpose of ascertaining the existence in different parts of the United Kingdom of any Erratic Blocks or Boulders, of indicating on Maps their position and height above the sea, as also of ascertaining the nature of the rocks composing these blocks, their size, shape, and other particulars of interest, and of endeavouring to prevent the destruction of such blocks as in the opi- nion of the Committee are worthy of being preserved. Drawn up by the Rev. H. W. Crosskey, Secretary. The Royal Society of Edinburgh has appointed a Committee for the special examination and description of Boulder or Erratic Blocks in Scotland ; and it wiU therefore not be necessary for this Committee to include Scotland in its investigations. Throughout England and Wales boulders and groups of boulders are scattered, among which the work of destruction is constantly going on. Groups of boulders are removed from the fields and built into waUs ; krge boulders are frequently blasted ; and during these operations the signs of ice- action are either rendered obscure or entirely removed. The geological importance, however, of obtaining the exact facts respecting the distribution of travelled boulders is increasing with an extended knowledge of the very complicated character of the phenomena of the glacial epoch. The dispersion of boulders cannot be traced to one sbvjJe period of that great epoch. Prof. Ramsay has pointed out that transported blocks have travelled in some instances over land higher than the parent beds from which they have been derived, thus affording support to the theory that oscillations of the land took place during the one great glacial period, which would neces- sai'ily be accompanied by a scries of dispersions of boulders*. The distances of the boulders from the rocks from which they were de- rived, the heights over which they have passed and at which they are found, the matrix (if any) in which they are imbedded, whether of loose sand, gravel, or clay, will form elements in determining at what period in the gla- cial epoch their distribution took place. As the dispersion of boulders cannot be traced to one single period, neither can it be referred to one single cause. The agency of land-ice, the direction in which icebergs would float during the depression of the land, the power of rivers in flood to bring down masses of floating ice, must be taken into account. It will not be the office of this Committee to oifer theoretical explanations, but to coUect facts, although the bearing of these facts upon debatable geo- logical problems may from time to time be not unjustly indicated. WTiile the dispersion of boulders can neither be traced to one single period nor referred to one single cause, in some cases boulders distributed at different periods and by different causes may have become intermixed. This possibility, of course, largely adds to the complexity of the problems in- volved, and to the difficulty of assigning to various isolated boulders and groups of boulders their definite place in a great series of phenomena. The following circular has been distributed by the Boulder Committee of the Royal Society of Edinburgh : — * Quart. Jouri). Gcol. Soc. toI. xiix. p. 360. ON ICRRATIC BLOCKS OH BOULDERS. 189 '" If there are in t/cmr Parish any Eruatic Blocks or Boulders, — i. e. Masses of Rock evidenfly transported from some remote locality, and of a remarkable size, say containing above 10 citbio yards — i. e. about 20 totts, — please to answer the following Queries : — ' Queries. 1. What is name of the Parish, Estate, and Farm on which Boulder is situated, adding name of Proprietor of Estate, and Tenant of Farm? 2. What are dimensions of Boidder, in length, breadth, and height, abore ground ? 3. Is the Boulder, in shape, rounded or "1 angular ? J 4. If the Boulder is long-shaped, what is direction by compass of its longest axis? Answers. 5. If there are any natural ruts, groovings, "> or striations on Boulder, state — (1) Their length, depth, and number (2) Their direction by compass (3) The part of Boulder striated, viz. whether top or sides 6. If the Boulder is of a species of rock differing from any rocks adjoining it, state locality where rock of the same nature as the Boulder occurs, the dis- tance of that locality, and its bearings by compass from the Boulder ? 7. What is the nature of the rock com- posing Boulder, giving its proper Geo- logical or Mineralogical name, or other description ? 8. If Boulder is known by any popular name, or has any legend connected with it, mention it. 9. What is the height of Boulder above the sea ? 10. If Boulder is indicated on any map, 1 state what map. j 11. If Boulder is now, or has been, used to mark the boundary of a County, Parish, or Estate, explain what boundary. 12. If there is any photograph or sketch of the Boulder, please to say how Com- mittee can obtain it. 13. Though there may be no one Boulder "j in your Parish so remarkable as to deserve description, there may be groups of Boulders oddly assorted ; if so, state where they are situated, and how grouped. Sometimes they form lines more or less continuous, — some- times piled up on one another. 14. If there are in your Parish any " Kanies," l or long ridges of gravel or sand, state j- their length, height, and situation. J 190 REPORT— 1873. It is proposed by the Committee to issue a similar circular, with some modifications, to Secretaries of Field-clubs and local Geological Societies in England and Wales, and others who may be willing to assist in their work. The Committee would especially in^-ite the cooperation of the various field-clubs of England and Wales, whose members, in their various excur- sions, enjoy singular opportunities of becoming acquainted with the boulders of the country. Chaenwood-Forest Bouldeks. The railway-cutting at Hugglescote, approaching Bardon Hill, passes through an immense number of striated and polished boulders. Mr. Plant, of Leicester (who has investigated the boulders of this district, and furnished us with considerable information), describes this cutting at Hugglescote as 30 feet deep. The diift-gravel is a hard cemented mass, with hundreds of erratics, at all heights, sticking not on their longer faces, but sometimes on end, distinctly proving that the ice melted in situ, and left the materials to find their own bearings. One, of which he saw the fragments, had to be blasted to get it out, and was estimated by the engineer to weigh 10 tons. All the boulders (except one, a peculiar millstone-grit) were derived from the Charnwood-Forest range, the most travelled from a distance of 30 miles, the nearest about 2 miles. Some of the boulders were upwards of 5 tons in weight, and were striated and polished frequently on more than one side. Many were angular and subangular. They were very irregularly dispersed through an unstratified matrix of sand and clay. The whole distance from the vast accumulation in the cutting to Bardon Hill, the nearest point of Charnwood, a distance of about 2 miles, is covered with trails of boulders. The jagged edges of the Bardon-Hill rock, 854 feet above the sea-level, indicate the way in which boulders would be broken off, supposing the hill itself covered with ice. During some part of the glacial epoch Charnwood Forest was evidently a centre from which highly glaciated boulders were distributed. Mr. Plant reports that a great south front of igneous rock has been broken down and distributed, east, south, and south-west, 10, 15, and 20 miles, in direct lines. An area of 10 miles N.N.W. and 20 miles S.S.E. and S.W., is covered with boulders derived from Charnwood Forest, fi-om 2 cwt. up to 10 tons. Centuries of cultivation (he adds) have been occupied more or less in clear- ing the surface of these boulders. They are still found in great numbers, 2 to 3 feet deep ; but the surface-boulders are found in the walls of village houses, churches, farm-houses, and other old structures, all over the county. Four large blocks from the railway-cutting at Hugglescote have been removed, and placed in the grounds of the Leicester Museum. One of these is a fine example of a polished rock, and is full of ice-grooves. Its dimen- sions are : — 6 ft. high, 3 ft. 2 in. broad (or thick), 3 ft. wide ; weight nearly 4 tons. It consists of "porphyritic greenstone" from Charnwood Forest, grey felspathic base (dolerite), with crystals (| to f on face) of quartz. Through long chemical action in the drift the felspar has been decomposed, aud left the crystals standing out all over the surface, except on the polished side. The other three blocks are neai'lj- of the same size and composition. It is intended to remove other blocks to the museum-grounds for preser- vation. ON ERRATIC BLOCKS OR BOULDERS. 191 Charnwood Forest and other Boidders, beneath marine sands and gravels, 357 feet above the sea. At the base of Ketley gravel-pit, near "Wellington (Shropshire), is a bed of very fine sand, containing a remarkable group of large angular and sub- angular boulders. The sands and gravels extend to heights of from 25 to 30 feet, and yielded 13 species of raollusca, chiefly in fragments. Dentalium ? (very worn). Turritella terebra, Linn. Natica eroenlandica, Beck. Cardium edule, Linn. ■ echinatum, Litin. Cyprina islandica, Linn. Astai-te borealis, Chemnitz. sulcata, Ba Costa. Tellina balthica, Linn. Mactra solida, Li7in. Buccinum undatum, Linn. Trophon truncatus, Strom. Nassa reticulata, Linn. It wiU be observed that only one of these species is extinct in British waters, viz. Astarte borealis. Throughout the sands and gravels waterworn pebbles are found, with occasional masses of larger size, composed of the same material as the larger boulders beneath. Beneath the marine sands and gravels some of the boulders are 8 feet by 5 feet, and theii- sides are planed very smoothly, and they have a subangular shape. Out of 100 specimens, 80 per cent, consist of Permian sandstones from the immediate neighbourhood. From the immediate neighbourhood also there are boulders of Mountain Limestone. I Silurian Limestone. Old Eed Sandstone. | Greenstone. The travelled boulders consist of Various granites, both red and grey (very numerous), probably from Cumberland or Scotland. * Eocks of Charnwood Forest, from a distance of 50 miles. One remarkable feature of this group of boulders is the intermixture of boulders from the neighbourhood with those that have travelled from different points of the compass, the whole group being buried beneath marine sands and gravels, at the elevation of about 300 feet above the sea.. The elevation of Ketley village is 357-319 feet above the sea. For the boulders of the neighbouring drift of the Severn valleys reference may be made to an exhaustive paper by Mr. G. Maw (Quart. Journ. Geol. Soc. vol. XX. p. 130). The Geological Section of the Birmingham Natiu-al-History Society has commenced a systematic examination of the boulders of the Midland district, and has favoured the Committee with the following preliminary lleport : — " The Ordnance Map of the neighbourhood of Birmingham has in the first place been divided by ruled lines into squares of one inch side, each square enclosing a representation of one square mile of country. Enlarged maps, on the scale of six inches to the mile, were prepared from this ; and on these enlarged maps the boulders were to be marked by circles, the number of concentric circles representing the diameter of the boulder in feet. For col- lecting specimens of the rocks of which the boulders are composed, bags were made, and numbered corresponding to each square on the map ; at the same time notes were to be made of any specimen that was of unusual interest. errs, '4i^ f^^^Y^ V, J^M A^ f^<JZ /> l^f' 192 REPORT— 1873. Finally, it was proposed to represent, on a duplicate map, the number, of boulders and character of the rocks by disks of colour, so that a graphic re- presentation of the boulders, as to position, numbers, and kind of rock, would be given, and the source of any class of boulders (as granite e. g.) could be readily traced. It was further proposed to number a rough rehef-map of the district, so as to judge in what way the configuration of the country had affected the distribution of the boulders. " Considerable information has been already obtained, of which the follow- ing is a summary : — ■ •' A difficulty was experienced in defining the term boulder ; and, after much discussion, it was thought that for the district the following definition would serve : — ' A boulder is a mass of rock which has been transported by natural agencies from its native bed.' Respecting the size at which a rock may be called a boulder, it is thought better not to assign any very definite limit. Some specimens, measuiing not more than a foot in some one direc- tion, are both transported from great distances and glaciated, and fairly fall into the categorj' of boulders. " Distribution of the Boulders. — The district has not as yet been sufficiently examined to report fully on this question. There are unquestionably some places where great accumulations have taken place, separated by country with only a few boulders per square mile. The places where large accumu- lations (a thousand or so) occur, as far as has yet been ascertained, are : — 1. Tettenhall. 2. Bushbury. 3. Cannock. Places where moderate accumulations (50 to 100 or 200 per square mile) occur : — Penkridge. Shareshill. Brewood. Codsall. Stone. ShifnaU. Harborne, near Birmingham. Bridgenorth. " The southernmost point where boulders have been observed is on the left of the lane leading from Bromsgrove Station to the town, the most eastern at Bugeley, where only two or three occur. " It has been suggested that the cause of accumulations of boulders is due to the stranding of an iceberg at the place in question ; but at present there is not sufficient evidence to form any satisfactory opinion as to the cause of the accumulation. " The boulders of the Midland district seem originally to have been im- bedded either in clay or drift-sand ; but it is quite the exception to find them in situ. They seem commonly to be disturbed bj- farmers in the district, who meet with them when ploughing. If the boulder be of manageable size, it is at once dug up and turned into the nearest ditch, or sometimes is buried, or, it may be, carried to the road-side, and broken up for road-purposes. Farmers find some of the boulders useful as horse-blocks, or for protecting gate-posts or the corners of v/aUs and buildings ; and it is thus that many are preserved. If the boulder be a very large one, it is generally left in the ground, and the plough carried on each side of it. Since a plough may pass over a boulder several times before the men will take the trouble to remove the obstruction, there is eveiy chance for the boulder to become marked by striations ; and hence much care is required in forming a judgment as to the origin of striae which may be found upon it. It should be mentioned here that boulders gradually 'work up' to the surface. This is due no doubt to ON ERRATIC CLOCKS OR BOULDERS. 193 the denudation which is taking place. In a field near Red-Hill Farm, be- tween btafFord and Stone, is one of the largest boidders of the district. This boulder was not noticed until some twenty years ago, when it was found to obstruct the plough, although still some depth underground. The obstruc- tion became more and more serious each year, until a few years ago, when, because of this impediment, the field was turned from an arable to a grazing one. At this time the boulder rises about one foot above the level of the field. The part exposed measures 6 feet by about 5, and evidently extends under the turf for a much greater distance. This boulder is composed of the grey granite of which so many other boulders in the neighbourhood consist. " The boulders consist mainly of white granite and of felstone ; but many other rocks occur, as may be seen by inspecting the specimens collected. In the neighbourhood of Tettenhall there is a large percentage of granite boul- ders ; but south of here there are very few indeed, the boulders being mainly of felstone. In the Harborne district only one granite boulder has been observed, while there are a hundred or so boulders of other rocks. The contrast between the immense accumulation of granite boulders in the Wolverhampton district and their comparatively small size and rarity around Birmingham is most remarkable." Granite Boulder on the shore of Barnstaple Bay, North Devon, Mr. Pengelly reports the following particulars respecting this boulder, upon which the raised beach on the northern side of Barnstaple Bay rests. So far as it is visible, it measures 7'5 x 6 x 3 ft., and therefore, containing upwards of 135 cubic feet, cannot weigh less than 10 tons. It appeal's to have been first described by the late Eev. D. Williams, in 1837, as " flesh-coloured, like much of the Grampian granite " and, in his opinion, " neither Lundy, Dartmoor, nor Cornish granite." In 1866 Mr. Spence Bate, believing that very similar granite existed in Cornwall, expressed the opinion that it was not necessary to go so far as Aberdeen, but that some transporting power must have been required to bring it even from the nearest granite district, and that it without doubt occupied its present position before the deposition of the beach resting upon it. Recently Mr. Pengelly has been informed that red granite occurs on Dartmoor, and therefore has no disinclination to say, with Mr. Bate, that we need not go as far as Aberdeen to find the source of the boulder, although it nevertheless may have come from the Grampians. Assuming that the block may have come from Lundy, twenty miles towards the west, or down the valley of the Torridge from the nearest point of Dart- moor, thirty miles off as the crow flies, its transport in either case must have been due to more powerful agencies than any now in operation in the same district. Between Barnstaple Bay and Lundy there are upwards of 20 fathoms of water, a depth at which no wave that ever entered the Bristol Channel would probably ever move the finest sand. Again, as the highest part of Dartmoor is but 2050 ft. above mean tide, a straight line from it to where the boulder now Hes would have a fall of 1 in 77 only, down which the Dartmoor floods would certainly not transport a rock upwards of 10 tons in weight. The foregoing considerations apply, of course, with at least equal force to the hypothesis of any more distant derivation. That such a block might have been brought from Dartmoor down the Tor- 1873. 194 REPORT— 1873. ridge to the place it now occupies, had the actual heights been the same as now and the climate as cold as that of Canada at present, wiU be ob- vious to every one conversant with that country. It is only necessary to suppose that the block fell from a cliff into a stream where the water was at least sometimes of sufficient depth that when frozen round the mass the latter would be lifted by the buoyancy of the ice. On the breaking up of the ice the floods would transport the rock so long and so far as its ice-buoy was capable of supporting it ; and though the distance accomplished in a single journey might, and probably would, be inconsiderable, by a repetition of the process season after season it would become equal to any assigned amount. Blocks of great size have been in this way transported in Canadian rivers for 100 miles or more. Again, were Lundy Island capable of generating a glacier and launching it into the sea as an iceberg, there would be no diffi- culty in supposing that any number of boulders might be transported thence to the mainland of Devon. In short, whether the boulder came from Dartmoor or Lundy or any more distant source, it must have been transported by ice-action ; and hence its presence where it now lies is good evidence of a cUmate in this country much colder than that which at present obtains. From the foregoing considerations it will be seen that, if the mass were ice-borne, the land could not have been higher above the sea during the era of the boulder than it is at present. There is nothing, however, to prevent its being lower. The boulder may have been dropped by an iceberg on or near the spot it now occiipies when that spot was covered with deep water. The only stipulation to be made on this point is, that the land which fuirnished the mass was capable of supplying it with an ice-body. For example, if the boulder was derived from Dartmoor, Devonshire as a whole could not have been any thing like 2050ft. lower than at present; for that would have been to submerge the entire country, whereas there must have been subacrial land sufficient to form the ice-raft whose buoyancy floated the boulder. It is hoped that the steps proposed to be taken by the Committee will enable the boulders of one or two districts at least to be systematically mapped, and the existence of other such remarkable boulders as the granite boulder on the shore of Barnstaple Bay to be recorded. Any attempt at systematic classification, however, must necessarily be deferred until the facts are more largely accumulated. Fourth Report on Earthquakes in Scotland, drawn up by Dr. Bryce, ■F.G.S. The Committee consists of Dr. Bryce, F.G.S., Sir W. Thomson, F.R.S., Geo. Forbes, F.R.S.E., and Mr. J. Brough. The conjecture hazarded in last Report, that " the state of quiescence " therein referred to was " not likely to continue," received a speedy fulfil- ment. In a postscript to the Report, which was not, however, forwarded in time to be read at the Meeting, it was noticed that " while the Associ- ation was in Session at Brighton an earthquake of considerable severity " had " occurred in the Comrie district ;" and in April of the present year another ON-EATHQUAKES IN SCOTLAND. 195 took place in the south of Scotland. Of these an account has now to be given. — A few days after the occurrence of the earthquake, the Member of Committee resident at Comrie communicated with me ; and having seen in the newsjiapers notices of other places where the earthquake had been felt, I entered into correspondence with gentlemen in the various districts. In the end of September I visited several of these districts, and made inquiries in person. From the facts thus made known to me the following account has been drawn up ; but before proceeding with it, I have to express my obligations to the following gentlemen for the kind manner in which they complied with my request, and communicated at once all the observa- tions made by themselves, and facts collected from others on whom they could depend : — Dr. Campbell and Eev. James Muir, Bridge of Allan ; Eev. "William Blair, Dunblane ; Mr. J. Stirling Home-Drummond, of Ardoch, Braco ; Dr. William Bryce and Mr. David Cousin, both from Edinburgh, the former happening to be at Crieff at the time, and the latter at Bridge of Allan ; Mr. P. Macfarlane and Mr. J. Brough, Comrie ; Sir David Dundas, of Dunira, Comrie ; and llev. J. E. H. Thomson, B.D., Blair Logie. Dr. Campbell's evidence is especially valuable, as he resided for some time in Upper Strathearn, where earthquakes are of frequent occurrence and were often experienced by him, and as he is in the constant practice of accurate every-day observations of meteorological instruments for a register kept by him at the usual hours. Mr. Macfarlane and Mr. Brough at Comrie possess, of course, like advantages. Mr. Cousin also had the advantage of previous experience in observations of this kind, an earthquake having occurred while he was resident in Algeria. A similar advantage was enjoyed by the Eev. J. E. H. Thomson ; at the instant when the shock occurred he was in con- versation in his own house with two ladies, one of whom had resided for some years in Valparaiso, where earthquakes are of very common occurrence, as is well known. The earthquake took place on the 8th of August, 1872, at from 8™ to 10" past 4 o'clock in the afternoon. The day was warm and perfectly still. In the early part of the day there had been alternations of a cloudy and clear sky ; but at the hour mentioned only the western part of the horizon showed cloudy masses, the sky overhead and eastwards was free from cloud of any kind. The barometer rose slightly during the day, from 29-800 at 10 a.m. to 29-975 at 10 p.m. The maximum temperature of the day, in the shade, was 64°-3 F. ; the minimum temperature of the night preceding was 53°-8 F., of the night following 51° F. No perceptible change in the temperature or character of the atmosphere as to wind and cloud took place after the shock. The successive phases, according to almost all the observers, were : — a noise or sound, loud, heavy and rumbling; a shock with a shaking and rattling of objects ; and a wave-like motion of the ground. The noise or sound is compared to the sound of thunder, to that made by a, heavy waggon on a stony street, to the emptying of a cart of small stones or rubbish, to the noise one hears when under a bridge over which a heavy train is pass- ing. Many who were within doors supposed that a heavj^ piece of furniture had fallen on the floor of an adjoining room. A clergyman was standing on the hearthrug in his study, and, hearing a sudden noise or crash, imagined a chimney-stack was falling, and rushed instantly into a position of safety. Finding this surmise incorrect, he referred the noise to the faU of a ward- robe in the next room. This surmise also proving incorrect, he went imme- diately down stairs and found his servants panic-stricken. In the nursery the nurse had i-ushcd to the window and screamed in alarm to her mistress, o2 196 REPORT— 1873. who was in the garden. So strong, indeed, and concurrent is the evidence on this point that no doubt can remain about it, in regard to almost all the localities from which communications have been received ; the slight discre- pancy among the witnesses to the fact may be accounted for by some of them being resident on a soil composed of soft alluvium, and others upon a rocky surface. Some of the witnesses notice that the sound was instanta- neously repeated with even greater violence. The shock instantly followed the noise or sound ; and its occurrence was marked in many ways : houses were shaken, doors and windows made to rattle, suspended objects to oscillate ; in one house bells were set a-ringing with violence, in another they were strongly agitated ; jugs, basins, and water-glasses in bedrooms, apothecaries' bottles, phials, and pots, the glasses in the pump-room at Bridge of AUan Spa were heard to knock against one another and seen to move ; a chimney-mirror, loosely fastened, was thrown down ; and chimney ornaments were dashed upon the floor. Next succeeded that most appalling of all the attendant circumstances of an earthquake, the sensation as of a heaving impulse or wave, giving the idea of a crest and declivity, instantly followed by a double vibration, the whole duration being from three to four seconds. The statements of the ob- servers (as made known in the various reports) on whom one feels that most reliance is to be placed from their previous expei'ience, habits of close ob- servation, and the circumstances in which they were placed at the time, all go to show that the undulation came from a direction W. or N.W., some observers making the direction exactly opposite by not distinguishing the first impulse from the recoil or restoration of the wave-surface. One ob- server, on whom the utmost reliance can be placed, had the most distinct feeling of vertigo or dizziness arising from the undulation, a sensation so strong that a few moments' continuance of it would have produced nausea — a strong testimony to the reahty of the wave-motion. The extent of country throughout which this earthquake was felt is greater than that of any which has occurred since this inquiry was under- taken. The limits are marked by Stirling and Blair Logic on the S.E., and St. Fillans on Loch Earn and Glen Lednock on the N.W. The shock was feebler at these limits than in the parts intermediate, as Bridge of Allan, Dunblane, Greenloaning, Ardoch, and Crieff. In regard to the breadth of country agitated, I have been unable to determine that it extended more than two or three miles from the vaUey of the Allan Water, the concussions recorded being greater to the east of that valley than in the opposite direc- tion, while in the village of Doune, four miles west, they do not seem to have been noticed. The want of self-recording instruments, the extreme difficulty of determining the exact instant of the occurrence of an event so sudden and startling, render it impossible to attempt any definite statement as to the progress of the wave, which, so far as instrumental indication can serve us, seems to have emanated from near Comrie. All the observers who have attempted to specify an exact time have, to aU appearance quite inde- pendently, agreed that it was, as above stated, at 10" past 4'' p.m. Persons trained to observe, or self-recording instruments, alone can furnish reliable data in such a case for indicating the time occupied in the undulation pass- ing from point to point. The intensity upon the Comrie scale, which ranges from 1 to 10, was of a medium force, about 4. The geological formation of the tract of country embraced wdthin the above limits varies greatly. The lower part of the village of Bridge of Allan is situated upon the alluvium of the I'orth valley, in which, as far up from ON EARTHQUAKES IN SCOTLAND. 197 the present channel of the river as the streets of the lower part of the village, skeletons of whales have been found. The upper or northern part of the village stands upon a high terrace of Old Eed Sandstone, traversed by whin dykes, alongside one of which its famous Spa is discharged. The front of this terrace runs east and west, and forms the former sea-margin, hewed out by the waves of the old estuary, against which the alluvium rests to an unknown depth. The town of Stirhng stands upon the south side of the Forth valley, partly on alluvium and partly on a trap ridge erupted through Old Eed Sandstone. Eastwards from Bridge of Allan by Blair Logic and Dollar, the Ochill HiUs, of which the terrace at Bridge of Allan is the first ridge or step, are composed of the same Old Sandstone, broken through and overlain by a vast body of trap rocks, clay-stones, and porphyries, and pre- sent a waU-like front to the Forth valley on the south. They completely cut off the Coal-measures, tilting up the strata at a high angle, altering the coal to the state of coke, shale to Lydian stone, and sandstone to quartzite. Dunblane, Greenloaning and Ardoch, and the wild moorlands N.W. to Crieff' are composed of Old Red Sandstone pervaded by traps ; and Crieff and Comrie are close upon the junction of the sandstone and old slates of the mountain-region. Glen Lednock and a large area E. of it towards Crieff are occupied by an eruptive granite which sends veins into the slate, and whose outer edge approaches close to the boundary of the slate and sandstone. Whatever the cause of this earthquake may have been — masses of rock fall- ing from the roof of a vast cavern, or a sudden impact of high-pressure steam emanating from the nether depths — all the strata were affected by it, and sent the awful tremor, yet with varying intensity, alike along beds of rock and alluvial strata. The particulars in regard to the earthquake in the south of Scotland have been kindly supplied by Dr. Grierson and Mr. Henrison, Thornhill, Dum- fries, and Mr. J. Shaw, Tyrnon parish. The earthquake took place on the 16th of April, 1873, at 9^ 55"" p.m. A smart concussion, producing a con- siderable sound, noise, or crash, as it is variously described, and causing a perceptible movement in fixed objects, and an oscillation of those suspended, was experienced in the parishes of Tyrnon, Glen Cairn, Keir, Penpont, Mor- ton, Closeburn, and Balmaclelland. Doors and windows were made to rattle ; there was a sensible vibration of walls and floors in many places; and objects near one another (as glasses and china on shelves) wore knocked together. In some cases alarm was shown by the lower animals. But the wave or undulation was not observed with any thing like precision, except in one case, in which a floor was distinctly seen to have such a movement. The late hour, however, was unfavourable for observation on the part of many persons. One only of the observers whose accounts have been fur- nished to me had any previous experience of earthquakes. This gentleman had resided for some time in the East. Another witness, in every way com- petent, experienced a repetition of the shock at Thornhill at S** 46" a.m. on the following morning ; but no information regarding this second shock has reached me from any other part of the district. 198 REPORT— 1873. Ninth Report of the Committee for Explorirtff Kent's Cavern, Devon- shire, the Committee consisting of Sir Charles Lyell^ Bart., F.R.S., Professor Phillips, F.R.S., Sir John Lubbock, Bart., F.R.S., John Evans, F.R.S., Edward Vivian, M.A., George Busk, F.R.S., William Boyd Dawkins, F.R.S., William Aysh- roRD Sanford, F.G.S., awe? William Pengelly, F.R.S. (Reporter.) The Committee, in opening this their Ninth Eeport, have to state that, since reporting at Brighton in 1872, the work has been continued without inter- mission, in the manner observed at the commencement. They have to add that whilst it is still conducted, under the Superintendents, by the same foreman (George Smerdon), the second workman (John Farr), believing that the Cavern work was prejudicial to his health, has obtained other employ- ment. Though reluctant to part with so satisfactory a workman, who had faithfully served them for upwards of five years, the Superintendents felt un- able to press him to remain under the circumstances ; and they had the satis- faction of engaging in his stead a man (John Cliunick) who has proved most efficient and trustworthy. As in former years, the cavern has been visited by a large number of persons, none of whom, when conducted by the guide only, has been allowed to be taken to the excavations then in progress. The Superintendents have had the pleasure of accompanying the following gentlemen during their visits : — Major-General R. C. Schenck, Minister for the United States of America to England ; Lord Chiford, of Chudleigh ; Sir R. Anstruther, Bart., M.P. ; Bev. Lord Charles Hervey, Rev. G. Butterworth, Rev. Dr. Hanna, Rev. C. N. Kelly, Rev. R. Locker, Rev. T. R. R. Stebbing ; Major-General Huyshe, Captain Lovett, Professor W. X. Clifford, Dr. B. Collenette, Professor W. King, Dr. R. Martin, Dr. W. Sharpey, Dr. Topham, Dr. C. Williams, of Burmah ; Mens. "Wyvekens, of Brussels ; and Messrs. A. T. Atchison, W. Babington, N". Bell, of Queensland, C. A. Bentinek, L. B. Bowring, W. BuUer, E. L. Corring, of V. S. America, J. A. Curtis, R. D. Darbishire, J. M. Dowie, B. J. M. Donne, E. A. Field, S. Gurncy, C. W. Hamilton, H. W. Haynes, of Boston, U. S., C. Sabapathi Jyah, of Madras, J. H. van Lennep, of Holland, C. Lister, P. C. Lovett, C. Meenacshaya, of Madras, P. H. Mills, A. G. Nathorst, of Lund, Sweden, P. Nind, A. Nesbit, A. Pengelly, of N.W.P. India, H. C. M. Phillips, C. H. Poingdestre, F. P. Purvis, T. Rathbone, Dr. Richardson, R. B. Shaw, British Commissioner, Ladak, J. H. Taunton, P. Watts, and J. E. Wolfe. A. R. Hunt, Esq., M.A., F.G.S., being about to assist in exploring a small cave on the coast of Kirkcudbright, visited the cavern in August 1873, for the purpose of studying the mode of working. As in former years, live rats have been observed from time to time in various parts of the cavern. As soon as they are seen, the workmen, having frequently suffered from such visits, set gins for them, and sometimes succeed in taking three or four in a week. On one occasion four (two old and two young ones) were found in the gin together. The adults were the extremes of the series, and, being caught by the neck, were dead ; whilst the others were held near the middle, and still alive. Though most prevalent near the entrances of the cavern, they have been frequently observed far in the interior ; and very recently they carried off a candle from a spot fully 300 feet from the nearest entrance. TJie Long Arcade. — The Committee stated in their last Report, bringing the work up to the end of July 1872, that they were then exploring the branch ON Kent's cavern, Devonshire. 199 of the cavern termed by Mr. MacEnery " The Long Arcade," and sometimes " The Comdor " *, and that they had expended about ten weeks' work on it f. The exploration of this great thoroughfare has been the work of the entire period since that date, and it is still in progress. The Arcade commences in the south-west corner of the " Sloping Chamber," and, after a length of about 252 feet, in a west-south-westerly direction, and almost in a straight line, terminates in the " Cave of Inscriptions," or " Cul-de- sac." Its height is variable — being in one place not quite 10, and in others upwards of 20 feet, the measurements being taken from the bottom of the excavation made by the Committee. The roof and walls are much fretted and honeycombed, except at one part not far within the entrance, where the fall of a very large block of limestone in comparatively recent times has left edges tolerably sharp and angular. Omitting blocks of limestone here and there, the surface of the deposit in the Arcade when the Committee commenced its exploration presented but few inequalities ; and when they had completed their excavation to the uniform depth of 4 feet below the under surface of the Stalagmitic Floor, and up to the distance of 134 feet from the entrance, the bottom of their section was no more than 40 inches above that at the commencement — a mean rise of no more than 1 in 40. At the point just specified, however, the passage was almost entirely closed with a vast mass of limestone in situ, covered in places by thick accu- mulations of stalagmitic matter, and rising to the roof apparently from the limestone bottom of the Arcade. The only opening in it was a narrow aperture adjacent to the right or northerly wall ; and to gain this it was necessary to climb to the height of 8 or 9 feet. It proved to be about 6 feet high, to have a floor of limestone, with occasional stalagmitic incrustations, extending for a length of fuUy 20 feet ; whilst very near the entrance, on the left or southerly side, was the elliptical mouth of a smoothly eroded tunnel, measuring 30 inches in horizontal and 27 in vertical diameter, and having the aspect of a watercourse. Beyond this tunnel, and also on the left side, lay in wild confusion several very large masses of limestone, which had fallen from the roof obviously in remote times ; and beyond these the deposit of Cave-earth again presented itself, but at a higher level than before. Assuming the tunnel just mentioned to have been a watercourse, the stream issuing from it must have had a sudden fall of several feet ; and it may not, perhaps, be without interest to state that on excavating the deposits in the Arcade, deep pot-holes were found in the right wall of the cavern, having the position and character such a fall would have produced. The tunnel, fuUy 60 feet long, terminates in a branch of the cavern known as '' The Laby- rinth," and in one part of its course is so small as to render it somewhat difficult for even a small man to force his way. It has long been known as " The Little Oven ; " and when the cavern was visited by merely the idly curious, it was regarded as an achievement to have made its passage. One of the results of the work during the last twelve months has been to show that the great mass of limestone, which, as already stated, almost com- pletely closed the Arcade, extended downwards, not to the limestone floor, but merely to the level of the earthy deposits which choked up the passage beneath. The 'loose and confusedly grouped blocks of limestone already spoken of have been blasted and taken out of the cavern ; the blocked-up passage has been reopened and is now the common thoroughfare ; the mass of rock overhead has been dignified with the name of " The Bridge," and the excavation has been completed far beyond it. * See Trans. Devon. Assoc, vol. iii. p. 235 (1869). t Erit. Assoc. Eeport, 1872, p. 44. 200 REPORT— 1873. The Arcade is very narroAv in proportion to its length. From 17 feet wide at the entrance, it narrows to 5 feet at about 27 yards within, then expand- ing to 11 or 12 feet, and again contracting until, at 42 yards, it is no more than 6 feet wide, it once more enlarges to an average width of 9 feet, and beyond the Bi'idge it becomes an irregular chamber, upwards of 30 feet long and about 15 wide. The exploration has been completed to the inner end of this chamber ; but the Arcade, again much contracted, has a further prolonga- tion of about 50 feet before reaching the Cave of Inscriptions. In the left or southerly waU of the chamber just mentioned is the entrance to the Labyrinth, and of another and smaller branch. Towards these the work- men are now directing their labours. As the earlier explorers had made some excavations here and there throughout the greater part of the Arcade, and thus deprived the Committee of the opportunity of studying it before disturbed by man, the following description, compiled from Mr. MacEnery's manuscripts, may be of interest : — The floor was in great disorder, strewn with rocks having between them in certain places natural reservoirs of water, and in others loose heaps of red marl overspreading the stalagmite and containing fossil bones. The first rhinoceros-tooth found in the cavern was met with in one of those heaps. A peculiarity of this passage was a profusion of a white crumbling substance not unlike half-slacked lime. Hock after rock, on being turned over, presented patches of it on its surface ; the loose mud also contained it ; and wherever stalagmite had formed between the rocks, it, when ripped up, exhibited large deposits of the same matter. In the crevices of the rock and near the surface of the marl it occurred in baUs partly crushed ; several balls were found in some instances pressed together, in others uninjured, adhering, and exhi- biting the tapering point they had when dropped by the animal ; and they were occasionally found singly. There was no doubt that they were copro- lites, and no difi'erence between these faecal deposits and those of the hysena in Exeter Change, except in the far greater size of the fossil balls. The osseous substance was the same in both ; undigested particles of bone and enamel were detected in some of them ; and the explorers were led to the con- clusion that the Arcade was the chosen resort of the Cavern-hytenas for purposes of cleanliness. In this they were subsequently confirmed by a letter from Captain Sykes to Dr. Bucldand, published in the Edin. Phil. Journal*, descriptive of a recent hyajna-cave in India, where, from the almost exclu- sive accumulation of faeces in particular spots, the writer inferred that certain chambers were dedicated to cleanliness. In these retreats few or no bones occurred, " This description," says Mr. MacEnery, " is in its details quite applicable to Kent's Hole. It appears to have been preserved to us in its actual state as when occupied by the extinct hyajna Whilst reading his letter, I imagined myself reading the history of another, sealed one — the duplicate of Kent's Cave, and not the account of a living hycena's den." "Wherever this substance was found accompanying remains, the latter were invariably broken, and always in the same uniform manner; and none of it was found where they occurred entire. Dr. Buckland, to whom the material was pointed out, gave the Arcade the name of the " Hyajnse Cloaca Maxima." About halfway in the length of the Arcade, and near the left or southerly wall, three circular hollows were observed in the floor, about 3 feet in dia- meter, lined down the sides with a thin waving crust. The greasincss of the earth, and the presence of single teeth of bear in different states of prescrva- * Vol. xvi. pp. 378-9 (1827). ON Kent's cavern, Devonshire. 201 tion, at first suggested the idea that they were the beds of that animal, whose habit it is to crouch in particular spots ; but the occurrence of charcoal, and other indications of the presence of man, in the vicinity of the hollows were thought rather to lead to the opinion that they were rude hearths or ovens scooped out by savages, around which they collected to cook and enjoy the spoils of the chase *. Before returning from this digression it may be well to offer a few remarks on two or three points in the foregoing description, on which the exploration now in progress is calculated to throw some light : — Ist. " The loose heaps of red marl " in all probability consisted of material deposited in the era of the Cave-earth, and over which no stalagmite had in those particular spots ever been formed. If, however, they were actually observed, and not merely inferred, to " overspread the stalagmite," the latter, there can be little doubt, was the " Crystalline Stalagmitic Eloor," older than the Cave-earth, of which the Committee have found numerous portions in the Arcade during the present year, as well as in other branches of the cavern in previous years, some of them in situ and others not. 2nd, The Committee have also found a considerable quantity of coprolitic matter in the Arcade, never, however, more than 12, and rarely more than 6 inches below the surface. This material has been met with in all parts of the cavern wherever the Cave-earth has presented itself, but in no instance in any older or more modern deposit, whether of mechanical or chemical origin. The " Lecture Hall " may perhaps be equally entitled to the name of the Hycence Cloaca Maccima f. 3rd. There seems no reason to doubt that the " three circular hollows," instead of being the " beds of bears " or " hearths or ovens scooped out by savages," were natural basins in the stalagmite, such as were described in the Committee's Eighth Report +; for, to say nothing of the fact that several such basins, even when not more than a very few inches in diameter, have con- tained charred wood, possibly washed into them in rainy seasons (when such basins are fuU to overflowing), or perhaps dropped into them accidentally by recent visitors, it is difficult to understand why a savage should have selected for his hearth a spot having nothing to recommend it but its darkness and inconvenience, whilst so many others, in every respect more eligible, were equally at his command. It is noteworthy that, in another part of his memoir, Mr. MacEnery, replying to Dr. Buckland's suggestion that " the ancient Britons had scooped out ovens in the stalagmite," says, " Without stopping to dwell on the difficulty of ripping up a solid floor, which, notwith- standing the advantage of undermining and the exposure of its edges, still defies all our efforts, though commanding the apparatus of the quarry, I am bold to say that in no instance have I discovered evidence of breaches or ovens in the floor " §, But waiving all this, the Committee, on March 31 , 1873, in the course of their work reached a hollow precisely similar to those Mr. MacEnery de- scribes. It was of oval form, 4 feet long, 2 broad, and 9 inches deep, and contained nearly ten gallons of beautifully pure water, but, instead of having been formed by a bear or a human being, it was an example of Nature's handiwork, and in such a position as to render it certain that the foreman of the exploration now in progress was the first human being who ever saw it. It was in the stalagmite covering the deposit, which, as already stated, com- * See Trans. Devon. Assoc, vol. iii. pp. 235-7, 253-4, 270, 290, and 302-5 (1869). t See Report Brit. Assoc. 1868, p. 49. X Ibid. 1872, p. 45. § See Trans. Devon. Assoc, vol. iii. p. 334 (1869). 202 REPORT— 1873. pletelj filled np the space beneath the Bridge, and was neither discovered nor discoverable until the workmen had advanced 11 feet in the difficult work of reopening this passage. At the entrance of the Arcade, the Granular Stalagmitic Floor was con- tinuous in every direction for considerable distances. At the right or northerly wall its thickness exceeded that hitherto found in any other part of the cavern, measuring fully 5 feet for a length of about 8 yards ; but at the opposite wall it was very rarely more than 2 feet thick. Beyond the point just specified it became gradually thinner, disappearing entirely at 37 feet from it on the right wall, but extending somewhat further on the left. Still further in, such floor as ever existed appears to have been but thin and occa- sional only, until reaching the Bridge, where it appeared again in considerable volume*. Almost immediately bej^ond this, there rose from the Stalagmitic Floor a large boss of the same material, in the form of a paraboloid, 2 feet high and 6 feet in basal circumference. As it* bore no inscription, and was in the direct line of the work, it was dislodged and broken up, when it was found to consist of pure stalagmite without any extraneous substance. In the earthy deposit adhering to its base were one tooth of bear, a fragment of bone, a ball of coprolite, and a few bits of charcoal. Not far beyond it, but near the right wall of the Arcade, a much larger boss presented itself, having near its summit the inscription " R. L. (or E.) 1604." The mass has been so mutilated by early visitors as to render it uncertain whether the remaining part of the second letter is the lower portion of L or E. The date, however, which is quite distinct, and appears not to have been noticed prior to June 6, 1873, is the oldest at present known in the cavern, though there are several others of the seventeenth century. In excavating, care was taken to leave the mass, as well as the deposit on which it was formed, intact and undis- turbed. The only objects found in the Granular Stalagmitic Floor, in the Arcade, since the Eighth Report was sent in, were a tooth of Hysena, a few bones and bone chips, a " charcoal streak " about 3 inches above the base of the floor, where its total thickness was 42 inches at one end and 10 at the other, a few pieces of charcoal, and a flint tool. The tool (No. 5990) is of very white flint, having, as shown by an accidental fracture, a very chalk-like texture. It may be described as a hammer-like " core," broad at one end, round-pointed at the other, and formed by several flakes having been struck from the original nodule. Its pointed end shows that it has been used as a hammer. It is 3-2 inches long, 2 inches in greatest breadth, 1*7 inch in greatest thickness, and was found August 19, 1872. As already stated, remnants of the old (the Crystalline) Stalagmitic Floor occurred in situ in various parts of the Arcade, all attached to the right or northerly wall, and above the level of the Granular Floor. The first of them, about 60 feet within the entrance and 6 inches thick, had between it and the Granular Floor an unoccupied space of 15 inches in height. The second, 20 feet further up the Arcade, was a very large mass displaying strikingly the characteristic prismatic crystalline structure ; it has suffered much at the hands of visitors ; and on one of its fractured surfaces is the date 1836. The * It is worthy of remark that at the entrance of the Arcade, where the Stalagmitic Floor is so very tliick, the drip of water from the roof is at present very copious in rainy seasons, and commences within a few hours of a great rainfall ; whilst those parts of the same branch of the cavern where there does not seem to have ever been any stalagmite are perfectly dry at all times and seasons. ON Kent's cavern, Devonshire. 203 tliird and most important, about 30 feet long, lined the entire lower surface of the mass of limestone forming the Bridge, and extended into the chamber beyond. The less ancient, or Granular Floor, -was in some places in contact with it, and in others as much as 8 inches below. Numerous stones and a few fragments of bone (representing the Breccia on which the Old Floor was formed) were found firmly cemented to this, as well as to the first remnant. The progress of the work has not rendered it necessary to remove or diminish either of them. The deposit below the Granular Stalagmitic Floor was typical Cave-earth to the depth of at least 4 feet *, from the entrance of the Long Arcade to about 24 feet within it, and contained a considerable number of blocks of limestone, several of them requiring blasting in order to be removed. Beyond the point just specified the deposit was everywhere " Breccia " (the oldest deposit the cavern is known to contain), except at most the uppermost foot, which consisted of Cave-earth. The two deposits lay one on the other with- out, as in the South-west Chamber f, any stalagmite between ; and though they are so very dissimilar in composition — the Cave-earth, or less ancient, being made up of small angular fragments of limestone mixed with light-red clay, whilst the Breccia, or older deposit, consists of rounded and subangular fragments of dark-red grit imbedded in a sandy paste of the same colour — it was not always, or, indeed, frequently, easy to detect a well-defined line of separation. Each, however, was, as elsewhere in the cavern, characterized by its distinct fauna — -the Breccia containing remains of Bears only without any indication of other genera, whilst the Cave-earth yielded bones and teeth of Hyaenas, with their teeth-marks and coprolites, as well as the osseous remnants of the animals usually associated with them. At the entrance of the Arcade Mr. MacEnery's diggings were carried to a depth of 3 feet below the bottom of the Granular Stalagmite ; they gradually became less and less deep until at a distance of 15 feet they ceased. They were resumed at 52 feet, and continued at intervals throughout the entire length of the Arcade so far as the Committee have at present explored. They were, however, on a very limited scale, never exceeding 18 inches, and com- monly not more than a foot in depth, did not always extend from wall to wall, and were not continuous. In short, he seems to have contented himself with occasionally digging a small shallow trial pit, and, meeting with no speci- mens, to have i^roceeded elsewhere ; and this is borne out by his own state- ment. " As we advanced in the direction of the Long Corridor," he says, " the bones became less and less numerous until they nearly disappeared, rendering it not worth our while to prosecute our researches further in that line " J. He must, however, in some instances have broken up portions of the Breccia as well as of the thin layer of Cave-earth lying on it ; for, as was his wont, the materials he dislodged were not taken out of the cavern, but merely cast aside ; and these, on being carefully examined by the Committee, were found to contain undoubted fragments of the older deposit, with bones and teeth of Bear firmly imbedded in them. The specimens recovered from this broken ground, and which had been neglected or overlooked, belonged mainly to the Cave-earth. They were 72 teeth, 4 astragali, 5 ossa calcis, 15 phalanges, 1 claw, 3 portions of jaws, 2 ver- tebraj, 1 portion of skull and 1 of antler, several fragments of bone, and 8 * The excavation is not carried to a depth exceeding 4 feet below the bottom of the granular stalagmite. f See Brit. Assoc. Export, 18G8, pp. 50-52. J See Trans. Devon. Assoc, vol. iii. p. 290, 204 REPORT 1873. flint flakes and chips. With them was a portion of an iron hammer, which, on becoming useless, MacEaery or his workmen had no doubt thrown away. Omitting those of Bear, at least some of which belonged to the era of the Breccia as already stated, the teeth may be distributed as in the following Table :— Table I. — Showing how many per cent, of the turbed material in the Long Arcade belonged Cave Mammals. Teeth found in the dis- to the diff'erent kinds of Hyaena 70 per cent. Horse 10 „ Rhinoceros 10 „ Deer 3 ,, Ox 8 per cent. Elephant 1-5 „ Fox 1-5 „ The flint flakes mentioned above were of little value when compared with