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{Office of the Association : 22 AtDraivRtT; BrREET, London, W.] 




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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 


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 



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- 


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 



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 


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' 







Address by Professor H. J. S. Smith, M.A., F.R.S., President of the Section 1 


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 



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 


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 


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 


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 



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 


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 


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 



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 


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 



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 


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 


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 


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 



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 


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 



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 


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 



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 


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 




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 


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 



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, 

Progress of chemistry, report of the Committee for superintending the monthly reports of 

the, 24. 

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. 



Illustrative of the Report of the Committee on the Labyrinthodonts of the 





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1873. 6 


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* 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 



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* Passed by the General Committee. Edinburgh, 1871. 

t This and the following sentence were added by the General Committee, 1871. 


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The Accounts of the Association shall be audited annually, by Auditors 
appointed by the General Committee. 


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REPORT 1873. 

Presidents and Secretaries of the Sections of the Association. 

Date and Place. 





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... 


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. 

Prof. Forbes, W. S. Harris, F. W. 

W. S. Harris, Rev. Prof. PoweU, Prof. 

Rev. Prof. Chevallier, Major Sabine, 

Prof. Stevelly. 
J. D. Chance, "W. Snow Harris, Prof. 

Rev. Dr. Forbes, Prof. Stevelly, Arch. 

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. 

Rev. H. Price, Prof. Stevelly, G. G. 

Dr. Stevelly, G. G. Stokes. 
Prof. Stevelly, G. G. Stokes, W. 

Ridout WiUs. 
W. J. Macquorn Rankine, Prof. 

Smyth, Prof. Stevelly, Prof. G. G. 

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., 

Rev. Prof. WheweU, RR.S 

Prof. rorbe.s, F.R.S 

Rev. Prof. Lloyd, F.R.S 

Very Rev. G. Peacock, D.D., 

Prof MCulloch, M.R.LA 

The Earl of, 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. 

Rev. W. Whewell, D.D., F.R.S., 

Prof. W. Thomson, M.A., F.R.S. 

The Dean of Ely, F.R.S 

Prof. G. G. Stokes, M.A., Sec. 



Date and Place. 


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 


Eev. S. Earnshaw, J. P. Hennessj, 
Prof Stevelly, H. J. S. Smith, Prof. 
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. 

W. Spottiswoode, M.A., F.E.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.,: 


1870. Liverpool...' J. Clerk Maxwell, M.A., LL.D.,iProf W. G. Adams, W. K. Clifford, 
" " -• ' Prof. G. C. Foster, Eev. W. Allen 

Prof W. G. Adams, J. T. Bottomley, 

1871. Edinburgh . 

1872. Brighton ... 

1873. Bradford ... 

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. 



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. 


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- 

Prof Johnston, Prof Miller, Dr. 

Prof 'Miller, E. L. Pattinson, Thomas 

Golding Bird. M.D., Dr. J. B. Melson. 

Dr. E. D. Thomson, Dr. T. Clark, 
Dr. L. Playfair. 

J. Prideaux, Eobert Himt, W. M. 

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. 


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 



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. 

Sir J. F. W. Hersohel, Bart., 

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.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., 

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. 

H. S. BlundeU, Prof. R. Hunt, T. J. 

Dr. Edwards, Dr. Gladstone, Dr. Price. 
Prof. Frankland, Dr. H. E. Roscoe. 
J. Horsley, P. J. Worsley, Prof. 

Dr. Davy, Dr. Gladstone, Prof. Sul- 
Dr. Gladstone, W. Odling, R. Rey- 
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. 

Prof. Liveing, H. L. Pattinson, J. C. 

A. V. Harcourt, Prof. Liveing, R. 

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. 

Dr. Mills, W. Chandler Roberts, Dr. 

W. J. RusseU, Dr. T. Wood. 
Dr. Armstrong, Dr. MiUs, W. Chan- 
dler Roberts, Dr. Thorpe. 



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 . 


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. 


Captain Portlock, T. J. Torrie. 
William Sanders, S. Stutchbury, T. J. 

Captain Portlock, R. Hunter. — Geo- 
graphy. Captain H M. Denham.R.N. 



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* 


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., 

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. 


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- 

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. 

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., 

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. 

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. 

Prof Harkness, Rev. J. Longmuir, H. 
C. Sorby. 

Prof Harkness, Edward Hull, Capt. 

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 


REPORT — 1873. 

Date and Place. 



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., 

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., 

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 

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. 




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 


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- 

Eev. Prof. Henslow, F.L.S 

Sir J. Eichardson, M.D., F.E.S. 

H. E. Strickland, M.A., F.E.S..., 


J. Curtis, Dr. Litton. 

J. Curtis, Prof. Don, Dr. Eiley, S. 

C. C. Babington, Eev. L. Jenyns, W. 

J. E. Gray, Prof. Jones, E. Owen, Dr. 

E. Forbes, W. Ick, E. Patterson. 

Prof. W. Couper, E. Forbes, E. Pat- 

J. Couch, Dr. Lankester, E. Patterson. 

Dr. Lankester, E. Patterson, J. A. 

G. J. Allman, Dr. Lankester, E. Pat- 

Prof. Allman, H. Goodsir, Dr. King, 
Dr. Lankester. 

Dr. Lankester, T. V. Wollaston. 

Dr. Lankester, T. V. Wollaston, H. 

Dr. Lankester, Dr. Melville, T. V. ' 

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. ... 



Date and Place. 














Belfast . 

Eev. Prof. Henslow, M.A., F.R.S, 
W. Ogilby 


Liverpool ... 
Q-lasgow . . . 



Aberdeen ... 




Newcastle . . . 



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. 
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. 

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.' 


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. 

Dr. T. S. Cobbold, Sebastian Evans, 

Prof. Lawson, Thos. J. Moore, H, 

T. Stainton, Rev. H. B.Tristram, 

C. Stanilaud Wake, E. Ray Lan- 

Dr. T. R. Eraser, Dr. Arthur Gamgee, 
E. Ray Lankester, Prof. Lawson, 
H. T. Stainton, C. Staniland Wake, 
Dr. W. Rutherford, Dr. Kelburne 

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, 



REPORT — 1873. 

Date and Place. 

1873. Bradford ... 


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. 


Prof. Thiselton-Dyer, Prof. Lawson, 
B. M'Lacblan, Dr. Pye-Smith, E. 
Ray Lankester, F. W. Eudler, J. 
H. Lamprey. 







Dr. Haviland 

Dr. Abercrombie 

Dr. Bond, Mr. Paget. 

Dr. Roget, Dr. William Thomson, 



Liverpool . . . 

Newcastle . . . 
Glasgow . . . 


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.. 





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. 

Dr. J. Butter, J. Fuge, Dr. R. S. 

Dr. Chaytor, Dr. R. S. Sargent. 
Dr. John Popham, Dr. R. S. Sargent. 
I. Erichsen, Dr. R. S. Sargent. 


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. 


Glasgow . . . 



Aberdeen ... 


Manchester . 
Cambridge . 




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. 

[For Presidents and Secretaries for Geography previous to 1851, .see Section C, p. rrrii.] 

1846. Southampton 

1847. Oxford 

1S4S. Swansea ... 

1849. Birmingham 

1850. Edinburgh.. 


Dr. Pritchard IDr. King. 

Prof. H. H. Wilson, M.A IProf. Buckley. 

G. Grant Francis. 

IDr. R. G. 

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. 


Date and Place. 








Ipswich . . . 



Liverpool . . . 
Glasgow . . . 



Aberdeen ... 


Manchester . 
Cambridge . 




Norwich ... 


Liverpool . . . 
Brighton ... 
Bradford ... 

Sir R. I. Murchison, E.E.S., Pres 

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 

Sir E. I 

F.R.S. Sir James 

Ross, D.C.L., F.R.S. 
Sir E, L Murchison, 

John Crawfurd, F.R.S. , 

Murchison, G.C.St.S. 


Francis Gulton, F.R.S. .. 



Sir R. I. Murchison, 

Sir R. I. Murchison, 

Major-General Sir R. Rawlinson, 

M.P., K.C.B., F.R.S. 
Sir Charles Nicholson, Bart., 


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 

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. 

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. 


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. 



1833. Cambridge .IProf. Babbage, F.E.S 

1834. Edinburgh .{Sir Charles Lemon, Bart. ... 

J. E. Drin'cwater. 

Dr. Cleland, C. Hope Maclean. 


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. 


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 



Et. Hon. Lord Sandon 

W. E. Greg, W. Langton, Dr. W. C. 

W. Cargill, J. Heywood, W. E. Wood. 
F. Clarke, E. W. Eawson, Dr. W. C. 

C. E. Baird, Prof. Eamsay, E. W. 

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. 

Prof. Hancock, J. Fletcher, Dr. J. 

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.. 

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, 

Sir John P. BoUeau, Bart. 
His Grace 

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., 

Prof. J. E. T. Eogers 

Eev. C. H. Bromby, E. Cheshire, Dr. 
W. N, Hancock Newmarch, W. M. 
Prof Cairns, Dr. H. D. Hutton, W. 

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. 
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. 

G. J. D. Goodman, G. J. Johnston, 

E. Macrory. 
E. Birldn, Jun., Prof. Leone Levi, E. 

M. E. Grant Duff, M.P jProf. Leone Levi, E. Macrory, A. J. 


Samuel Brown, Pres. Instit. Ac-:Eev. W. C. Davie, Prof. Leone Levi, 
tuaries. I 



Date and Place. 



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. 





Bristol ... 
Liverpool ... 

1840. Glasgow 
















Plymouth . . . 
Manchester . 





Edinburgh .. 




Liverpool ... 

Glasgow ... 




Aberdeen ... 


Manchester . 

Cambridge .. 
Newcastle . . . 





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 

John Grantham, J. Oldham, J. Thom- 
L. Hill, Jun., William Eamsay, J, 

C. Atherton, B. Jones, Jun., H. M. 

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. 

P. Le Neve Foster, Rev. F. Harrison, 

Henry Wright. 
P. Le Neve Foster, John Eobinson, H. 

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.. 

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., 

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. ... 


REPORT 1873. 

Date and Place. 



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.... 

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. 

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. 

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 


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., 

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. 

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. 


Magnetic and Diamagnetic Pheno- 

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 

Tlie present stato of Photography. 



Date and Place. 

1854. Liverpool ... 

1855. Glasgow 

1856. Clieltenhani 


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- 


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., 

Prof. W. K. Chfford 

Anthropomorphous Apes. 

Progress of researches in Terrestrial 

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 
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- 
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- 
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 

Reverse Chemical Actions. 
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 

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. 


REPORT — 1873. 

Date and Place. 


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., 

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. 


Sunshine, Sea, and Sky. 

1-1 ^ o o •* 

0» CO O iO rt 

(M CI O -^ i-< 

%n CO o T)< 

-^ "^ I— t 







as iM 


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. 








Newcastle-on-Tyne .. 




Manchester ..., 






Birmingham .... 
Edinbiu-gh .... 











Manchester .... 
Cambridge .... 
Newcastle-on-Tyne .. 


Birmingham .... 
Nottingham .... 





Edinburgh .... 


Bradford , 



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 

New Life 





































































at Annual Meetings of the Association. 

Attended by 



during the 


Sums paid on 

Account of 

Grants for 



Old N 
Annual An 


nual Asso 

ciates. La 

dies. Fore 

igners. Total. 

Members. Men 


£ s. d. 

£ s. d. 



• • 





434 14 
918 14 6 









34 1438 

1-0 1353 


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 















35 1079 

36 857 
53 1260 
15 9^9 








22 1071 


159 19 6 





H 1241 

10S5 ° ° 

345 18 





37 7'° 


391 9 7 





9 1108 


304 6 7 





6 876 



121 1 




10 l8o2 


33° 19 7 

142 ] 

01 1 



26 2133 


480 16 4 





9 ii'5 


734 13 9 

156 ) 




26 2022 


507 15 3 





13 1698 


618 18 2 

125 1 

79 I 



22 2564 


684 II 1 





47 1689 


1241 7 

1S4 1 

25 I 



15 3139 


iiii 5 10 





25 1161 


1293 16 6 

154 : 

.09 I 

704 I 


25 3335 


1608 3 10 

182 1 

03 1 

119 I 


13 2802 


1289 IS 8 

215 1 




23 1997 


1591 7 10 

218 1 




II 2303 


1750 13 4 

193 J 

18 I 



7 2444 


1739 4 





45 t 2°°4 



229 ] 




17 1856 




95 1 



14 2878 


1472 2 6 





21 2463 


1285 ° 





43 2533 







11 1983 


* Ladies were not admitted by purcba.scd Tickets until 1843. 

t Tickets for admission to Sections only. J Including Ladies. 

xlvi REPORT — 1873. 



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. 



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. 


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. 


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. 


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. 


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. 


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. > > , > , 


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. 


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. 


LL.D., F.E.S. 


The Eight Hon. the Eael of Enniskillen, D.C.L., 

The Right Hon. the Eabl of EoSSE, F.E.S., 

Sir EiCHAED Wallace, Bart,, M.P. 

The Eev. Dr. Henby, President of Queen's College, 

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. 



Dr. P. Eedfebn. 

T. Sinclaib, Esq., J. P. 

William J. C. Allen, Esq. 

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. 


Maxwell, Professor J. Clebk, P.H.S, 
Meeeifield, C. W., Esq., F.E.S. 
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. 


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. 


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. 

Geokge Gbiffith, Esq., M.A., F.C.S., Harrow-on-the-hill, Middlesex. 

William Spottiswoode, Esq., M.A., LL.D., F.E.S., F.E.G.S., TO Grosvenor Place, London, S.W 


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 


" 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 

" 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 

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 : — 


" 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 


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 



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. 



Professor E. CrouUebois. 
Professor G. Devalque. 
M. "VV. de Fonvielle. Paris. 
Professor Paul Gervais. Paris. 
Professor James Hall. Albany, New 

Mr. J, E. Hilgard. Coast Survey, 

M. Georges Lemoine. 

Eichter. St. 

Professor Victor von 

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 

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 

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. 


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 

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 

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 

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 


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 

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- 

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- 

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- 

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 


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- 

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. 


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 

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. 


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. 


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 


*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 


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 






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 


Statistics a^ul Economic Science. 
Houghton, Lord. — Economic Effects of Trades Unions 25 


*Froude, Mr. W. — Instruments for Measuring the Speed of 

Ships and Currents (renewed) 50 

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, 


REPORT — 1873. 

General Statement of Sums which have been paid on Account of Grants 

for Scieiitific Purposes. 

TiJo Discussions 20 


TiJe Discussions ■ fi2 

r.ritisli Fossil Ichthyology lOj 



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 


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 


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 


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 


















£1595 11 


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 


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 









£ 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 


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 


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 





8 4 


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 


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 

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 

British Association Catalogue of 

Stars 1844 211 15 

Fossil Fishes of the London Clay 100 




£ 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 

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 

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 


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 

Maintaining the Establishment at 

Kew Observatory 255 18 

Transit of Earthquake Waves ... 50 

Periodical Phenomena 15 

Meteorological Instruments, 

Azores 25 

£3 IT) 18 b 



£ s. d. 
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 


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 


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 



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 

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 

Maintaining the Establishment at 
Kew Observatory : — 

1854 £ 75 01 ,,. . . 

1855 £500 oM''' " " 


£ ». 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 


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 

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 

Maintaining the Establishment 

of Kew Observatory 500 

Dredging near Belfast 16 6 

Dredging in Dublin Bay 15 



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 

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 

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 

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 

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 

Maintaining the Establishment 

of Kew Observatory 600 

Balloon Committee 100 o 

Hydroida 13 



£ 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 

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 



£1750 13 4 


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 

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 

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 


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 



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 


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 

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 

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 






2 6 


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. 






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 

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 


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 

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 


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- 

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 

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 

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 

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 


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 

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 

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- 


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 

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 

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 

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 

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 

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 

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 


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 

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- 

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 

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 

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- 


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. 


"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 

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 

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 

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 


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 

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 


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. 




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, 


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 


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. 


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 

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/) 

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, 


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. 

(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.' 


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 

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 

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 

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 

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 


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 

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, 



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. 


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 


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 . . ,). 


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 

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 


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 


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 


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 

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 


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 

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) 


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 

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 

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 

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 

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. 


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); 


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 

[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 


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. 


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 : 





























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. 


" 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 

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. 


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 

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 

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. 


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 

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 

In a note on p. siv of the Introductio to the ' Supplemental he (Felkel) 


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 

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. 


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- 

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 

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 


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 


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 

[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. 


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 

[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 ; 


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 

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 

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 

*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 

We have also seen advertised ' Tafcln zur Bercchnung goniometrischer 
Co-ordinaten,' by F. M. Clouth — no doubt a German edition of the same 

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.] ; 


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 

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) 


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 

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 


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. 


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- 

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 


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 

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 


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 

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. 


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 

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 


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, „ 


„ 8539 

„ log 118538, „ 


„ 8576 

220, log 127340, „ 


„ 9648 

312, log 173339, „ 


., 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. 

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. 


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. 


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. 

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- 

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, 


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 


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, 

The tbree Tables are separated in the following : — (To 5 places) Mackay, 
T. XLYIII.-L. ; Moore, 1814, T. XXIII. ; Norie, 1836, T. XXYII.- 

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 


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 

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 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 

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 


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 


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 


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 


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 



19- 19-12 



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 

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, 



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 ^ 


" 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 

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, 

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. 


(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 

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 


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 

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. 


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 

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 


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. 


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., 
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- 

[T. III.] Log sines, cosines, tangents, cotangents, secants, and cosecants for 
every quarter point, to 5 places. 

86 REPORT — 1873. 


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 

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 

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. 


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- 

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- 

[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 

[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 


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 

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 


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 

[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 


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 

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 


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. 


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 

• 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 

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). 


[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 

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- 

[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. 


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 

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. 


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- 


[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- 

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- 


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 

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 

[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- 

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 

No editors' names aj»pcar ia the work i but Lalande (Bibliog.Astron. p. 516) 


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 

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 

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 

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 

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 

T. XIII. Areas of segments of circles for diameter unity to 6 places : the 


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 

[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 

D. Three tables to express (I.) hundredths, (II.) thousandths, (III.) ten- 
thousandths of 90°, in degrees, minutes, and seconds (sexagesimal). 


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 

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- 

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 

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" 


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 

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- 


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 

T. II. Logarithms to 1000, and thence for odd numbers to 1199, to 20 

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 

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 


■- 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 

[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 

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 

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 

[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. 


1.3.5 1 

T. XI. Decimal values of certain coefficients, such as — , , 2 . 4 . r>' 


-, «S:c., with their logarithms. There arc 40 in all ; and the talkie 

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 

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 


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 





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. 




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 


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 

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." 


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. 


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 

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 ^ . 


[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 

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 

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 


: parts. The three tables aro sexmnitecl, as is now usual (see Mack ay, § 4, 

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 

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 


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 

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. 


[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 


printed at full length ; the proportional parts are given at the foot of tho 

T. 6-5. Log sines, tangents, and secants to every quarter point, to six 

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 

^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 : — • 


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 

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. 


[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 

[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 


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 

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 


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. 


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 

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- 


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 

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 

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 


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- 

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- 

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. ■ . ■• . ■ -..-.. 


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 

[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 

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 

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. 


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. 


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 

[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 

[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 ] 
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 

[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- 

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 


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 

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. 


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 

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 


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 


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. 


*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. 


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). 


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. 


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, 


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. 


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). 

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, 

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. 


[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 

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 


(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, 



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 


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). 


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 


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, 


164 REPORT — 1873. 

07 4-1 

Weidenbach. Tafel um den Logarithmen von ' zu finden wenn der 


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. 


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 

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 


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. 


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 


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 

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 

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(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 


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- 

The formulae are : — 

e — -= 1 — 2o cos 2a; + 27* cos 4.r—2o^ cos 6.r+. . ., 

^ 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 + . . .), 


= k'^ (1 + 22 cos 2^+22* cos 4:X-[-2q^ cos 6a? + , . .) ; 

80 that 

2X.V ^ 2Kr . ^2Kx 
Sin am = Q^ —Q , 


cos am = 0., -i- 6 , 

TT ' TT It 

2Xx' „ 2K.i; . ^2K.r 
A am =63- — ~" J 



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 


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- 

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 

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 



§ 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 



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; 




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 


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 

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 

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' 

" 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 

" 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.'' 


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 

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- 



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 

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 



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 


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 


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- 

Maintenance at Id. per ton 

Eedemption of capital at 2d. per ton .... 






1 14 



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 

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- 


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. 


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. 


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 

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. 



'" 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 : — ' 


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 


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. 


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- 



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 

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 : — 




Harborne, near Birmingham. 


" 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 


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 

" 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- 

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 


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, 


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 


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- 

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, 


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 
many found in the Cave-earth. 

Up to the end of August 1873, the Cave-earth which the Committee found 
intact in the Long Arcade had yielded, when the few mentioned in the Eighth 
Report (1872) are included, about 280 teeth, which may be apportioned as in 

the following Table : — 

Table II. — Showing how many per cent, of the Teeth found in Cave-earth in 
the Long Arcade belonged to the difl'erent kinds of Cave Mammals. 

Hyaena . .\ 40 per cent. 

Horse 24 „ 

Rhinoceros 11 „ 

Bear 9 „ 

Fox 5 „ 

Pig 3 „ 

Deer 2-5 per cent. 

Megaceros 1'5 „ 

Elephant 1-5 „ 

Dog? 1-5 „ 

Lion 1-0 „ 

Machairodus only 1 incisor. 

On comparing the foregoing Tables with those in previous Reports, the 
following facts present themselves : — 

1st. That Hyaena is everywhere the most prevalent animal of the Cave- 
earth era, and is followed by the Horse and Rhinoceros without any consider- 
able variation in their ratios. 

2ud. That the Bear is relatively more prevalent in the Long Arcade than 
in any other part of the cavern explored by the Committee. 

3rd. That teeth of Wolf, Badger, Rabbit, Reindeer, and Sheep * — all of 
which presented themselves in the various branches of the Eastern Division 
of the cavern — have not hitherto been met with in the Long Arcade. 

None of the animal remains found in the Cave-earth during the last twelve 
months require detailed description or special remark. Many of the bones 
had been gnawed by the Hyaena ; some were much decayed ; a few small 
fragments had been burnt ; and one (a phalanx) exhibited marks of disease. 
The few remains of the Mammoth were those of immature animals ; one 
canine of Lion (No. 6020) was worn almost to the fang ; and a right lower 
jaw of Pig (No. 6098)t, found March 26, 1873, without any other specimen 
near it, contained eight teeth, some of which had not risen quite above the jaw. 

Including the two (Nos. 5819 and 5829) mentioned in the Eighth Report 
(1872), the Cave-earth in the Long Arcade has, up to the end of August 

* The remains of Sheep are probably such as had been recently introduced by foxes and 
other animals frequenting the cavern, 
t This specimen has a very fresh aspect. 

ON Kent's cavern, Devonshire. 205 

1873, yielded 25 flint implements and flakes, without counting those found 
in Mr. MacEneiy's dislodged materials. Though many of them would have 
attracted a large share of attention a few years ago, a description of a very 
few will suffice at present : — 

No. 6082 is a light-grey flint having a sharp edge all round its perimeter. '"' -^ 
It is nearly flat on one side, and sUghtly convex on the other, from which 
four principal longitudinal flakes have been dislodged. It belongs to the lan- 
ceolate variety of implements, is about 3-5 inches long, 1-2 inch in greatest 
breadth, and -25 inch in thickness. It was found February 22ud, 1873, without 
any animal remains near it ; and no stalagmite had ever covered the deposit 
in which it lay. 

No. 6086 may be said to belong to the same type; but it is more massive, • v^" 

and is abruptly truncated at each end. It is 3-5 inches long, 1-6 inch ^i/ u^ 
in greatest breadth, '0 inch thick, very concave on the inner face, on ' / 

which the " bulb of percussion " is well displayed near what may be termed 
the point ; and the outer very convex face has been rudely fashioned. It does 
not appear to have been used ; its edges are quite sharp and not serrated 
or chipped. It was found March 4, 1873, with a tooth and a gnawed scapula 
(No. 6086). 

As in all other parts of the cavern in which it has occurred, the Breccia 
in the Long Arcade difli'ers from the Cave-earth not only in the mineral and 
mechanical characters of its materials, as already pointed out, but also in the 
absence of those films of stalagmite which so frequently invested bones and 
stones at all levels in the less-ancient accumulation. 

The deposits resembled each other in being entirely destitute of any ap- 
proach to a stratified arrangement ; and the incorporated fragments of stone 
lay with their longest axes in every possible direction. 

Up to the end of August 1873 there had been found in the Breccia in the 
Long Arcade upwards of fifty teeth, together with a considerable number of 
bones, of Bear. As they were much more brittle than those found in the 
Cave-earth, probably from their highly mineralized condition, and almost 
invariably occurred where the materials were firmly cemented together, 
it was impossible to prevent their being injured in the process of extraction. 
Not unfrequently bones or teeth were found broken but having the parts in 
contact and juxtaposition in the concrete