^_/?ie ve^CyUo/rWi University of California • Berkeley University of California • Berkeley y THE PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY OF LONDON, IKOM THEIR COMMENCEMENT, IN 1665, TO THE YEAR 1800; ^ilbvitisetr, WITH NOTES AND BIQGRAPHIC ILLUSTRATIONS, BY CHARLES HUTTON, LL.D. F.R.S. GEORGE SHAW, M.D. F.R.S, F.L.S. RICHARD PEARSON, M.D. F.S.A. VOL. IV. FROM 1694 TO 1702- LONDON; PRINTED BV AND FOR C. AND R. BALDWIN, NEW BRIDGE-STREET, BLACK.FRIARS. 1809. /i/Gtf, LOAN STACK CONTENTS OF VOLUME FOURTH. DPage R. WALLIS, on Extracting Surd Roots. . 1 E. Halley, Sun's Entrance into the Tropics . . 3 Dr. Baynard, on Rheumatism and Suppres- sion ot Urine 9 Dr. Connor, on the Vertebrae firmly united . . 10 Biographical Account of Dr. Connor ibid On a Water-Spout, by Mr. Zachary Mayne 12 Account of Ridley's Anatomy of the Brain . . 13 Burning Mountain in the Isle ofTernate .... ibid The same in the Isle of Sorea ibid Demoivre, on the Use of Fluxions, ^c 14 Biographical Notice of Mr. Demoivre ibid Mr. Halley, on Constructing Logarithms .... 18 . , on a Proposition in Gunnery .... 27 Account of Dr. 'Wallis's Mathemat. Works . . 29 of Dr. Grevv's Tract, de Salis Cathar. 31 Dr. Sampson, on a prematuie Child ibid On Tadraor, or Palmyra, by Wm. Halifax ... 33 Dr. Wailis, on Quadrable Cycloidal Spaces . . jp Account of Woodward's Natural History. ... 41 Biographical Notice of Mr. Woodward ..... ibid Account of Dr. Pitcairn's Dissert, de Febribus 46' Biographical Notice of Dr. Arch. Fiicairn . . . ibid Mr. Halley, on Cycloids and Epicycloids .... 47 Two Journeys from Aleppo to Pdlmyra .... 49 E. Halley, on the ancient Slate of Palmyra . . 6o Dr. Wailis, on a horse slaked in ihe Stomach 04 Acco. of Agost. Sciila's La Vana Speculat. ike. 1)6 E. Halley, Analogy of Log. Tang, to Meridian 68 Ac. of Da. Gregory's Catopt. and Dioptr.Elem. 77 Dr. Connor's Dis. Medico-Pliysicae . . ibid On a Substance like Butter, by Mr. R. Vans 78 On ihe same, by the Bishop of Cloyne ibid Sir \^ m. Bteston, on the Barometer, &c 79 Dr. Ed. Smyih, Soap-f^arth, near Smyrna .. . 80 Mr. Wilham Cowper, on Chylification .... 81 Catal. of Plants at Tangier, by Mr. Spotlswood 85 Mr. Ch. Bernard, two large Stones cut out, &c. 86 Mr. J. Harris, on Animalcules in Water, &c. 89 Account of Kennet's Parochial Aniiquities. ... 92 Dr. T. Robinson, on Henry Jenkins's Age . . . ibid Leuwenhoeck, on Eels, Mites, Seeds, Sec. . . 94 Staph. Gray, Microscopical Experiments, &c. 97 Dr. Ed. Smyth, Use of Opium by the Turks 10! Child born with a Wound on the Breast 102 Account of Dr. Sloane's Catalogus Plantarum 103 Dr.Sloane, Beans cast on shore at the Orkneys ibid Mr. Wm, Cowper, on a large diseased Kidney 105 Dr. Ch. Preston, on Stones of ihe Bladder ... 109 Dr. Garden, on great Thunder near Aberdeen ibid On a Foetus without the Uterus 110 E. Halley, on a Whelp voided by a Dog .... ibid Ra. I'horesby, on a Roman Pottery Ill Sir R. Sibbald, on Shells in Scotland ibid Account of J. Celsus de Vita et Rebus, &c. ..113 Dr Ch. Preston, on the Dropsy, &c 114 Cutting for the Stone in the Kiiiney 1 16 Chr. Merret, on antiquities in Lincolnshire . . 117 VOL. IV. * Pa^e Stephen Gray, on the Water Microscope .... 120 Quantity of Rain at Greshani College 121 Account of Seller's History of Palmyra 122 Dr Ch. Preston, opening the Body of a Boy ibid Dr Lister, on the Juices of 1 'lants 1 23 Isaac Newton's Solution of two Problems proposed by John Bernouilli \og Biographical Notice of John Bernouilli ibid Samuel Doudy, on an Hydrops. Pectoris .... 13 1 Account of Musei Pet veriani Centuria, &c. . . 132 Biographical Notice of James Petiver ibid M. Giles, History of a Tumor, &c ibid Dr Molyneux, on a Scolopendra Marina . . . . 133 Dr. Lister, on Grasses useful for Hay 136 , on the long Worm in India 137 Dr. Ch Preston, Internal Parts of Fish 138 Isaac Newton, on Descents in a Cycloid, &c 140 Account of Dampier's Voyage round the \\ 'dl41 Biographical Notice of William Dampier .... ibid Account of Plukenei's Ahnagestum Botan. ..ibid of the New Map of France 142 E. Halley, on the True Theory of the Tides. . ibid Dr. Ch. Preston, on a Child without a Brain . 149 J. M. Lancisi, on Malpiglii's Death 151 Biographical Notice of John Maria Lancisi . . ibid M. Giles, Origin of a Polypus 152 Ace. ol W. Cockburn on Diseases of Seamen 154 Continuation of the same Subject . . ibid D. Papin's Collec. of New Machines ibid Dr. Ch. Morley, on Uones voided per Anurn I55 Dr. Molyneux, large Horns dug up in Ireland 156 On some Magnetical Experiments 161 Account of Potter's Lycophronis Chalcid. Alex. ibid Biographical Notice of abp. John Potter .... ibid Dr. Richardson, on Fossil Wood in Yorkshire 162 Nic. Witsen, further Account of the Burning Mountains in the Moluccas 163 Dr. Tyson, on a Brain depressed into the Neckl64 , on a Sphacelated Brain, &c 165 Oct. PuUeyn, on Ancient Inscript. at Rome ibid Stephen Gray, on an Optic Lens of Water, and a Natural Reflecting Microsc.^ .... 166 Edward Coles, a Red Colour by a Mi vure &c. )67 Mr. Hill, on Henry Jenkins's Ag ibid Pitch, Tar, and Oil, from a stone, by Mr. Ele 168 Account of Malpighi's Posthumous Works . . ibid Dr. Slare, on the N° of Births, Deaths, &c.. . 169 Dr. Wailis, on the Antiquity of the Cycloid . . ibid Dr. Lister, on the Dissection of the Scallop . . 170 E. Halley, on large H^il at Chester 171 Another Account of the same 172 R. Taylor, Hail Siorra in Hertfordshire .... ibid A great Hail SiDrm in Herefordshire 173 On the same in Monmouthshire ibid Dr. Preston, on a Foetus voided by the Navel. . ibid E. Halley, Measurement of Snowden-hill by the Barometer 174 Account of M. ie Comte's Memoirs of China 175 CONTENTS. Page Mr. Locke, Person with Homey Excrescencesl76 Dr. T. Smith, Voyage from England to Con- stantinople ibid De Moivre, on hi* Multinomial Theorem ..ibid Wm. Molyneu.K, on an Error in Surveys ... 180 De Vallemont, on a small Egg in another .... 182 J. Ray. on Hemlock and a poisonous Root . . 183 Da. Gregory, on the Catenarian Curve 184- Dr. Waliis, on Hail, Thunder, and Lightning 196 Mr. Scarburg, on a Storm in America 198 Sir Rob. Southwell, on the Isle of Portland . . . ib'd Mr Thoresby, on Iwo Roman Altars ibid Benj. Allen, on the Generation ot Eels 19,'' Account of Cluverius's Geography 200 Dr. H. Sloane, Tongue of a Pastinaca Marina ibid Ja. Peliver's Catalogue of Guinea Plants .... 201 J. Hillier's Observations at Cape Corse 201 John Craig on the Quadrature of Curves. . . . 202 Rob. Tredevey, on a piece of Ambergris. . . . 205 Account of a Moving Bog in Ireland 20() Wm Molyneux, on the same Bog ibid M. Gaillard, Observations on Maladies 207 M. Poupai t, Anatom. Hist, of the Leech 209 Biogra|)hicul Noiice of Francis Ponpart ibid Dr. Mart Lister, on Venom in a PorpusToolh21 1 Mr. Aubrey, on a Medicated Spring ibid Mr. Martin, on the Western Isles of Scotland 212 Dr. Waliis, Additions on Thunder, &c ibid Account of Bilbery's Refraction of the Air, &c. 213 Ramazzinion the Wells in Mo(lei:aibid Biographical Notice of Bernard Ramazzini. . . ibid Mr Thorevby on Roman Antiquities 215 T. Molyneux, on swarms of Insects in Ireland2l6 E. Tentzel, on the Bones of an EKphant 218 Biographical Notice of Ant. Magliabechi .... ibid Rob. Clarke, on a Polypus of the Lungs, and Death of a Dog by a sudden Noise 221 Wm.Byrd, on a Negro Boy, with white Spots ibid Rob, Mawgridge, on a Stroke of Lightning . . 222 Edm. Halley's Observat. on a Lunar Eclipse . . ibid Stephen Gray, on Concave Specula ibid M. Leuwenhofck, on the Eggs of Snails 223 Wm. lierharn, on Portable Barometers, &c. . . 224- Biographical Notice of the Rev. Dr. Derham. . ibid Dr. Waliis, Effects of Lightning at Everdon . . 2:6 Tho. Molyneux, on the Stone of the Bladder 227 Ja. Cassini, Observ. of a Lunar Eclipse 228 Biographical Notice of James Cassini ibid Ace. of Commelin's Hortus Medicus Amst. . . ibid Biographical Notice of John Commelin ibid » of Frederick Ruysch .... 229 SirRob. Southwell, on preserving Flowers, &c. 230 Wm. Derham, on Measuring the Height ot the Barometer, by a Circular Plate 231 Geo Dampier, on the Biie of Mad Animals . . 232 Jas. Cassini, on Chinese Astron. Observations 233 Dr. Geo. Garden, on Caierpillars in IVees . . ibid Obseivatio de Fof-mina, qu,T, non obstante Vaginae uteri Coalesceiilia, lufantem pepcrit 231' Account of J. Evelyn's Numisniata, &c 235 . Casp. Bailholin's Specim. Phil. Nat. 23() P. J. Hailinann's Hist, et Exp. Fig. ibid Use of Ipecacuanha lor Loosenesses 237 Dr. Hans Sloaiie's Notes on the same 239 Page ob. Southwell, on a Monstrous Calf .... 240 Dr. Waliis, on the Division of the Monochord ibid Dr. Vaughan, on the Poison of Hemlock, &c. 2+2 Sir Rob. Southwell, on Tincturing Waters .... 243 Mr. Dale, on the Generation of Eels ^44, Account of Connor's History of Poland .... 24-7 Dr. Tyson, on the Anatomy of an Opossum. . 248 Benj. BuUivant's Ob.serv. in New ftlngland. . . . 26'7 XL Leuwenhoeck, on the Eyes of Beetles. . . . 26'8 Steph. Gray, on enlarging the Barom. Divisions 2()Q Dr. Musgrave, on Respiration 070 , on syringing Water into the Thorax 27 1 Dr. Plot, on the Substance called Black Lead 272 Dr. Musgrave, Irish Giant Edm. Malone . . . '. 273 Dr. Chr. Pitt, on Mercury injected into a Dog ibid Dr Waliis, on Easter-Day, &c ibid De Moivre, on the Root of an Infinite Equat. 275 Dr. Hallev, on an Extraordinary Rainbow .... 277 Account of Travels in South America 278 — — Alliot's 1 raite du Cancer, Sec. . . . 279 Ra. Thoresby, on .1 Roman Shield ibid Dr. Tho. Molyneux, on the Giants' Causeway 281 R. Vieusseiis, on the Human Blood 283 Account of J. D. Cassini's Meridian Line, &c. 286 Dr. Lister, on the Bite of a Mad Dog jbid Dr. Waliis, on the Imperfections in an Organ 287 Dr. Musgrave, on a Periodical Palsy 293 On an extraordinary Posture Master 294 James Newton, on the Papaver Comic. Lui. . . 295 Sir Rob. Sibbald, on Stones voided by a Boy. . ibid M. Witsen, on an Inundat. at the Mauritius. . 297 Concerning Irisli Slates 298 Observations on the East-Indies ibid Dr. Pitt, on the siomach and Guts 300 Edw. Lhwyd, on Figured Stnnes, and Lang.. . ibid Signior Redi, on Factitious baits 3(11 Dr. Rob. Conny, on a Shower of Fishes .... ."^os "I'hn. Bent, on making Pitch, Tar, &c ihici On voiding the Bones of a Foetus 303 Dr. Ashe, on the Barometer, &c ihij Concerning Rusma and Alcanna 304 Sir R. Southwell, giving Iron a Copper Colour ibid ■ , on Gilding Gold upon Silver 305 Dr. Waliis, on the Effects of Ancient Music ibid Account ot Chr. L. Welsch's Basis Botanica 307 J. Pflugk's Catal. Bibl. Budensis. . ibid Marsigli s Dis on Bologna Stone \b\^ Biograph. Noiice of Count Louis F. Marsigli ibid Ra. Thoresby, on a Roman Coffin, &c 3()g Sam. Brown, on some Indian Plants 310 Mr. Buttei field, on Magnelical Sand ibid Dr Lister's Object, to Leuwenhoeck's Hvpot. ibid Account of Fryer's Reflec on India and Persia3Ii A. Goodvear, on the Bile of a Serpent ibid Dr. Waliis, instructing Deaf and Dumb Persons.! 12 Mr. Witsen's Observations on New Holland 3l6 SirRob. Southwell's I'hilosoph. Exptrane ,s:il7 Dr. Cay, on Mineral Waiers at Egliiigham . . ibid M. Cassini, on the Longitude of Canton 313 J. Craig's Quadrature of the l.ogar. Curve. . . . 313 Benj. Allen, on the Gal) Bee gig , on the D.aih- Waich ibid Dr. Rob, St. Clair, on auLruplion of Fire, &c. 320 CONTENTS. Ill Page Dr. Rob. St. Clair, on a new kind of Lamp . . ibid R. P. on an Eruption of Water 322 Mr. Desmasters, Experiments on Freezing . . . ibid Dr. Plot's Catalogue of Electrical Bodies , . . 32.1 Account of Tournefort's Histoire des Plantes .ibid Biographical Notice of Jos. de Touniefort. . . . ibid Mr. Buckly, on a China Cabinet 32+ Ja. Petiver, on some Maryland Animals, &c. . ibid Capt. Lnngford's Observations on Hurricanes. . 330 Mr. Ballard, on the Magneti.sm of Diills .... 332 George Lewis, on some Indian Manuscripts . . 334. Dr. E. Baynard, Swallowing of 2 Farthings . . 333 R. Sault on the Curve of swiftest Descent .... ibid M. GeoftVoy, on Mineral Waters, &c 336' Biographical Notice of Ste. Francis Geoffroy ibid Experiments and Observations on Sounds. . . . 337 Mr. Desmasters, further Exper. on Freezing. . 340 Mr.Boaavert, a Stone at theRoot of the Tongue ibid Dr. Musgrave, on a Piece of Antiquity, &c. . .341 On the Oxford Catalogue of Manuscripts .... ibid Dr. Sloane, further Ace. of the China Cabinet 345 Ace. of Boccone's Museo di Plante Rara, &c. 34ti Geo. Camelli, on the Tugusor Amomum .... 347 Ph. Ja. Hartman's Account of Amber ibid D. Caslone, on the Generation of Flees 348 M. Geoffroy, on mixing Inflammable Liquors ibid Dr Sloane, again on the China Cabinet 349 Wm. Derham, on the Barom. and Weather . . ibid Rd. Townley, on the Quantity of Rain, &c. . . 350 Mr. Dale, on several Insects ibid Ra Thoresby, on a Man killed by Lightning. . 351 Account of Boccone's Museo di Fisica, &c. . . ibid . Dr. Herman's Paradisus Batavus 352 Dr. Sloane, again on the Chinese Cabinet .... ibid Mr. Maleverer, on Coal-Bormgs 353 M. Cassini, on a Comet observed at Paris . . . 354 Dr. Cay, on the Virtues of the O-tracites . . . 355 Fr. Joannes, on the Faba Sancti Ignatii 356 Fa. Camelli, on the same Seed ibid Wm. Clerk, on Stones in the Stomach, &c. . . 357 M. Bussiere, on Cutting for the Stone 358 Mr. Petto, on Some Solar Parhelia 36l SirPaul Ricault, on Sable Mice in Lapland. . . . ibid Dr. Sloane, on Some Plants in Jamaica 362 Dr. Fern, on a Fcetus without the Womb .... 365 Mr. Stephen Gray, on Solar Parhelia 367 Dr. Tho. Molyneux, on Scolopendra Marina. . 368 M. Dupre, on the Muscles in the Neck .... ibid , on a Deformed Human Skull .... 372 M. Bussiere, on a Child without a Brain .... 373 M. Geoffroy, Regulat. of the Royal Acad 374 Ace. of Allen's Nat. Hist, of Chalyb. Waters 375 Dr. Sloane, on a Dropsy in the Ovary ibid Wm. Cowper, on curing a cut Heel Tendon. . 376" Dr. Cockburn, on a Blister in Fevers 378 Wm. Aglionby, on the Nature of Silk 380 Two Propositions proposed to be answered . . ibid Mr. Llwid, on a Figured Stone 381 Sir CharUs Holt, on Swallowing Stones ibid Dr. John Woodward, Exptr. on Vegetation . . 382 Account of Savery's Sieam Engine 398 James Eraser, Account ot Loch-Ness ibid Dr. Clopion Havers, on Concoction 400 Jer. Jones, on the Moorish Cookery 407 * a Page Account of Dr. Wallis's Opera Math. vol. 3.. 410 Leuwenhoeck, on the Animal, in Sem. Masc. .412 Dr. Wallis's Correspond, with M. Leibnitz ... 413 Account of Charmoye's Origin of Nations. . . . ibid Dr. Wallis's Answer to Leibnitz's Letter .... 414 ■ on the MeridianLine 415 Tho. Luffkin, on the Numeral I'igures ibid James Petiver, on the Virtues of Herbs 4l6 James Cuninghame's Catalogue of Shells, &c 418 Leuwenhoeck on the Animal, in Sem. Human. 419 Ph. Lloyd, on Diseases in the North Nations 420 John Houghton's History of Coffee Ibid Dr. Freind, on a Hydrocephalus 423 Biographical Notice of Dr. John Freind ibid J. Cunningham, Barometer and Weather .... 426 Dr. Dan. Giegory, on a SolarEclipse ibid Dr. Lister, on the Origin of white Vitriol. . . . 427 Mr. Thoresby, on Mr. Greatrix's Cures ibid Account of Pat. Gordon's Geography 428 Huygens's Celestial Worlds .... 429 Dr. Tyson's Orang Outang .... 431 J. Lowthorp, on the Air's Refraction 432 Dr. Wallis, on the Julian and Gregorian Cal. . 434 Ld. Burleigh's Report on the Calendar 437 John Greaves's Reflections on the same ibid Credibility of Human Testimony 438 Account of Swammerdam de Apibus, &c. . . . 442 D'Amerique's Analysis Geomet. . ibid Biographical Notice of Swammerdam ibid Wm. Cowper, on 2 new glands 445 Dr. Vieussens de Organo Auditus 44S Dr. Wm. Musgrave, on Laryngotomy ibid Tho. LutFkin, on Cupping Glasses 451 Quadrature of Hippocrates's Lunula, by Wallis, Perks, Da. Gregory, and John Caswell .... 452 Dr. Da. Gregory's Defence of the Catenary 456 Sir J. Floyer, on Monstrous Pigs, &c 458 Biographical Notice of Sir JohnFloyer ibid Hugh Jones's Account of Maryland 460 On the Quadratix to the Circle 462 Leuwenhoeck, Circulat. in Tadpoles 46^ Dr. Molyneux, on a Bodkin in the Bladder. . 468 Sir Theo. Mayerne. on the Viper and Poisons 469 Account of a double Pear 470 Number of Births, Burials, sec. in March .... ibiJ Tho. Povey, on making Brass ibid Account of Geudron's Cure of Cancers ibia Dr. James Brewer, on beds of Oyster-Shells. . 471 Dr. Tho. Molyneux, on Giants ibid Number of Births, Burials, &c. in Branden. . 477 Leuwenhoeck, Worms in Sheeps' Livers, &c. ibid Capt. South, on the Houses, &c. in Dublin . .481 , No. of Seamen, &c. in Ireland. . ibid , No. of People in Ireland 4-82 , No. of Romish Clergy in Ireland ibid Dr. Wm. Sherard, on China Vanishes ibid Mr. Derham's Observations on the Weather . 483 M. Homberg, on Acid salts ibid Biographical Notice of M. Homberg ibid Mr. Gray, on a Parhelion and Halo 436 Dr. James Wallace, Account of Darien, &c. . 437 Ace. of Dr. Wallace, on the Orkney Islands. . ibid M. Bussiere, on a Polypus in the l.ungs 488 Dr. Wallis, on measuring curved Figures. . ibid 2 IV CONTENTS. Page Leuwenhoeck, Circulation of the Blood 491 Abr. de la Pryme, on Roman Antiquities. . . . 49+ Biographical Notice of Mans, de Chirac 497 Account of Chirac de Motu Cordis, &c. . . . ibid An Incubo Ferrum, &c 49S -_ An Passioni lliacae, &c ibid Steenvelt de Ulcere Verniinoso,&c. ibid G. Bidloo de Aniinalculis in Ovino, &c 499 Ace. of N. Chevalier on a Piece of AmbergrisodO Ja. Cunningham, on the Thermometer, &c. ibid Ra. Thoresby. on Thunder and Lightning .... ibid On some Indian Plants, Drugs, &c 501 M. Witsen, on Ratavian Mountains, &.c 50'2 Dr. James Bunough, on a Bulimia 503 J. M Laiicisi, on Acid Spirit in Blood ibid Account if Van Reverhorst de Motu Bills. . . . ibid . Van Kessel's Pharm.ic. Harlem. . 501 Silvesire, on New Boolis in Italy ibid Mr. Greenhlll's 4 Medico-Surgic, Cases ibid J. P. Worzelbaur, on a Solar Eclipse, &c. . . . ibid Demolvre, on the Solids of the Lunula 505 Dr. P. Silvesire, on the Learning of Italy .... 506 Leuu'enhoeck, on Worms in tlie Teeth 509 John Mourn, Catacombs in Italy 511 Account of Vnlckamer's Flora Noribergens . . 514 Chr. Hunter, on Koman Inscriptions Ibid Leuwenhoeck, Insects on Fruit Trees ibid Charles, King, on Crabs' Eyes 519 M .Poupart, on the Libella ibid Dr. WaDis, on the Numeral Figures 5'21 Abf de la Pryme, Shells, &c. in Lincoln .... ibid Dr. James Wallace, Stone from the Bladder. . 524 Dr. George Garden, on the same 525 Dr. Wm. Musgrave, on a Polypus in a Dog. . Ibid Rev. Mr. Gordon on the Cataract in Gottenburg River, and Tycho Brahe's (Jbservatory. . . . ibid Sir Rob. Slbbald, Stones and Plants in Scotland 5'2b' M. Lasage, Aneurism of the Arterla Aorta . . . ibid Ace. of Mr. Brown's '^d Book of India Plants 527 E. Halley, on the Rainbow, &rc ibid The Method of Colouring Marble 533 John Marshal, on the Indian Bramins, &c. . . 534 Leuwenhoeck, Animal, in Semine Masc. . . . 541 Rev. Jo. Craig, on the Solid of Least Resist- ance, and Curve of Quickest Descent .... 542 M. Bussiere, on a triple Bladder, &c 545 Dr. Manginot, on an unusual Medical Case. . 547 Mr. Clark, on some Roman Antiquities 548 Stephen Gray, on Fossils and the Merld. Line 549 Dr. Wallls, on Feeding oa Fiesh 550 Dr. Tyson, on the same Subject 552 Dr. Wallls, again on the same 55fJ Leuwenhoeck, Excrescences on Leaves 557 Dr. Sylvester, Dissecilon of a Woman, Jic. . 560 E. Halley, on Hook's Marine Barometer .... 561 Wm. Cowper on the Vena Pulmonalis 563 Dr. Freind on uncommon Convulsions 5(i4 Pair. Gordon, on a Water-Spout il)id J. Banister, on Insects in Viiginia 565 Stephen Gray, on a Meridian Line 568 Dr. Lister, on Powders passing the Lacleals. . .'170 A Scale of Degrees of Heat 572 Account of Cockburn, on Loosenesses 575 ■ — — — — Sanctoiius de StaticaMedicina. . . . 576 Page Accouut of Ray's General History of Plants. . 576 Wm. Derham, on the Death-Watch ibid Dr. Hale, on the Human Allantois 577 J. Petlver on Brown's 3d Book of India Plants 586 Dr Musgrave, Hemorrhage in the Thumb . . . ibid Leuwenhoeck, on the CEconomy of Spiders . . 587 Strange Bones dug up at Canterbury 599 Mr. Locke, extraordinary Mental Numbering 600 Dr. Davles, on voiding Hydatldes by Urine. . 601 Leuwenhoeck, on Tastes of Waters, &c ibid severalMicroscoplcal Ubserv. . . 602 M. Reneaume, on Walnut-Trees 6"03 J. Clamplni, on ihe Asbestos 604 Account of Gavell, on Fevers 606 Sanguineti, Dissert. Jatroph ibid J Luftkin, on large Bones, evc ibid J. Petlver, on Brown's 4th Book of India Plants 6O8 Dr. J. del Papa, EtFecIs of Indian Varnish . . . ibid M. Geotfrov, on Cold Solut. and Ferment . . . 61I Dr. Davles, on an unusual Cullc 618 Dr. Wallls, Isthmus between Dover and Calais ibid Abr. de la Pryme, on Subterraneous Trees . . . 624 Sir Cha. Holt, on Intestines in the Thorax, &c.6'30 Dr. Musgrave colouretl Liquor in Lacieals. . 632 Opium taken without causing Sleep 634 Chr. Birbeck, on a Foetus voided by the Navel ibid Mr. Wilson, on the Asbestos 6"35 J. Petlver, on Brown's 5th Bonk of India Plants636 Dr. Wallls, again on the Dover Isthmus. . . . 437 , on the Mariners' Compass ....ibid Account of Marslgll Operls Prodrom 640 Mr. Strachan on taking Elephants 641 J. Petlver, on Brown's6th Book of India Plants643 J Kay, on a strange Cancer jbid Ra.Thoresby, on seveial Curiosities • 64.4. Abr. de la Pryme, on Subterraneous Trees . . 645 .Alex Stuart on some Water-Spouts 647 Account of Dickenson's Physica Vetus et Vera 650 Mr Strachan's Observations at Ceylon ibid M. Blondel, on the Royal Academy at Paris . . 651 Dr. Wallls, on Magnetism 555 Demolvre on squaring Curves 653 E. Halley, on several Parhelia ggj, Chr. Hunter, on floman Antiquities (,'(;g Mr. Strachan, on Tobacco at Ceylon (j^j Leuwenhoeck, on Anlmalc. in Semine Masc. 668 Mr. Thoresby, on sonii- Roman Coins, &c. . . . 675 Sir J. Floyer, on Sweet Tastes (ij(j Dr. Charles Leigh, on Epileptic Fits 679 Mr. Cowper, on the Arteries and Veins .... 680 James Cuiniingham, on Chinese Customs. . . . 693 James Yonge, on the Internal Use of Cantha- ridts 696 Abr. de la Pryme, on Vegetation 6Q7 Dr. Drake, Motion of the Heart 698 On Mr. Wilson's Microscopes jQg Abr. de la Pryme, on a Water- Spout jbid H. Vanghan, on Swallowing F'rnit Stones. ... 710 Mr. Strachan's t_)bservations at Ceylon ^u J. Petlver, on Brown's 7th Book of India Plants7l2 Dr. Molyncux, on the Ancient Lyre jbjd Ja. Yonge, on a Plum-Stone in the Bowels . . 715 Dr. Sloane, on Swallowing Fruit Stones .... "17 Mr. Thoresby, Vestiges of a Roman Town , . 7JS THE CONTENTS CLASSED UNDER GENERAL HEADS. Class I. Mathematics. 1. Arithmetic, Political Arithmetic, Numbers of Persons, Annuities. Et-age XTRACTING Surd Roots, Dr. Wallis . . 1 Construction of Logarithms, by Halley 18 Old Jenkins's Age, by Dr. Robinson 92 same by Mr. Hill 167 Number of Births, Deaths, &c. Dr. Slare. . . . 169 The Numeral Figures, Tho. LufFkin 415 Credibility of Human Testimony 438 Number of Births, Deaths, &c. in March 4/0 Page Numb, of Births, Deaths, &:c. in Brandenbtirg 477 Houses, &c. in Dublin, Capt. South 481 Seamen in Ireland, by the same . . ibid People in Ireland 482 Romish Clergy ibid The Numeral Figures, Dr. Wallis 521 '2. Algebra, Analysis, Fluxions. Use of Fluxions, by Demoivre 14 Dr. Wallis's Mathematical Works 29 Solution of Two Problems, by Newton 129 The Multinomial Theorem, Demoivre 1 76 Roots of Infinite Equations, Demoivre 275 On Wallis's Opera Mathematica 410 3. Geometry. Quadrable Cycloidal Spaces, Wallis 39 Cycloids and Epicycloids, Halley 47 Merid line and Log. Tangents, Halley 68 Two Problems solved, by Newton . . 129 Descents in a Cycloid, inc. Newton 140 Antiquity of the Cycloid, Wallis 169 Errors in Surveys, Molyneux 180 The Caten.irian Curve, Da. Gregory 184 Quadrature of Curves, Craig 202 Quadrature of the Logarith. Curve, Craig. . . .318 Curve of Swiftest Descent, Savdt 335 D'Amerique's Analysis Geomet 442 Quadrature of Lunes, Wallis, Perks, &c. . . .450 Defence of the Catenary, Gregory 456 Quadratix to the Circle 462 Measure of Curved Figures, Wallis 488 Solids of the Lunula, Demoivre 505 Solid of Least Resistance, Craig 542 Curve of Quickest Descent, Craig ibid Quadrature of Curves, Demoivre 658 Class I J. Mechanical Philosophy. ] . Astronomy. The Sun's entering the Tropics, Halley, .... 5 Dampier's Voyage round the World 141 Voyage to Constantinople, Dr. Smith 176 On a Lunar Eclipse, Halley 222 Chinese Astronomical Observations, Cassini. 233 On finding Kasler-Day, &c. Wallis 273 The Meridian Line, Cassini '286 The Longitude of Canton, Cassini 318 A new Comet observed, Cassini 354 On the Meridan Line, Wallis 415 On a Solar Eclipse, Gregory 426 On Huygens's Celestial Worlds 429 Julian and Gregorian Calendar, Wallis 434 Report on the Calendar, Lord Burleigh 437 Reflections on the Calendar, Greaves ibid A Solar Eclipse, &c. Worzelbaur 504 On a Meridian Line, Ste. Gray 549 On the same, by the same 568 2. Projectiles. A Proposition in Gunnery Vi CONTENTS. 3. Hydraulics. Page Pa Re On a Water Spout, Patr. Gordon 56'4. On a Water Spoilt, Abr. de la Pryme 709 On Sonne Water Spouts. Alex. Stuart 6i7 4. Pneumatics. On the Barometer, &c. Sir Wm. Beeston ... 79 Portable Barometers, Derham 224 Measure of Snowden Hill by the Barometer, Circular Barometer, Derham 231 Halley 174 Refraction of the Air, Lowthorp 432 Refractionof the Air, &c. Bilbery 213 Hook's Marine Barometer, Halley 561 5. ylcouslics. Music. Division of the Monochord, Wallis 240 Experiments and Observations on Sounds. . . . 337 Imperfect, in an Organ, Wallis 287 On the Ancient Lyre, Dr. Molyneux 712 On the Ancient Music, Wallis 305 6. Optics. On Gregory's Catoptrics and Dioptrics 77 On Concave Specula, Gray 222 Microscopical Experiments, Gray 97 Refraction of the Air, Lowthorp 432 The Water Microscope, Gray 120 Microscopical Observations, Leuwenhoeck . . . 6'02 A Water Lens and Microscope, Gray \66 On Mr. Wilson's Microscopes 709 Refraction of the Air, Bilbery 213 7. Magnetism. Some Magnetical Experiments 161 Magnetism of Drills, Ballard 332 Errors from the Magnet. Variat. Molyn 180 The Mariner's Compass, Wallis 639 Magnetical Sand, Butterfield ... 3 10 On Magnetism, Dr. Wallis 655 Class III. Natural History. 1 . Zoology. Animalcules in Water, Harris 89 Animal, in Sem. Masc. Leuwenhoeck 412 On Eels, Mites, &c. Leuwenhoeck 94 Animal, in Sem. Hum 419 The Scolopendra Marina, Dr. Molyneux 133 Swammerdam de Apibus 442 The Long Worm in India, Dr. Lister 137 Monstrous Pigs, &c. Sir J. Floyer 453 Generation of Eels, B. Allen 196 On Giai ts. Dr. Tho. Molyneux 47 1 Insects in Ireland, Tho. Molyneux 2l6 Worms in Sheep's Livers, Leuwenhoeck .... 477 Bones of Elephants, E. Tentzel 218 Animalcula in Ovino, &:c. Bidloo 499 Caterpillars in Trees, Dr. Garden 233 Worms in the Teeth, Leuwenhoeck 509 On the Gall-Bee, Benj. Allen 319 Insects on Fruit Trees, the same 519 On the Death Watch by the same ibid On Ciabs" Eyes, Cha. King ibid Maryland Animals. Ja. Peliver 324 Animal, in Sem. Masc. Leuwenhoeck 541 On several Insects, Mr. Dale 350 Excrescences on Leaves, the same 557 Lapland Micf, Sir Paul Rycault 36 1 Ins ects in Virginia, J. Banister 565 Extra-uterine Foetus, Dr. Fern 365 On the Death Watch, Derham 576 Scolopendra Marina, Dr. Molyneux 368 Animal, in Sem. Masc. Leuwenhoeck 668 2. Botany. Catalogue of Tangier Plants, Spotswoood ... 85 On Rusma and Alcanna 304 Catalogus Plantarum, Dr. Sloane 103 Basis Botanica, C'hr. L. Welch 307 Beans at the Orkneys, Dr Sloane ibid On some India Plants, Brown 310 On Grasses for Hay, Dr. Lister 136 Hisioire des Plantes, Tnurnefort 323 Almagestum Botan Plukenct 141 On the Tugiw or Aniomnm, Camelli 347 On Hemlock, &c. J. Ray 1 83 Paradisus Balavus, Dr. Herman 352 Catalogue of Guinea Plants., Petiver 201 Faba baiicii Ign^lii, Fr. Joannes, . 356 CONTENTS. Vll Page On the same, by Fa. Camelli 356 On Jamaica Plants, Dr. Sloane 3t)2 Question proposed on Plants 380 History of Coffee, Houghton 4'20 Some India Plants, Drugs, &c 501 Flora Noribergens, Volckamer 514 Plants, &c. in Scotland, Sir R. Sibbald 526 Brown's 2d Book of India Plants, Petiver. . . . 527 Page History of Plants, Ray 57 6 Brown's 3d Hook of India Plants, Petiver .... 586 On Walnut Trees, Reneaume 603 Brown's 4th Book of India Plants, Petiver . . . 6()8 5th ditto 636 6th ditto 643 On Tobacco at Ceylon, Strachaii 667 Brown's 7ih Book of India Plants, Petiver . . . 712 3. Mineralogy. On Woodward's Natural History 41 A Butter-like Substance, K. Vans 78 On the same, by the Bp. of Cloyne ibid Soap-Earth near Smyrna, Smith 80 On Shells in Scotland, Sir R. Sibbald Ill Fos'iil Wood in Yorkshire, Richardson 162 Pitch, Tar, &c. from a Stone, Mr. Ele 168 Piece of Ambergris, Tredevey 205 Moving Bog in Ireland 206 On the same, Wm. Molvneux ibid On Black Lead, Dr. Plot' 272 On i rish Slates 298 On Factitious Salts, Redi 301 On Rusma and Alcanna 304 The Bologna Stone, Marsigli 307 Mineral Waters, Dr. Cay 3 1 7 Mineral Waters, by M. Geoffrey 336 Account of Amber, Hartman 34.7 On Coal-borings, Maleverer 353 Virtues of Ostracites, Dr. Cay 355 On a Figured Stone, Llwid 381 Catalogue of Shells, &c. Cuninghame 418 Origin of White Vitriol, Dr. Lister 427 Beds of Oyster-shells, Dr. Brewer 47 1 On a Piece of Ambergris, Chevalier 500 Shells, &c. in Lincolnshire, de la Pryme 521 Stones, &c. in Scotland, Sir R. Sibbald 526 On some Fossils, Ste. Gray 549 On the Asbestos, Ciampini 604 Subterranean Trees, de la Pryme 624 On the Asbestos, Mr. Wilson 635 Subterranean Trees, de la Pryme 645 4. Geography and Topography. Burning Mountain in Ternate 13 The same in the isle of Sorea ibid On Tadmor, or Palmyra, Halifax 33 Journey from Aleppo to Palmyra 49 Ancient State of Palmyra, Halley 60 New Map of Fr..nce 142 Burning Mountains in the Moluccas ]63 On the Isle of Portland, Sir R. Southwell 198 On Cluverius's Geography 200 Observations on Cape Corse, Hillier 201 Moving Bog in Ireland, Wm, IVlolyneux .... 206 Western Isles of Scotland, Martin 212 History of Poland, Connor 247 Observations on New England, BuUivant. . . . 267 Travels in South America 278 Inundation at the Mauritius, Witsen 297 Observations on the East Indies 298 On India and Persia, Fr5 er 311 Observations on New Holland, Witsen 3l6 Account of Loch-Ness, Ja. Fraser 398 On Mr. Pat. Gordon's Geography 428 Account of Maryland, Jones 46O Account of Darien, cVc. Wallace 487 On the Orkney Isles, Wallace ibid Cataract at Gotlenburg, Gordon 525 Tycho Brahe's Observatory, Gordon ibid The Dover Isthmus, Dr. Wallis 61 8 On the same by the same 637 5. Hydrology. Theory of the Tides, Halley , 142 A Medicated Spring, Aubrey 211 The Wells in JModena, Ramazzini 213 Inundation at the Mauritius 297 On Mineral Waters, Dr Cay 317 Eruption of Water, by R. P. 322 On Chalybeate Waters, Allen 375 Tastes of Waters, Leuwenhoeck 601 Class IV. Chemical Philosophy. 1 . Chemistry. On Tincturing Waters, Sir R. Southwell 243 Origin of White Vitriol, Dr. Lister 427 Expei iuients on Freezing, Desmasters 322 On Acid Salts, by M. Homberg 483 Further Experiments on ditto, by the same . . . 340 On Ambergris, by N. Chevalier 50O Mixing Inflammable Liquors, Geotfroy 348 Acid Spirit in Blood, Lancisi 503 Natural History of Chalybeate Waters, Allen 375 On Colouring Marble 533 vm CONTENTS. Scale of Degrees of Heat Solutions and Fermentations, del Papa Page , 57'2 On Sweet Tastes, Sir J. Floyer. ,6U Page , 67f> 2. Meteorology. On a Water Spout, by Z. Mayne 1'2 Great Thunder at Aberdeen, Dr. Garden, . . . IO9 Quantity of Rain at Gresham College 121 Large Hail at Chester, Halley 171.172 Hail Storm in Hertfordshire, Taylor ibid Hail Storm in Herefordshire 173 On the same in Monmouthshire ibid On Hail and Thunder, Dr. Wallis 196 A Storm in America, Scarburg 198 Hail and Thunder, Dr. Wallis .212 On a Stroke of Lightning, Mawgridge 222 On Lightning at Everdon, Wallis 226" Uncommon Rainbow, Halley 277 An Eruption of Fire, Dr. St. Clair 320 Catalogue of Electrical Bodies, Plot 323 Observations on Hurricanes, Langford .330 The Barometer and Weather, Derham J49 On the Quantity of Rain, Townley 350 Death by Lightning, Thoresby 351 On Solar Parhelia, Gray 367 The Barometer and Weather, Cunningham . . 426 On the Weather, Wm. Derham 483 A Parhelion and Halo, Gray 480 On Thunder, &c. Ra. Thoresby 500 On the Rainbow, &c. E. Halley 527 On Some Parhelia, E. Halley GiSi '- 3. Geology. On the Giant's Causeway, Molyneux 281 On Batavian Mountains, Witsen 502 Class V. Physiology. 1. Analomy. Anat. of the Brain, Ridley 13 A Brain depressed into the Neck 164 A Sphacelated Brain, &c. Tyson i6j Tongue of a Pastinaca Marina 200 Anatomy of an Opossum, Tyson 248 On the Stomach and Guts, Pitt 300 The Scolopendra Marina, Molyneux 368 A Deformed Human Scull, Pitt 372 A Child without a Brain, Bussiere 373 A Prop, proposed to be resolved 380 The Orang Outang, Dr. Tyson 431 On the Ear, by Dr. Vieussens 448 On a Triple Bladder, Bussiere 545 The Vena Pulmonalis, Cowper 563 On the Human Allantois, Dr. Hale 577 Strange Bones at Canterbury i^<^ On some Large Bones, Luff kin 6'06 Intestines in the Thorax, &c. Holt 630 On the Arteries and Veins, Cowper 680 2. Physiology of Animals. Rheumatism and Suppression of Urine 9 On Vertebrae firmly united, Connor 10 A Premature Child, by Dr. Sampson 31 On Chvlification, Wm. Cowper 81 Child born with a Wounded Breast 102 On a Diseased Kidney, Cowper 105 On Stones of the Bladder, Preston 109 On an Extra-uterine Foetus 110 A Whelp voided, &c. Halley ibid On the Dropsy, &c. Dr. Preston 114 The Hydrops Pectoris, S. Doudy 131 History of a Tumor, &c. Giles 132 The Long Worm in India, Lister 137 Internal Parts of Fish, Preston 138 A Child without a Brain, Preston 1 49 Orign of a Polypus, Giles 152 Bones voided per Anum, Morley 155 Large Horns dug up in Ireland 156 A Brain depressed into the Neck l6"4 A Sphacelated Brain, &c. Tyson i6.5 Dissection of the Scallop, Lister 170 Foetus voided by the Navel 173 Horny Excrescences on a Person 176 A small Egg within another 183 The Generation of Eels, Allen 199 Anatom. History of the Leech, Poupart 209 Venom in a Porpus Tooth, Lister, 211 Polypus of (he Lungs, R. Clarke 221 Dog killed by a Sudden Noise, Clarke ibid A Negro Boy with White Spots ibid On the Eggs of Snails, Leuwenhoeck 223 The Stone of the Bladder, Molyneux 227 Caterpillars in Trees, Dr. Garden 233 De FcEmina qu.p, non obstante Vaginse Uteri Coalescentia, Infantem peperit 234 On a Monstrous Calf, Southwell 240 Generation of Eels, by Dale 244 On the Eyes of Beetles, Leuwenhoeck 268 CONTENTS. IX Page On Respiration, Dr. Musgrave '^70 On Malnne tlie Irish Giant 273 Mercury injected into a Dog ibid On the Human Blood, Vieussens 283 Bite of a Mad- Dog, Dr Lister 286 A Periodical Palsy, Musgrave 293 Extraordinary Posture- Master 294. Stones voided by a Boy, Sibbald 295 On the Stomach and Guts, Dr. Pitt 300 On a Shower of Fishes, R. Coning 302 On Voiding the Bones of a Fcetus 303 Object, to Leuwenhoeck's Hypoth 310 The Bite of a Serpent, Goodyear .Ul The Gall-Bee, Benj. Allen 310 The Death- Watch, by the same ibid On Swallowing Farthings, Baynard 335 Generation of Fleas, Cestone 348 Stones in the Stomach, &c. Clerk 357 Prop, proposed to be answered . , 380 Page On Concoction, by Dr. Havers 400 The Animal, in 'em. Masc. Leuwenhoeck ..412 The same in Sem. Hum 4.19 On two New Glands, Covvper 445 Circulat. in Tadpoles, Leuwenhoeck 464 On Vipers and i^oisoos, Mayerne 469 Circulat. of the Blood, Leuwenhoeck 491 De Motu Cordis, &c. Chirac 497 On a Bulimia, Dr. Burrough 503 Acid Spirit in Blood, Lincisi ibid De Motu Bills, Van Reverhorst ibid On Feeding on Flesh, Dr. Wallis 550 On the same, by Dr. Tyson 552 On the same, by Dr. Wallis 5j6 On the Death-Watch, Derham 576 Economy of Spiders, Leuwenhoeck 587 Inject. Liquor into the Lacteals 632 On the Elephant, Strachan 6'41 Motion of the Heart, Dr. Drake 698 3. Physiology of Plants. The Juices of Plants, Lister 123 Grasses useful for Hny, Lister 136 Hemlock and a Poisonous Root 183 Hortus Medicns Amst. Commel 228 The Poison of Hemlock, Vaughan 242 The Papaver Cornic. Lut 295 Exper. on Vegetation, Woodward 382 On the Virtues of Herbs, Petiver 4l6 Account of a Double Pear 470 On Vegetation, Abr. De la Pryme 6^7 4. Medicine. De Sails Calhar, Dr. Grew 31 Dissert, de Febribus, Pitcairn 46 Dissert. Medico-Physicae 77 On Chyhfication, Wm. Cowper 81 Use of Opium by the Turks 101 Diseases of Seamen, Cockburn 154 Malpighi's Posthumous Works 168 Observ. on Maladies, Gaillard 207 Hortus Medicus Amst. Commelin . .... 228 Bite of Mad Animals, Dampier 232 Use of Ipecacuanha for Looseness 237 Notes on the same, by Dr. Sloane 239 Bite of a Mad-Dog, Lister 286 A Periodical Palsy, Musgrave 293 Diseases of Northern Nations 420 On Greatrix's Cures, Thoresby 427 An Incubo Ferrum, &c. Chirac 498 An Passioni Iliacae, &c. Chirac ibid Pharmacop. Harlem. Van Kessel 504 Four Medico-Surgical Cases ibid Unusual Medical Case 547 Uncommon Convulsions 564 On Loosenesses, Cockburn 575 De Statica Medicine, Sanctorius 576 On levers, by Gaveti 606 Effects of India Varnish 608 An Unusual Colic, Dr. Davies 618 Opium without causing Sleep 634 Physica Vetus et Vera, Dickenson 650 On Epileptic Fits, Dr. Leigh 67Q Internal Use of Cantharides, Yonge 6QIS 5. S. urgery. A Horse Staked in the Stomach 64 Two large Stones extracted, &c 86 On a large Diseased Kidney, Cowper 105 On Stones in the Bladder, Preston 109 On the Dropsy, &c. Dr. Preston 114 Cutting for the Stone in the Kidney 1 16 On Opening the Body of a Boy 122 On an Hydrops Pectoris, Doudy 131 History of a Tumour, &c. Giles 132 On Malpighi's Death, &c 151 Qrigin of a Polypus, Giles 152 nes voided per Anum, Morley 155 VOL. IV- Dissection of the Scallop, Lister . . . . , 170 Fostus voided by the Navel, Preston 173 Polypus of the Lungs, Clarke 221 Stone of the Bladder, T. Molyneux 227 Syringing Watir into the Thorax 271 Treatise on the Cancer, &c. Elliot 279 On Voiding the Bones of a Foetus 303 Stone at the Root of the Tongue 340 Stones in the Stomach, &c. Clerk 357 On Cutting for the Stone, Bussiere 353 Dropsy in the Ovary, Dr, Sloane 375 Curing a cut Heel-Tendon, Cowper 37t)' b CONTENTS. Page A Blister in Feveis,Cockbnrn 378 On Swallowing Stones, Sir Cha. Holt 3S1 On the HydrocephiiUis, Dr. Freind 423 On the Ear, by Dr. Vieussens 448 On Cupping-Glasses, T. LufFkin 451 A Bodkin in the Bladder, Molyneux 46'8 Cure of Cancers, by Gendron 470 Polypus in the Lungs, Bussiere 488 De Ulcere Verrainoso, &c. Steenvelt 4.98 Four Medico-Surgical Cases, Greenhill 304 Page Stone of the Bladder, Ja. Wallace 524 On the same, by Dr. Geo. Garden 525 Hemorrhage in the Thumb, Musgrave 586 Voiding Hydatides by Urine, Davits 601 Foetus voided by the Navel, Birbeck 634 An unusual Cancer, J. Ray 643 On Swallowing Fruii-Stones, Vaughan 7)0 A Plum -Stone m the Bowels, Yonge 713 On Swallowing Fruit-Stones, Sloane 717 Class FI. The Arts. 1. Mechanical. Collection of Machines, D. Papin 154 Enlarging the Barom. Divisions, Gray 26"9 Gilding upon Silver, Southwell 305 Philosophical Experiments, Southwell 317 A New kind of Lamp, St. Clair 320 On a China Cabinet, Buckly 334 On the same. Dr. Sloane 34 5, 349, 352 On Savery's Steam-Engine 393 On several Curiosities, Thorcsby 644 Roman Antiquities, Chr. Hunter 606 1. Chemical, A Red Colour by a Mixture, &c. Coles I(i7 On making Pitch, Tar, i;c. Tho. Bent 302 Gilding upon Silver, Southwell 304 On the Moorish Cookery, Jones On making Brass, Tho. Povey. . . .407 .470 3. The Fine Jrts. On Preserving Flowers, &c. Southwell' 230 4. Ant Journeys from Aleppo to Palmyra 49 Ancient Slate of Palmyra, Halley 60 On Kennel's Parochial Antiquities 92 On a Roman Pottery, Ra. I'horesby ill Antiquities in LincobTiliire, Mirrtt 1 l7 On Seller's History of Palmyra 122 Musei Petiveriani Centuria, \c 132 Large Horns dug up in Ireland 156' Fossil Wood in Yo\ Ushire l62 Ancient Inscriptions at Rome 165 Memoirs of China, Le Comte 175 Two Roman Altars, Thoresby 198 On Roman Antiquities, Ihoresby 215 Bones of an Elephant dug up, Icntzel 21S iqjiilies. On Evelyn's Nnmismata, &c 235 On a Roman Shield, Ra. Thoresby 279 Figured Stones and Languages 30O On a Roman Coffin, &c. Ra. Thoresby 309 A piece of Antiquity, ic. Musgrave 341 Origin of Nations, Charmoye 413 Roman Antiquities, Abr. De la Pryme 494 Catacombs in lisly, John Monro 511 Roman Inscriptions, Chr Hunter 514 The Numeral Figuies, Waliis 521 Shell ■, &c. in Lincolnshire, De la Pryme .... 521 Roman Antiquities, Clark 543 Roman Coins, i.S:c. Ra. Thoresby 675 Vestiges of a Roman Town, Thoresby 71 8 Class Vn. Education, Liteuakv Chap.acters, Mokal Philosophy. Manners, Customs. On Languages, &c. Edward I.hwyd 300 Teaching the Deaf and Dumb, Waliis 312 On Indian Manuscripts, Lewis 334 Catalogues of Oxford Manuscripts 341 Regulat. of ihe Royal Academy . 374 Correspondence with Leibnitz, Waliis 413 Leibnitz's Answer to Waliis 414 On Uie Numeral Figures, LulTkin 415 New Books in Italy, Silvestre . . On the Learning of Italy, Silvestre The India Brahmins, iS.c.Marbh;il On Mental Numbering, Locke . . Observ. at Ceylon, Sirachan .... Royal Academy ai Paris, Blondel ("hinese Cusloms, i\c. Cunningham. Observ. at Ceylon, Strachan . . .504 ,506 , 534 .600 650 .651 .'11 CONTENTS. XI Class Fill. Bibliography ; or, Account of Books. Page Bartholin's Specim. Pliil. Nat 236 Boccone's Museo di Plante Rare 346" I Museo di Fisica, &c 3a 1 Connor's Dissert. Medico- Pliysicse 77 Celsus de Vita et Rebus, &c 113 Cockburn, Diseases of Seamen 154 Le Comte, Memoirs of China 175 Cluverius-'s Geography 200 Commelin, Hortus Medicus Amst 228 Connor, History of Poland 247 Cassini, Meridian Line, &c 286 Chirac, De Motu Cordis, &c 497 , An Incubo Ferrum, &c 498 — , An Passioni Ihacae, &c. , ibid Chevalier, on a Piece of Ambergris 500 Cockburn, on Loosenesses 575 Dampier, Vo; age round the World 141 D'Amerique, Analysis Geometrica 442 Dickenson, Pbysica Vetus et Vera 650 Evelyn, Numismata, &c 235 Grew, Tract, de Salis Cathar 31 Gregory, Catoptr. et Dioptr. Elem 77 Gordon, Geography 428 Gaveti, on Fevers 606 Hartmann, Hist, et Explic. Fig. &c 236 Herman, Paradisus Batavus 352 Huygens, Celestial Worlds 429 Page Kennet, Parochial Antiquities 92 Kessel, Pharmacop. Harlem , 504. Malpighi, Posthumous Works 168 Marsigli, Operis Prodrom 640 Pitcairn, Dissertat, de Febribus 46 Plukenet, Almagestum Botan 141 Papin, Collection of Machines 154 Potter, Lycophronis Chalcid. Alex l6l Pflugk, Catal. Bibl. Budensis 307 Ridley, Anatomy of the Brain 13 Reverhorst, De Motu Bills 503 Ray, General History of Plants 576 Scilla, La Vana Speculat. &c dQ Sloane, Catalogus Plantarum 103 Seller, History of Palmyra 122 Swammerdam, De Apibus, &c 442 Silvestre, New Books in Italy 504 Sanctorius, de Statica Medicina 576 Sanguineti, Dissertat. Jatroph 606 Travels in South America 278 Tournefort, Histoire des Plantes 323 Tyson, Orang Outang 431 Wallis, Mathematical Works 29 Woodward, Natural History 41 Welsch, Basis Botanica 307 Wallis, Opera Math. vol. 3 410 Class IX. BiOGKAPHY ; or, Account of Authors. Page Bernouilli 129 Connor 10 Cluver 200 Cassini 228 Commelin ibid Chirac 497 Page Demoivre l-t Dampier 141 Derham 224 Freind 423 GeoftVoy 336 Homberg 483 Page Lancisi 151 Magliabechi 218 IMarsigli 307 Pitcairn 46 Petiver 132 Potter l6l Page Poupart 209 Ramazzini 213 Ruysch 229 Swammerdam .... 4+2 Tournefort 323 Woodward 41 REFERENCES TO THE PLATES IN VOLUME IV. Plate 1, Fig. I, p. 6; II l6;lU, IV, 17; V, 18; VI, 39; VII, 39; VIII, 40; IX, 40; X, 47; XI, XII, 70; XIII, 88. II, . . I to VII, 66 ; VIII to XIV, 67. III, ., I, 93; II, 100; III, 120; IV, V, 130; VI, 135; VII, 136; VIII, 140; IX, 145; X, 146; XI, 157- IV, .. I, l66; II, 171; III, 181; IV, 184; V, 187; VI, 191. V, ..I, 202; II, 203; III, 207; IV, 231; V, 269; VI, 267; VII, 318; VIII, IX, 320 ; X, 335. VI, . . I to XVI, 266. VII, .. I to VI, 348; VII to IX, 358; X to XIII, 357;* XIV, 363. ,. VIII, .. I, 370; II, III, IV, 371 ; V, 376; VI, 378; VII, VIII, 381. IX, . . I, II, 39s ; III, IV, V, 432. X, .. I, II, 415 ; III, 417; IV, 415 ; V to XIII, 451; XIV, 452 ; XV, XVI, 472; XVII, 473. XI, .. I, 452; II, III, 453; IV, 455; V, 456; VI, VII, 457; VIII, 463; IX, 464; X, 465; XI, 466; XII, 466; XIII, 467. XII, .. I, 478; 11,479; HI, 484; IV, 487 ; V to VIII, 492; IX to XI, 493; XII to XIV, 489; XV, XVI, 505; XVII, XVIII, 528; XIX, 541. .. XIII, .. I, 542; II, 543; III, 544; IV, 546; V, 557; VI to X, 559, XI, 560; XII, 560 ; XIII, 562. , . XIV, . . I, 564 ; II, 563 ; III, 564 ; IV, V, VI, 662. XV, .. I, 581; II, 582; III, 583 ; IV, V, 588; VI, 589; VII, 590; VIII, IX, 592; X, XI, 596; XII, 632. XVI, .. I, 616; II to X, 649; XI, 664; XII to XVI, 669 : XVII, 671 ; XVIII, 672 j XIX, 673. XVII, .. I, II, 686; 111,689; IV, 69O; V, VI, VII, 691 ; VIII, 692. * ERRATUM. Page 357, line 5 from the bottom, add, Plate VII, fig. 10, 11 represent the leaves of this plant; fig. 12 the flower ; and fig. 13 the fruit. THE PHILOSOPHICAL TRANSACTIONS OF THE ROyAL SOCIETY OF LONDON; ABRIDGED. On the Methods of approximation in the Extraction of Surd Roots. By John Jfallis, S. T. D. and Saviiian Professor of Geometry at Oxford. N° 213, p. 2. Fol. XIX. J. HE several methods of approximation, which have been mentioned of late years, for extracting the roots of simple or affected equations, give occasion to say somewhat on that subject. It is agreed by all, and I think demonstrated by the Greeks long ago, that if a number proposed be not a true square, it is in vain to hope for a just quadratic root of it, explicable by rational numbers, in- tegers, or fractions. And therefore, in such cases, we must content ourselves with approximations, without pretending to accuracy. And so for the cubic root, of what is not a perfect cube. And the like for superior powers. Now the ancients had their methods of approximation in such cases ; some of which have descended down to us. But since the methods of decimal frac- tions have come into practice, it has been usual lo prosecute such extractions in the places of decimal parts, to what accuracy we please. Mr. Newton's method of approximation for extracting roots, even of affected equations, I have given some account of in my English Algebra ; and somewhat more fully in the Latin edition ; where I gave an account also of Mr. Raphson's method. Since which time, M. de Lagny has published his Method of Ap- proximation, principally for single equations, or extracting the root of a single power. And Mr. Halley has since improved this method, with a further ad- vantage, especially as to affected equations. These may all, or any of them, be of use for making more speedy approaches, and by greater leaps, in many cases, than Vieta's method, pursued and improved by Mr. Oughtred and Mr. Harriot of our own country, and by others abroad ; VOL. IV. B 2 PHILOSOPHICAL TRANSACTIONS. [aNNO l6g4-5. especially as to simple equations, if we suppose such extractions to be pursued to the full extent. But if we make use of Mr. Oughtred's expedient, for multiplication, divi- sion, extraction of roots, and other like operations, by neglecting so much of this long process, as is afterward to be cut off and thrown away as usekss which I think is generally practised, the work will be much abridged, and the advantage of the other methods much less considerable. And if we further consider, what preparative operations are to be made in some of those other methods, before we come to the prescribed division for giving the root desired ; the advantage will not be so great as may at first be apprehended. But, without disparaging these methods^ what I here intend, is, to show the true foundation of the methods used by the ancients, and the just improvement of them. Which though anciently scarce applied beyond the quadratic, or perhaps the cubic root, yet are equally applicable, by due adjust- ments, to the superior powers also. . I shall begin with the square root, for which the ancient method is to this purpose : from the proposed non-quadrate, suppose n, subtract the greatest square in integers, suppose a^. The remainder, suppose b = 2 A E -(- e'^, is to be the numerator of a fraction, for designing the near value of e, the re- maining part of the root sought, viz. a -|- e = v' n, whose denominator or divisor is to be 2 A, or 2 A -|- 1, the true value falling between these two ; some- times the one, sometimes the other, being nearer to the true value. But the latter is commonly directed. This method M. de Lagny affirms to be more than 200 years old : and it is so ; for I find it in Lucas Pacciolus, otherwise called Lucas de Bergo, or de Burgo Sancti Sepulchri, printed at Venice in the year 1494, if not even sooner, for I find there have been several editions of it; and it seems much older, for he does not deliver it as a new invention of his own, but as a received practice, and derived from the Moors or Arabs, from whom they had their algorism, or practice of arithmetic by the ten numeral figures now in use. And it is continued down hitherto in books of practical arithmetic in all languages, which teach the extraction of the square root, and this method of approximation, in case of a non-quadrate. The true ground of the rule is this : a'^ being the greatest integer square contained in n, it is evident that e must be less than 1. Now if the remainder B = 2 A E -f e'- be divided by 2 a, the result will be too great for e ; but if we diminish the quotient, by increasing the divisor, adding 1 to it, it then becomes too little ; because the divisor is now too great. For, e being less than 1, 2 a -|- 1 is more than 2 A -{- k, and therefore too great. As (or instance ; if the non-quadrate proposed be n = 3, the greatest integer square is a* = 4, VOL. XIX.] PHILOSOPHICAL TRANSACTIONS. 3 the square of a or '2 : which being subtracted, leaves n — a' =5 — 4 = 1 = B = '2 A E + E^. Which divided by 2 a = 4, gives 4- : but divided by 2 a + 1=4+1=5, gives 4- : that too great, and this too little for e. And there- fore the true root (a + e = y^ n) is less than 24- = 2.25, but greater than 2; = 2.2. And this was anciently thought an approach near enough. If this approach be not now thought near enough, the same process may be again repeated ; and that as often as is thought necessary. Thus, take now for a. 24-=: 2.2, whose square is 4.84 = a^ ; this, subtracted from 5.00, leaves .16 2 a new remainder b = O.16: which divided by 2 a = 4.4, gives 7^ = — =: ' '^ 4.40 55 16 8 0 03636 +, too much ; but divided by 2 a + 1 = 4.5, it gives ^-~ = — - = 0.03555 +, too little. The true value, between these two, being 2.236 proAiiTie, whose square is 4.999696. If this be not thought near enough, subtract this square from 5.000000; the remainder b = 0.000304, divided by 2 a = 4.472, or by 2 a + 1 = 4.473, gives, either way, OOOOO68 — ; which added to a = 2.236, makes 2.236o68 — , somewhat too large ; but 2.236067 + would be much more too little. Proceed we now to the cubic root. For which, the rule is this : from the non-cubic proposed, suppose n, subtract the greatest cube in integers, sup- pose a^ ; the remainder, suppose b = 3 a'^ e -|- 3 a e^ -|- e^ is to be the numerator of a fraction for designing the value of e, the remaining part of the root sought A + e = ^ n. To this numerator, if, for the denominator or divisor, we subjoin 3 a'^, the result will certainly be too great for e, because the divisor is too little. If, for the divisor, we take 3 a'^ 4- 3 a -f- l, it will certainly be too little, because the divi>or is too great. It must therefore, be- tween these limits, be more than this latter. And therefore this latter result being added to a, will give a root whose cube may be subtracted from the non- cubic proposed, in order to another step. This approach I find in Wingate's Arithmetic, published in the year l630, and must therefore be at least so old; how much older I know not. But if for the divisor we take 3 a'- -j- 3 a, the result may be too great ; or, in case b be small, it may be too little. Thus, for instance ; if the non cube proposed be 9 = n. The greatest in- teger cube is 8 = a^, whose cubic root is a = 2 ; which cube ^ubtracted, leaves 9 — 8 = 1 = b = 3 a'^ e + 3 a e'- -|- e\ This divided by 3 a' = 12, gives -fV = 0.08333 -|-, too great for e. But the same divided by 3A--f3A + l = l2-f6+l =19, gives -^ = 0.05263 +, too little. Or, if but by 3 a' + 3 a = 12 -J- 6 = 18, it gives -pL = -J>- = 0.05555 -f-, yet too little. For the cube of a -|- O.06, = 2.06, is only 8.742 — , which is short of 9. And so much short of it, that we may safely take 2.07 as not too £ 2 4 PHILOSOPHICAL TRANSACTIONS. [anNO 1694-6. great; or perhaps i.OS ; and upon trial it will be found not too large ; for the cube of '2.08 is but 8.998912. If this first step be not near enough, this cube subtracted from 9.0{)0()00, leaves a new b = O.OOI088, which divided by 3 a-= 12.9796, gives 0.000084 — ; which will be somewhat too great but not much. So that if to 2.08 we add 0.000084 — , the result 2.080084 will be too great, but 2.080083 will be more too little. So that either of them, at the second step, gives the true root within an unit in the sixth place of decimal parts. Hitherto I have pursued the method most affected by the ancients, in seeking a square or cube, and the like of other powers, always less than the just value, that it might be subtracted from the number proposed, leaving b a positive remainder ; thereby avoiding negative numbers. But since the arithmetic of negatives is now so well understood, it may in this be advisable, to take the next greater, in case that be nearer to the true value, rather than the next less. According to this notion, for the square root of 5, I would say, it is 2 +> somewhat more than 2, and inquire, how much more ? but for the square root of 8 I would say, it is 3 — , somewhat less than 3 ; and inquire, how much less? taking, in both cases, that which is nearest to the just value. And what is said of these is easily applicable to the higher powers. I shall omit that of the biquadrate, because here perhaps it may be thought most advisable to extract the square root of the number proposed, and then the square root of that root. But if we would do it at once, we are from n to subtract a^, the greatest biquadrate contained in it, to find the remainder b = 4 A^ E + 6 A^ e" + 4 A E^ + E^ ; which remainder, if we divide by 4 a% the quotient will certainly be too large for e ; if by 4 a^ + 6 a^ + 4 a + 1, it will certainly be too little ; and we are to use our discretion in taking some inter- mediate number. And if we chance not to hit on the nearest, the inconve- nience will be only this, that our leap will not be so great as otherwise it might be. Which will be rectified by another b at the next step. For the sursolid of five dimensions, we are, from n to subtract a\ the greatest sursolid therein contained, to find the remainder b = 3 a^ e -|- 10 A' E- -\- 10 a' e' + 5 a e* + e' ; which if we divide by 5 a\ the result will be somewhat too great, if by 5 a* -|- 10 a^ + 10 a' -|- 5 a + 1, the result will cer- tainly be less than the true e. The just value of e being somewhat between these two ; where we are to use our discretion, wiiat intermediate number to take. Which, according as it proves too great or too little, is to be rectified at the next step. If, to direct us in the choice of such intermediate number, we should mul- tiply rules or precepts for such choice, the trouble of observing them would be 3 VOL. XIX.] PHILOSOPHICAL TRANSACTIONS. 5 more than the advantage to be gained by it. And, for the most part, it will be safe enough, and least trouble, to divide by 5 a*, which gives a quotient some- what too great ; which we may either rectify at discretion, by taking a number somewhat less, or proceed to another b, affirmative or negative, as the case shall require, and so onward to what exactness we please. Which is, for sub- stance, in a manner coincident with Mr. Raphson's method, even for affected equations. How far this method may be coincident with some of those before- mentioned, I do not trouble myself to inquire, nor whether, or for what causes, all or any of those may be more eligible. My design being only to show the true natural ground from whence such rules of approach are, or might have been derived, and by which they may be examined. In affected equations, especially where the coefficients are great, and some affirmative, others negative, the cases will be more perplexed. And to multi- ply rules for each case would increase the trouble, with no great advantage. Which therefore I leave to the prudence of each to take some intermediate, between a greater and a less. Or, if they please, to accommodate that in my Commerc. Epistol. to the present case, which is there applied to a case not less intricate. Or to make use of some of the methods delivered by others to this purpose. Where this is to be considered, that such affected equations are capable of more roots than one, according to the number of dimensions to which they arise. A Method of discovering the true Moment of the Suns Ingress into the Tropical Signs. By E. Halley. N° 215, p. 12. It may perhaps pass for a paradox, if not seem extravagant, if I should assert that it is an easier matter to be assured of the moments of the tropics, or of the times of the sun's entrance into Cancer and Capricorn, than it is to observe the true times of the equinoctials, or ingress into Aries and Libra. I know the opinion both of ancient and modern astronomers to the contrary, as Plolemy, Riccioli, &c. and this because of the exceeding slowness of the change of the sun's declination on the day of the tropic, being not a quarter of a minute in 24 hours. This indeed would make it very difficult, nor would any instru- ments suffice to do it, were the moment of the tropic to be determined from one single observation. But by three subsequent observations made near the tropic, at proper intervals of time, I hereby design to show a method to find the moment of the tropics, capable of all the exactness the most accurate can desire ; and that without any consideration of the parallax of the sun, of the refractions of the air, of the greatest obliquity of the ecliptic, or latitude of the place ; all which are required to ascertain the times of the equinoctials from 6 PHILOSOPHICAL TRANSACTIONS. [aNNO 1694-5. observation, and which being faultily assumed, have occasioned an error of near 3 hours in the times of the equinoctials deduced from the tables of Tycho Brahe and Kepler ; the vernal beinfj so much later, and the autumnal so much earlier, than by the calculus of those famous authors. Now before we proceed, it will be necessary to premise the following lem- mata, serving to demonstrate this method, viz. 1. That the inotion of the sun in the ecliptic, about the time of the tropics, is so nearly equable, tiiat the difference from equality is not sensible, from 5 days before the tropic, to 5 days after ; and the difference arising from the little inequality that there is, never amounts to above 4- of a single second in the declination, and this by reason of the nearness of the apogoeon of the sun to the tropic of Cancer. 2. That for 5 degrees before and after the tropics, the differences by which the sun falls short of the tropics are as the versed sines of the sun's distance in longitude from the tropics ; which versed sines, in arciies under 5 degrees, are be)ond the utmost nicety of sense, as the squares of tliose arches. From these two follow a third ; 3. That for 5 days before and after the tropics, the declination of the sun falls short of the utmost tropical declination, by spaces which are in duplicate proportion, or as the squares of the times, by which the sun is wanting of or past the moment of the tropic. Hence it is evident, that if the shadows of the sun, either in the meridian or any other azimuth, be carefully observed about the time of the tropics, the spaces by which the tropical shade falls short of, or exceeds, those at other times, are always proportionable to the squares of the intervals of time between those observations and the true time of the tropic; and consequently if the line, on which the limits of the shade is taken, be made the axis, and the cor- respondent times from the tropic expounded by lines be erected on their res- pective points in the axis, as ordinates, the extremities of those lines shall touch the curve of a parabola, as may be seen in fig. 1, pi. l. Where a, b, c, e, being supposed points observed, the lines ah, b c, ca, ep, are. respectively pro- portional to the times of each observation before or after the tropical moment in Cancer. This premised, we siiall be able to bring the problem, of finding the true time of the tropic by three observations, to this geometrical one, having three points in a parabola a, b, c, or a, f, c, given, together with the direction of the axis, to find the distance of those points from the axis. Of this thtre are two cases, the one when the time of the second observation b is precisclv in the middle time between a and c ; in this case putting i for the whole time be- tween a and c, we shall have At, the interval of the remotest observation a from the troj ic, by the following analogy : as 2 a i: ~ b c to 2 a c — -^ b c :: so VOL. XIX.] PHILOSOPHICAL TRANSACTIONS. 7 is -L z or A E : to A c, the time of the remotest observation A froin the tropic. But the other case, when the middle observation is not exactly in the middle between the other two times, as at f, is something more operose, and the whole time from a to c being put = t, and from a to p = *, c e z=z c, and ftp I I, ft s s b c ■= b, the theorem will stand thus — —r- z=z a c, the time sought. ' 2 t c — 2 u s To illustrate this method of calculation, it may perhaps be requisite to give an example or two, for the sake of those astronomers that are less instructed in the geometrical part of their art. Anno 1500 Bernard Walther, in the month of June, at Nuremburg, observed the chord of the distance of the sun from the zenith by a large parallactic instrument of Ptolemy, as follows : June 2, 45467 June 8, 44975 June 9, 44934 and June 12, 44883 June 16, 44990 June 16, 4499O. In both which cases the middle lime is exactly in the middle between the extremes, and therefore in the former three, a c ^ 533, b c = 477, and t the time between, being 14 days; by the first rule, the time of the tropic will be found by this proportion, as 589 to 827-i- :: so -|- t or 7 days, to 9 days 20h. 2m. whence the tropic. Anno 1500, is concluded to have fallen June lid. 20h. 2m. In the latter three, a c is = 107, and b c = 15, and the whole in- terval of time is 8 days = t ; whence as 199 : to 206^^ :: so is 4 days to 4d. 3h. 37m., which taken from the l6th day at noon, leaves lid. 20h. 23m. for the time of the tropic, agreeing with the former to the third part of an hour. Again, Anno l63d, Gassendus, at Marseilles, observed the summer solstice by a gnomon of 55 feet high, in order to determine the proportion of the gnomon to the solstitial shade ; and he has left us these observations, which may serve as an example for the second rule. June 19, N.S. shadow 31 766 parts, whereof the gnomon was 89428. June 20, 31733 June 21, 31751 June 22, 31759- These being divided into two sets, of three observations each, viz. the 19th, 20th, and 22d, and the igth, 2ist, and 22d ; we shall have in the first three, c = 1 3, and b = 7, t = 3 days, * = 1 ; and in the second, c = 15, and b = 7, t = 3, and s = 2 ; whence, according to the rule, the 19th day at noon the sun wanted of the tropic a time proportionate to one day, as tlc — ssb to Itc — lbs, that is, as 1 10 to 64 in the first set, or 107 to 62 in the second set; that is, id, 17h. 15m. in the first, or id. 17h. 25m. in the second set; 8 PHILOSOPHICAL TRANSACTIONS. [aNNO ] 694-5. SO that we may conclude the moment of the tropic to have been June lod. 17h. 20m. in the meridian of Marseilles. Now that these two tropical times thus obtained, will be found to confirm each other's exactness from their near agreement, appears by the interval of time between them, viz. id. 2h. 30m. less than 136 Julian years; of which id. ih. Sm. arises from the defect of the length of the tropical year from the Julian, and the rest from the progression of the sun's apogeeon in that time ; so that no two observations made by the same observer in the same place, can better answer each other, and that without any the least artifice or force in the management of them. What were the methods used by the ancients to conclude the hour of the tropics, Ptoleuiy has no where delivered ; but it were to have been wished that they had been aware of this, that we might have been more certain of the mo- ments of the tropics we have received from them, which would have been of singular use to determine the question, whether the sun's apogaeon be fixed in the starry heaven ; or if it move, what is its true motion ? It is certain, that if we take the account of Ptolemy, the tropic said to be observed by Euctemon and Meton, Junii 27 mane, Ann. 432 ante Christum, can nowise be recon- ciled, without supposing the observation made the next day, or June 28 in the morning. And Ptolemy's own tropic observed in the third year of Antoninus, Anno Christi 140, was certainly on the 23d, and not the 24th day of June ; as will appear to those that shall duly consider and compare them with the length of the year deduced from the diligent and concordant observations of those two great astronomers, Hipparchus and Albatani ; established and confirmed by the concurrence of all the modern accuracy. For these observations give the length of the tropical year such, as to anticipate the Julian account only one day in 300 vears ; but we are now certain that the said period of the sun's revolution anti- cipates very nearly 3 days in 400 years ; so that the tables of Ptolemy founded on that supposition, err about a whole day in the sun's place, for every 240 years. Which principal error in so fundamental a point, vitiates the whole superstructure of the Almagest, and serves to convict its author of want of dili- gence, or fidelity, or both. But to return to our method : the great advantage we have hereby, is, that any very high building serves for an instrument, or the top of any high tower or steeple, or even any high wall that may be sufficient to intercept the sun, and cast a true shade : nor is the position of the plane on which you take the shade, or that of the line therein, on which you measure the recess of tiie sun from the tropic, very mriterial ; but in what way soever you discover it, the said recess will be always in the same proportion, by reason of the smallness of TOL. XIX.] PHILOSOPHICAL TRANSACTIONS. Q the angle, which is not six minutes in the first five days: nor need you inquire the height or distance of your building, provided it be very great, so as to make the spaces you measure large and fair. But it is convenient that the plane on which you take the shade be not far from perpendicular to the sun, at least not very oblique, and that the wall which casts the shade be straight and smooth at top, and its direction nearly east and west, for reasons that will be well under- stood by a reader skilful in the doctrine of the sphere. And it will be requisite to take the extreme greatest or least deviation of the shadow of the wall, be- cause the shade continues for a good while at a stand, without alteration, which will give the observer leisure to be assured of what he does, and not to be sur- prised by the quick transient motion of the shade of a single point at such a distance. The principal objection is, that the penumbra, or partile shade of the sun, is, in its extremes, very difficult to distinguish from the true shade, which will render this observation hard to determine nicely. But if tlie sun be transmitted through a telescope, after the manner used to take his species in a solar eclipse, and the upper half of the object-glass be cut off by a paper pasted thereon, and the exact upper limb of the sun be seen just emerging out of, or rather continging the species of the wall, the position of ihe telescope being regulated by a fine hair extended in the focus of the eye-glass, I am assured that the limit of the shade may be obtained to the utmost exactness : and of this I design to give a specimen by an observation to be iDade in June next, by the help of the high wall of St. Paul's church, London, of which some follow- ing Transaction may give an account. In the mean time what I have premised may suffice to set others at work, where such or higher buildings are to be met with. I shall only advertise, that the winter tropic, by this method, may be more certainly obtained than the summer's, by reason that the same gnomon afl^ords a much larger radius for this kind of observation. On the probable Cniises of the Pain in Rheumatisms ; and on the Cure of a total Suppression of Urine, not caused by a Stone, by the use of yields. By Dr. Edward Baynard, Fellow of the Col!, of Physicians. N" 215, p. \Q. This ingenious physician was always of opinion, that the pains in a rheuma- tism were not caused by any saline or acid particles in the blood, &c. but rather by its clamminess and density distending the channels through which it passes, which distension produces those sharp and pungent pains which rheumatic per- sons so generally complain of; for although the proper coats of the veins and arteries seem to be insensible in themselves, yet those thin membranes which beset them are of most exquisite sense, and full of lymphaeducts, which being dilated and stietched, cause an inflammatory symptomatic fever, with continual VOL. IV. C 10 PHILOSOPHICAL TRANSACTIONS. [aNNO 16Q4-5. sweats ; the blood being glutinous and sizy, as in quinsies and pleurisies, and all Other inflammatory distempers. The fever being increased by the great quantity of alkaline corrosive salts lodging in the blood, causing thirst, &c. and not diluted and washed off by urine, which urine is always thick, turbid, and high coloured, and almost, if not totally devoid of any saline impregnations, as was proved upon analysis. A patient had laboured for 7 or 8 days under a total suppression of urine. Upon trying with a catheter, there was not the least appearance of any stone, nor a drop of water in his bladder ; upon which Dr. Baynard caused the patient to take a quantity of acids, which produced a great discharge of urine, and restored him to health. ^n Extract of a Letter from Bernard Connor,* M. D. to Sir Charles fValgrave, giving an Account of an extraordinary Human Skeleton, having the Vertehrce of the Back, the Ribs, and several Bones down to the Os Sacruin, all firmly united into one solid Bone, without Jointing or Cartilage. N°2l5, p. 21. This was not an entire skeleton, but consisted only of the os ilium, the os sacrum, the 5 vertebrae of the loins, 10 of the back, 5 entire ribs on the right side, and 3 on the left ; the bottoms or ends of the other were closely united to the transverse apophyses of their vertebrae. The vertebrae of the neck, the claviculae and sternum were wanting. All these bones which are naturally distinct from each other, were here so straightly and intimately conjoined, their * Dr. Bernard Connor was a native of Ireland, and is supposed to have been born in 1666. In l686 he went to Montpellier, where he prosecuted his medical studies, and from thence he removed to Paris. While he was here he was fortunate enough to be appointed travelling physician to the two sons of 'the chancellor of Poland, whom he accompanied to Warsaw, passing through Italy, Austria, Moravia, and Silesia. Not long after his arrival at Warsaw, the King of Poland made him one of his physicians. This honour was conferred upon him when he was only 28 years of age. During bis stay at Warsaw he evinced his medical skill and penetration in several difficult cases, whereby he acquired a great degree of professional celebrity ; but being fond of travelling, and at the same time desirous of returning to England, we are told that he gladly embraced the opportunity of accom- panying, as physician, the King of Poland's daughter, Princess Theresa, who had been married to the Elector of Bavaria, to Brussels. This journey being completed, he quitted the service of the princess, and proceeded through Holland to England. He now resided parlly at Oxford and partly in London, delivering lectures, at both places, on the animal economy. These he rendeied highly in- teresting by ihe introduction of much new information on anatomical, physiological, and chemical lubjects, which he had collected during his travels on the continent. These lectures were afterwards read at Cambridge. At this time he published his Dissertatioiies Medico-physicae de Aniris lethi- feris, de Montis Vesuvii Incendio, &c. of which an account will be found in a subsequent number of these Transactions. In l697 he published a small treatise, entitled Evangelium Medici, which raised him many enemies, especially among the clergy, it being considered as " an attempt to account for the miracles of the Bible upon natural principles." (Gen Biog. Diet.) His last publication was a History of Poland. Dr. Connor died of a fever iCyS, before he had attained his 33d year. VOL. XIX.] PHILOSOPHICAL TRANSACTIONS. 11 ligaments so perfectly bony, and their articulations so effaced, that they really made but one uniform continuous bone ; so that it was as easy to break one of the vertebra into two, as to disjoint or separate it from the other vertebrae, or the ribs, or the os sacrum from those of the ilia. Nor could I observe any greater distinction between all these bones than is usually seen in adult persons between the os pubis, the ischion, and ilium, which are but one entire bony substance. The roots of all the ribs made but one equal, smooth, and plain superficies with the vertebrae and their apophyses. The oblique apophyses of all the vertebrae were so confounded and lost, that it was not possible to observe any marks of them. The cartilaginous edge of the vertebrae themselves was become perfect bone. In short, they were as entire as a skeleton cut out of the same piece of wood by a carver. Being willing to see if these vertebrae were united throughout their whole diameter, or at the edges only, I sawed two of them asunder at the commissure, and found this uniting did not enter above 2 lines deep, and that afterwards their middles were separated as they usually are, and touched each other only at the edges. On the left side at half a finger's breadth from the vertebrae, two ribs were joined together for the space of an inch, and afterward ran separated and parallel, like the rest, to the sternum. The figure of this trunk was crooked, making part of a circle. The spine forming the convex, and the inside of the vertebrae the concave part of this segment. If the other vertebrae of the back and neck had been preserved, and had bent in the same curve, they would have made near the half of a circle. The direction of the ribs was unnatural, for instead of terminating at the sternum in parallel semicircles nearly horizontal, their extremities where they reached the sternum, dipped so much down towards the hypogastrium, as to touch the sides of the ossa ilii. This trunk seemed to be of a grown person, the bones being of a proportion and thickness equal to those of old men. The vertebrge of the loins were larger than those of the back, as they naturally are; there was no unnatural bunching out, their joining together being very regular, no one vertebra stand- ing out beyond the other, either before, behind, or on the sides. The cavity for the spinal marrow had no fault but its bending figure. The bones of the OS pubis were separated as usual. The socket or cavity of the last spurious rib on the right side, being smooth and polished, seemed as if that rib had not been so firmly united as the rest. In the extremity of the ribs next the sternum, the usual cavities for the cartilages to move in were observable, which as it seems by this were not bony, nor continuous MMth the ribs. It was a surprising sight to see the sport of nature in the fabric and hardening of these bones, which naturally move upon each other, are separated by carti- lages, and held together only by cords and ligaments, and chiefly that the ribs c 2 12 PHILOSOPHICAL TRANSACTIONS. [aNNO 1 094-5. should be thus joined with the rest, which are perpetually raised in respiration, and whose motion is upon the vertebrae as its centre; and we see motion hinders the lips of a wound from closing, and a broken bone from uniting. The author supposes the bones to have been thus united in the fcetal state. He further remarks, that necessarily the body of this person must have been immoveable, that he could neither bend nor stretch himself out, rise up nor lie down, nor turn upon his side, having only the head, feet, and hands moveable. The great difficulty seeming to be in the respiration, how that could be per- formed when the ribs were thus immoveable : he endeavours to obviate this by observing, first, how little motion of the breast is necessary to continue the motion of the blood through the lungs, as is visible in hysteric fits, &c. Again, the ribs of his skeleton, though fixed at the centre, might yet be moved at the extremities, and so the thorax be enlarged by a much less strength than that of the muscles used for that purpose; besides, the diaphragm, the chief organ of respiration, was in this subject free in its acting. But it is likely this person breathed very short, the quickness of the returns supplying the defect of a large draught of air at once. And possibly the foramen ovale might continue open, and by it and the arterial canal the blood might pass from the cava to the aorta, and but a part of it pass through the lungs. He confirms this by an observation he lately made in a girl of 4 or 3 years old, in whom the foramen ovale was but half closed up, and in the form of a crescent. To this our author adds another observation of the bones of the thigh and leg growing together in an adult person, the place of their joining being much more solid than any other part. These bones were so bent at the knee as to make an acute angle, yet were they without any exostosis, rottenness, fracture, or unnatural figure. It is more surprising to find the knee, whose motion is free and large, to be thus united than that of the ribs of the skeleton, whose motion is obscure, and scarcely sensible. Concerning a Spout of Water that happened at Topsham, on the River between the Sea and Exeter. By Mr. Zachary Mayne. N° 215, p. 28. These phaenomena are very frequent abroad, yet rarely if ever seen with us, though some pretend to have seen them in the Downs. The French call them trombes, I suppose from their figure, and the noise they make, that word signifying a sort of humming top. They are certain elevations of water, during storms and tempests, reaching from the surface of the sea to the clouds. They happen several ways ; sometimes the water is seen to boil, and raise itself for a considerable space round, about a foot from the surface, above which appears, as it were, a thick and black smoke, in the middle of which is observed a sort of stream or pipe resembling a tunnel, whicli rises as high as the clouds ; at VOL. XIX.] PHILOSOPHICAL TRANSACTIONS. 13 other times these pipes or tunnels are observed to come from the clouds, and suck up the water with great noise and violence. They move from the place where they were first collected, according to the motion of the wind, and dis- charge themselves sometimes into the sea, to the unavoidable destruction of such ships as are in their way, if they be small vessels, and to the damaging of large ones : sometimes they fall on the shore, beating down all they meet with, and raising the sand and stones to a prodigious height. It is said, that vessels that have any force usually fire their guns at tliem, laden with a bar of iron, and if they be so happy as to strike them, the water is presently seen to run out of them with a mighty noise, without farther mischief. yin Account of a Booh, viz. — The Anatomy of the Brain, containing its Me- chanism and Physiology, together with some neiv Discoveries and Corrections of Ancient and Modern Authors upon that Subject: to luhich is annexed, a parti- cular Account of Animal Functions and Muscular Motion. By Hen. Ridley,* Col. Med. Land. Soc. N° 215, p. 32. Account of the upper Part of the Burning Mountain in the Isle of Ternata. N"" 216, p. 42. This is a tedious uninteresting narrative. Account of the sad Mischief which befel the Inhabitants of the Isle of Sorea, near the Moluccas, for which they have been forced to leave their Country. In a Letter to M. Nicholas IVitzen, of Amsterdam, 1693. And communicated to Dr. Lister, S.R.S. N°2l6, p. 49. In the beginning of the easterly season, the isle Sorea, situated towards the south-east of these islands, consisting for the most part of one mountain, which now is more terribly shaken than ever before, casting out abundance of fire and smoke, only with some short intermissions. And when the easterly wind had blown about 6 or 7 weeks, till about the 4th of June, the inhabitants being almost so far used to the trembling and casting up of fire that they were careless, the mountain Sorea began early in the morning to cast out more fire than ordinary, which continued for 5 or 6 days ; during which it was dark and cloudy weather, till at last it brought forth not only a most prodigious flame, but also such a black and sulphureous vapour, that the inhabitauts of Hislo, a village in the western part of the island, and nearest to the opening of the mountain, were wholly covered by it ; and afterwards followed a whole stream of burning brimstone, which consumed many that could not escape. After- * This work of Ridley's on the brain, wherein several (hings are noticed which had escnped others, deserves to be numbered among those productions of the English anatomists of the 17ih century, which are distinguished for their uiihty and ingenuity. The plates, however, as Haller has re- marked, are upon too small a scale. 14 PHILOSOPHICAL TRANSACTIONS. [aNNO I695. wards the inhabitants perceived that a part of the mountain was sunk down, and 3 or 4 days after another part ; and so from time to time, until the burning lake was become almost half the space of the island. Wherefore the inhabitants went on board their vessels and boats, from whence they perceived that huge pieces of the mountain fell into this fiery lake, as into a bottomless pit, with a most prodigious noise, as if a large cannon were discharged. It was re- markable, that the more vehement the fire was, the less the island was shaken. The inhabitants of another town, called Woroe, upon the east side of the island, not thinking themselves in so great danger, the opening or fiery lake being yet at some distance, remained a month longer, until they saw the same continually approaching them : they observed that when great pieces fell down, and that the lake became wider, the noise was so much the greater : so that they saw no likelihood but that all the island wf)uld be swallowed up. Where- fore they unanimously transported themselves to Banda, leaving all their move- ables for want of vessels. Several burning mountains have now been filled up, and quenched ; others have begun to open themselves, and to cast out fire, as in the isle Chiaus. There is likewise a burning mountain on the island Celebes. And in an infinite number of places there is hot water found if you dig but 10 feet deep. In the mountains of Ternata is always heard a terrible noise, as ot the crying of a great many people, caused by the fire ; it often throws out stones, and jjrobably is exceedingly deep, and the rather because it is likely that the several burning mountains of the Molucca islands are beneath consumed by the same tire, which joins the spacious openings together. The burning mountain on Banda throws out a vast quantity of smoke and ashes, often much fire ; and makes a noise as if many of the largest cannon were heard all at once. This mounfain has cast out so many stones, and some near 6 feet long, that the adjacent sea, which has been 40 or 50 fathoms deep, is not only filled up there, but become many fathoms higher than the water. Specimens of the Use of Fluxions in the Solution of Geometric Problems. By Jllr. ylbr. Demoivre* N° 21 6, p. 52. Transldted from the Latin. You have here a method for the quadrature of curve figures ; for the measure * Abraham Demoivre was originally a Fri'iichman, and born at Vitry in Champagne, 1667; but on ihe revoralion of ih^ edict of Nantes he came to England, where he continued to study tlie ma- thematics, of which he became a celebrated master, and iaui;ht it (or liis living; and aUo oi ca^i.■n- ally answered for gentlemen curious and difficult quesiioiis in chances, games, and other s.ubj;ct.s. He died at London in 1754, at 87 years of age. Mr. Demoivre wa'. admiited a member ol the Royil Society, and contribuied a number of papers to the Pbilosophicairran tactions, lie wa-i :\'.>o author of several other works ; as, 1. \nimad' er^ions on Dr. Chene's Fluxio.is. Svo. I7i'-1; '' Doc- trine of Chances, 4to. 1718, which has been much esleemedj J. Miscellania Anaiytica, 4to. 17ilO, VOL. XIX.] PHILOSOPHICAL TRANSACTIONS. 15 of the surfaces and solidities of solids formed by the rotation of planes ; for the rectification of curves ; and for the calculation of their centres of gravity. But before proceeding further you must understand, that I assume what the great Newton has demonstrated, in p. 251, &c. of his Principia, concerning the momentary increments or decrements of quantities, which either increase or n decrease by perpetual flux, and especially that the moment of any power a™ n n is - a A"» ' Hence the fluxion - a a™ being given, on the contrary we n may find the flowing quantity a™, 1st, by taking a out of the fluxion ; 2d, by increasing the index of the fluxion by unity ; 3d, by dividing the fluxion by the index so increased by unity. In what follows, the absciss of the curve shall be denoted by x, its fluxion by .f, the ordinate by y, and its fluxion by i/. This being supposed, to proceed now to quadratures, first let the value of the ordinate be obtained, by means of the equation expressing the nature of the curve ; 2d, multiply this value by the fluxion of the absciss ; then the rectangle hence arising will be the fluxion of the area ; 3d, find the fluent of this fluxion of the area, and the required area will be found. Let there be proposed the equation x"' = yn, expressing the nature of any m paraboloid, in which the value of the ordinate h y ■=. xn; which being multi- plied by .r, the rectangle x" i will be the fluxion of the area ; therefore the area m TO required will be x" or xy, putting y for x" . Again, let there be proposed the curve whose equation is x* -\- a^ x"^ =z y^, which is the first of Mr. Craig's examples. Then talcing xV x x-\-aa=.y, the fluxion of the area will be x x '>^ x x -\- aa. Now since this is involved in a radical sign, suppose '/xa; + afl=:z; hence ar or + a o = z^, and therefore ar i' = z i ; hence putting z i and z for x x and V x x -\- a a, the fluxion freed from surds will be z^ i ; this reduced back to its origin will be ^ z^ ; which, by restoring '^ x x -\- a a for z, gives \ . x x — a a ^ x x — a a for the area re- quired, f. And that it may appear with what ease these quadratures may be obtained, take one example more, viz. let the equation of the curve be — — = y"^ ; therefore « = —=.=, and --^L^ is the fluxion of the area. Suppose V x + a = z ; Vx+a Vx+ a being a good treatise on infinite series^ and other miscellaneous tracts in mathematics ; 4. A Treatise on Annuities on Lives, &c. Mr. Demoivre was one of the commissioners of the Royal Suciely, who decided in Newton's favour in the celebrated dispute between that great man and Leibnitz, respecting the discovery of the doctrine of fluxions. l6 PHILOSOPHICAL TRANSACTIONS. [aNNO '16Q5. hence x = z" — a, and i- = "2 2 ir ; therefore , = 2 z'^ i — 2 a z, and hence Vx + a ^z^ — 1 uz or ^x — ^ a ^ X -{■ a will be the area sought. But it often happens that some curves, such as the circle or hyperbola, are of such a nature, that it would be in vain to endeavour to free their fluxions from surds, in which case the value of the ordinate must be reduced to an infi- nite series ; then every term of this series being multiphed by the fluxion of the absciss, as above, the fluent of every term must be separately found, and the new series thus arising will exhibit the quadrature of the curve proposed. With the same ease may this method be accommodated to the measures of solids formed by the rotation of a plane, viz. by assuming for their fluxion the product of the circular base into the fluxion of the absciss. Let the ratio of a square to its inscribed circle be 1 to n ; then the equation to the circle being y y z= dx — XX, therefore 4 ft. d x x — x"^ x is the fluxion of a portion of the sphere, and consequently 4 n.-f d x^ — -3- x^ is the portion itself. But the cylin- der circumscribed about this is 4 n.dx- — .r^ ; therefore the ratio of the portion of the sphere to the circumscribed cylinder, is as -i- c^ — -^ x io d — x. The rectification of curves will be obtained if the hyf>othenuse of the right- angled triangle, whose sides are the fluxions of the absciss and ordinate, be considered as the fluxion of the curve ; observing that, in the expression of that hypnthenuse, only one of the fluxions be retained, and only the indeter- minate quantity of the same ; as will be plain from the examples. From the right sine b c, (fig. 2, pi. 1) of the arc a c, being given, to find the arc. Putting A b = x, B c = j/, o a = r ; let c E be the fluxion of the absciss, D E the fluxion of the ordinate, and c d the fluxion of the arc ac. Now from the property of the circle, 1 r x — x x = y y; hence 1 r x — 2 x i' = 2 y i/, and X = -^-'^ ; but c D = w y 4- x x z=. 11 11 A 4-^ = y u A — i-^y— = r — X ^ ^ ' -^-'rr — Irx-'rxx ^ ~j ' j- y — y y S1JJ_ therefore c D = . -^ ; but . -'' is the product of .- or r r — y y V r r — y y v ;■ r — y y Vr r —j y r r — y y\~^ into 7;/: hence i f r r — y y\~ he thrown into an infinite series, and every term multiplied by r y, and the fluent of every product be found, there will be obtained the length of the arc a c. In like manner may the arc be found, from the versed sine being given. Thus, resuming the equation 2 /• x — 2 x x = ■! y y, it gives y = — ^ — ; but 0 . . , . . . . , rrxx — 2rxxx x- x x . ■ , r r i x — '2 r x x x + x- x C D= X X A- 1/ 1/ = X X A- = X X A ;r , or, reducing all to the same denominator, and expunging the p.irts which cancel each other, it is = „ , and hence c u = -^\ then, from wliat has 1rx — xx ^/^Irx-xx been done above, the length of the arc will be easily obtained. VOL. XIX.] PHILOSOPHICAL TRANSACTIONS. \>j Sometimes the fluxion of the curve is more easily found by comparing toge- ther the similar triangles c d e, c b o, making this proportion cb:co::ce:cd, that is, for the circle, ^^ 1 r x — x x : r:: i 'J 2 r X — X X The curve of the cycloid may be known in the same manner. Let a k l be a semicycloid, whose generating circle is a d l (fig. 3). Any point b being as- sumed in the diameter a l; let b i be drawn parallel to the base k l, meeting the circle in the point d ; complete the rectangle a e i b ; also draw f h parallel and infinitely near, to e i, cutting b i produced in g, and the curve a k in h. Then, putting A L = 624372950279032551 15309200128864190695864829866 7 0,8450980400142568307 122l6258592636l93483572396323965406503835 11 1,041392683158225040750199971243024241706702190466453094596539 13 l,1139-13352306837769206541S95G2624625456l 189005053673288598083 17 1,230448921378273028540169894328337030007567378425046397380368 19 1,278753600952828961536333475756929317951129337394497598906819 The next prime number is 23, which I will take for an example of the fore- going doctrine ; and by the first rules, the logarithm of the ratio of 22 to 23 will be found to be either 1 1,1 1,1. — ^ ^—^ -4- — -4- &c or 22 968 ' 319-i4 937024 ^ 25768I6O 1 J. _L. j_ _J_ _i. __L_ _!. L_ &c 23 ' 1058 ~ 36501 ^ 1119364 ^ 32181715 As likewise that of the ratio of 23 to 24 by a like process, 1 1,1 1,1a 23 1058 ^ 36501 1119364 ^ 32181715 1 , _L. , \ I 1 [_ 1 &c 24 ' 1152 "•" 41472 "•" 1327104 """ 39813120 And this is the result of the doctrine of Mercator, as improved by the learned Dr. Wallis. But by the second theorem, viz. ^ -j- — -f- — &c. the same logarithms are obtained by fewer steps. To wit, II I fij 45 "*" 273375 ~^ 922640625 "*" 26l5686l71875 2 2 2 2 _ 4. ^ 4. ~ _L Z gjc 47 ^ 311469 1146725035 ^ 3546361843241 which was invented and demonstrated in the hyperbolic spaces analogous to the logarithms, by the excellent Mr. James Gregory, in his Exercitationes Geome- tricae, and since farther prosecuted by the aforesaid Mr. Speidall, in a late treatise in English by him published on this subject. But the demonstration, as I conceive, was never till now perfected without the consideration of the hyper- bola, which in a matter purely arithmetical, as this is, cannot so properly be applied. But what follows I think I may more justly claim as my own, viz. 24 PHILOSOPHICAt TRAN^CTIONS. [aNNO 16Q5. That the logarithm of the ratio of the geometrical mean to the arithmetical, between 22 and 24, or of i/528 to 23, will be found to be either ^ _L , \ . 4. I I ____! fee or 1068 ' 1119364, ~ 888215334 ^ 6264878822*8 J_ . ! . __JL_ &c 1057 ' 3542796579 659676558485285 All these series being to be multiplied into 0,4342944819, &c. if you design to make the logarithm of Briggs. But with great advantage in respect of the work, the said 43429448 J 9 &c. is divided by 1057, and the quotient thereof again divided by 3 times the square of 1057, and that quotient again by 4- of that square, and that quotient by ^ thereof, and so on till you have as many figures of the logarithm as you desire. As for example, the logarithm of the geometrical mean between 22 and 24 is found by the logarithms of 2, 3, and 11, to be 1 .36 1 3 1 696 1 266906 1 2945009 1 726698O5 1057 ) 43429 &C. ( 41087462810146814347315886368 3 in 1117249 ) 41087 &c. ( 12258521544181829460074 f in 1117249 ) 12258 &c. ( 6583235184376175 4- in 1 1 17249 ) 65832 &c. ( 4208829765 ■f- in 1117249 ) 42088 &c. (______ 2930 Summa 1.36l7278360175928788677771 12251 17 Which is the logarithm of 23 to thirty-two places, and obtained by five divi- sions with very small divisors, all which is much less work than simply multi- plying the series into the said multiplicator 43429 &c. Before I pass on to the converse of this problem, or to show how to find the number appertaining to a logarithm assigned, it will be requisite to advertise the reader, tliat there is a small mistake in the aforesaid Mr. James Gregory's Vera Quadratura Circuli et Hyperbolae, published at Padua, Anno 1667, wherein he applies his quadrature of the hyperbola to the making the logarithms: in p. 48 he gives the computation of the Lord Napier's logarithm of 10, to 25 places, and finds it 23O2585O92994O45024O1787O instead of 2302585092994045684017991,* erring in the 18th figure, as I was assured upon my own examination of the number I here give you, and by comparison thereof with the same wrought by another hand, agreeing therewith to 57 of the 60 places. Being desirous to be satisfied how this difference arose, I took the no small trouble of examining Mr. Gregory's work, and at length found • This mistake was before noticed by Euclid Speidall, vis. in his Logarithms, published Anno 1688. VOL. XIX.J PHILOSOPHICAL TRANSACTIONS. 25 that in the inscribed polygon of 512 sides, in the 18th figure was a O instead of 9, which being rectified, and the subsequent work corrected therefrom, the result agreed to a unit with our number. x\nd this I propose, not to cavil at an easy mistake in managing, of so vast numbers, especially by a hand that has so well deserved of the mathematical sciences, but to show the exact coincidence of two so very differing methods to make logarithms, which might otherwise have been questioned. From the logarithm given, to find what ratio it expresses, is a problem that has not been so much considered as the former, but which is solved with the like ease, and demonstrated by a like process, from the same general theorem of Mr. Newton : for as the logarithm of the ratio of 1 to 1 + 9 was proved to be J + q\"' — 1 , and that of the ratio of 1 to I — 9 to be 1 — 1 — ^j"" *• so the logarithm, which we will from henceforth call l, being given, 1 -f- l will 1 be equal to I + ^|'"in the one case; and 1 — l will be equal to l — r^f in the other: consequently l + lV" will be equal to i -\- q, and I — l)"* to 1 — q; that is, according to Mr. Newton's said rule, I -\- nih -\- Vm" l^ + im^L^ + ^'-7n' L-* + -pl-m'L' &c. will be = 1 + y, and I — m l + -A-Tn'-L- — -^m^ v^ + -p-m* 1.* — -j^-g-m'* l^ &c. will be equal to 1 — q, m being any infinite index whatever; which is a full and general proposition, from the logarithm given, to find the number, be the species of logarithm what it will. But if Napier's logarithm be g ven, the multiplication by m is saved, which multiplication is indeed no other than the reducing the other species to his, and the series will be more simple, viz. 1 + L + 4-i-L-f- -^l^ + -Vl* + ^t.l'- &c. or 1 - L + iL L - 4-L^ + VtL^ — twL' &c. This series, especially in great numbers, converges so slowly, that it were to be wished it could be contracted. If one term of the ratio, whereof l is the logarithm, be given, the other term will be had easily by the same rule : for if l were Napier's logarithm of the ratio of a the less, to h the greater term, h would be the product of a into 1 + L + iLL + -iLLL &c. = a + a l + -la ll + ^a l' &c. But if h were given, a would be = i- — ^l + i^LL — -^^l^ &c. Whence, by the help of the chiliads, the number appertaining to any logarithm will be exactly had to the utmost extent of the tables. If you seek the nearest next logarithm, whether greater or less, and call its number a if less, or b if greater than the given l, and the difference thereof from the said nearest logarithm you call /; it will follow that the number answering to the logarithm l, will be eitlier a Into 1 + / -}- ^// -L ^lll -j- _•_/* -j- _4_/5 &c. or else h into i — / -j- j// VOL. IV. E 26 PHILOSOPHICAL TRANSACTIONS. [aNNO I695. — j^Ul + t't^* rW ^* &c. wherein as / is less, the series will converge the swifter. And if the first 20000 logarithms be given to J 4 places, there is rarely occasion for the three first steps of this series, to find the number to as many places. But for Vlacq's great canon of 100000 logarithms, which is made but to ten places, there is scarcely ever need for more than the first step a + al or a + malm one case, or else b — bl or b — mbl in the other, to have the number true to as many figures as those logarithms consist of. If future industry shall ever produce logarithmic tables to many more places, than now we have them ; the aforesaid theorems will be of more use to deduce the correspondent natural numbers to all the places thereof. In order to make the first chiliad serve all uses, I was desirous to contract this series, wherein all the powers of / are present, into one, wherein each alternate power might be wanting; but found it neither so simple nor uniform as the other. Yet the first step thereof is I conceive most commodious for practice, and withal exact enough for numbers not exceeding 14 places, such as are Mr. Briggs's large table of logarithms ; and therefore I recommend it to common use. It is thus: a + -■ _ ^ or b — , . will be the number answering to the logarithm given, differing from the truth by only one half of the third step of the former series. But that which renders it yet more eligible is, that with equal facility it serves for Briggs's or any other sort of logarithms, with the only variation of writing — instead of 1, that is, ./ , , - a + Ma -b — \ lb a + 7 and b j s or — ; and —. — - , - - il i + W --4^ ~ + ^l m m m ' m ' ' which are easily resolved into analogies, viz. As 4342Q&C. — 4- / to 43420 + i / :: So is a > ^ ,, , ,^ A .o.r,!^fi I i it^Aci.r,^^ , I c • ; r to the number sought, or As 43429 &c. + -^- / to 43429 — 4 / :: 00 is 0 ) ° If more steps of this series be desired, it will be found as follows, a -\ _ , ■ — 'f-^i + 7^0/ ^^- ^s ™^y easily be demonstrated by work- ing out the divisions in each step, and collecting the quotes, whose sum will be found to agree with our former series. Thus I hope I have cleared up the doctrine of logarithms, and shown their construction and use independent of the hyperbola, whose affections have hitherto been made use of for this purpose, though this be a matter purely arithmetical, nor properly demonstrable from the principles of geometry. Nor have I been obliged to have recourse to the method of indivisibles, or the arith- metic of infinites, the whole being no other than an easy corollary to Mr. Newton's general theorem for forming roots and powers. VOL. XIX.] PHILOSOPHICAL TRANSACTIONS. 27 A Proposition of General Use in the Art of Gunnery, showing the Rule oj laying a Mortar to pass, in order to strike any Object above or below the Horizon. By E. Halley. N" 2!6, p. 68. It was formerly the opinion of those concerned in artillery, that there was a certain requisite of powder for each gun, and that in mortars, where the distance was to be varied, it must be done by giving a greater or less elevation to the piece. But now our later experience has taught us that the same thing may be more certainly and readily performed, by increasing and diminishing the quantity of powder, whether regard be had to the execution to be done, or to the charge of doing it. For when bombs are discharged with great elevations of the mortar, they fall too perpendicular, and bury themselves too deep in the ground, to do all that damage they might, if they came more oblique, and broke upon or near the surface of the earth ; which is a thing acknowledged by the besieged in all towns, who unpave their streets, to let the bombs bury themselves, and thereby stifle the force of their splinters. A second conveni- ence is, that at the extreme elevation, the gunner is not obliged to be so curi- ous in the direction of his piece, but it will suffice to be within a degree or two of the truth ; whereas in the other method of shooting, he ought to be very exact. But a third, and no less considerable advantage is, in the saving the powder, which in so great and so numerous discharges, as we have lately seen, must needs amount to a considerable value. And for sea mortars, it is scarcely practicable otherwise to use them, where the agitation of the sea con- tinually changes the direction of the mortar, and would render the shot very uncertain, were it not that they are placed about 45 degrees elevation, where several degrees above or under makes very little difference in the effect. In N° 179 of these Transactions, I considered and demonstrated all the pro- positions relating to the motion of projectiles, and gave a solution to this pro- blem, viz. to hit an object above or below the horizontal line, with the greatest certainty and least force.* That is, that the horizontal distance of the object being put ^ 6, and the perpendicular height = //, the charge requisite to strike the object with the greatest advantage, was that which with an elevation of 45° would throw the shot on the horizontal line to the distance of y/ b b -\- hh -\- h when the object was above the horizon ; or if it were below it, the charge must be less, so as to reach on the horizon at 45° elevation, at no greater a distance than V' b b -{- h h — h, that is, in the one case, the sum of the hypothenusal distance of the object from the gun, and the perpendicular * See p. 270, vol 3, of these Abridgments. -E 2 28 PHILOSOPHICAL TRANSACTIONS. [^ANNO ]6Q5. height thereof above the gun ; and in the other case, when the object is below the horizon, the difference of the same, per 47, 1. Eucl. And I then showed how to find the elevation proper for the gun so charged, viz. As the horizontal distance of the object, to the the sum or difference of the hypothenusal distance and perpendicular height :: so radius to the tangent of the elevation sought. But I was not at that time aware that the aforesaid elevation did constantly bisect the angle between the perpendicular and the object, as is demonstrated from the difference and sum of the tangent and secant of any arcli being always equal to the tangent and cotangent of the half complement thereof to a qua- drant. Having discovered this, I think nothing can be more compendious, or bids fairer to complete the art of gunnery, it being as easy to shoot with a mortar at any object on demand, as if it were on the level ; neitlier is there need of any computation, but only simply laying the gun to pass, in the middle line between the zenith and the object, and giving it its due charge. Nor is there any great need of instruments for tliis purpose : for if the muzzle of the mortar be turned truly square to the bore of the piece, as it usually is, or ought to be, a piece of looking-glass plate applied parallel to the muzzle, will by its reflection give the true position of the piece;- the bombardeer having no more to do, but to look perpendicularly down on the looking-glass, along a small thread with a plummet, and to raise or depress the elevation of the piece, till the object appear reflected on the same point of the speculum on which the plummet f;ills; for the angle of incidence and reflection being equal, in this case a line at right angles to the speculum, as is the axis of the chase of the piece, will bisect the angle between the perpendicular and the object, according as our proposition requires. So that it only remains by good and valid experi- ments to be assured of the force of gunpowder, how to make and conserve it equal, and to know the effect thereof in each piece ; that is, how far different charges will throw the same shot out of it : which may most conveniently be engraven on the outside thereof, as a standing direction to all gunners, who shall from thenceforward have occasion to use that piece : and were this matter well ascertained, it might be worth the while to make all mortars of the like diameter, as near as may be alike in length of chase, weight, chamber, and all other circumstances. This discovery, that the utmost range on an inclined plane, is when the axis of the piece makes equal angles with the perpendicular and the object, com- pared with what I have demonstrated of the same problem in the aforesaid N° 17Q, leads to and discovers two very ready theorems; the one to find the greatest horizontal range at 45° elevation, by any shot made on any inclined plane, with any elevation of the piece whatever : and the other lo find the VOL. XIX.] PHILOSOPHICAL TRANSACTIONS. 2Q elevations proper to strike a given object, with any force greater than what suffices to reach it with the aforesaid middle elevation. Both which being per- formed by one single proportion, may be very serviceable to such as are con- cerned in the practice of gunnery, but are unwilling to trouble themselves with tedious and difficult rules. The two propositions are these : Prop. I. — A shot being made on an inclined plane; having the horizontal distance of the object it strikes, with the elevation of the piece, and the angle at the gun between the object and the perpendicular : to find the greatest hori- zontal range of that piece, laden with the same charge ; that is, half the latus rectum of all the uarabolae made with the same impetus. Rule. — Take half the distance of the object from the nadir, and take the difference of the given elevation from that half ; the versed sine of that differ- ence subtract from the versed sine of the distance of the object from the zenith : then shall the difference of those versed sines, be to the sine of the distance of the object from the zenith, as the horizontal distance of the object struck, to the greatest horizontal range at 45^. Prop. 2. — Having the greatest horizontal range of a gun, the horizontal distance and angle of inclination of an object to the perpendicular ; to find the two elevations necessary to strike that object. Rule. — Halve the distance of the object from the nadir, this half is always equal to the half sum of the two elevations we seek. Then say, As the greatest horizontal range is to the horizontal distance of the object, so is the sine of the angle of inclination, or distance of the object from the perpendicular, to a 4th proportional ; which 4th being subtracted from the versed sine of the distance of the object from the zenith, leaves the versed sine of half the differ- ence of the elevations sought ; which elevations are therefore had by adding and subtracting that half difference to and from the aforesaid half sum. I shall not need to speak of the facility of these solutions, I shall only ob- serve that they are both general, without exception or caution, and derived from the knowledge that these two elevations are equidistant above and below the line bisecting the angle between the object and the zenith. An Account of Books, viz. I. The Mathematical TForks of Dr. John JVallis, Savilian Professor of Geometry at Oxford, and F. R, S. 2 Fols. fol. Oxon, N°2l6, p. 73. In the former of them are contained, 1. His Inaugural Oration, when he entered on that employment, Oct. .31, 1649. 2. His Mathesis Universalis, or Opus Arithmeticum ; ccjniainingnot only numeral arithmetic, but the specious and algebraic, or the calculus geometricus, with many discourses or smaller tracts 30 PHILOSOPHICAL TRANSACTIONS. [aNNO I695. intermixed, relating to the same. 3. A Treatise concerning Proportions ; with a Preface concerning Cubic Equations. 4. A Treatise of Conic Sections, in a new and easy method ; considered as plain figures, out of the cone. 5. His Arithmetic of Infinites, being a new method of investigation, or inquiry into the quadrature of the circle and other curve-lined figures ; and many other mysteries in mathematics. 6. An Observation of a Solar Eclipse at Oxford, Aug. 2, 1654. O. S. 7. A Treatise of the Cycloid, with the bodies and surfaces thence derived. 8. A Treatise of the Cissoid ; and the Rectification and Complanation of Curve Lines and Surfaces. 9. His Mechanica, or a large Treatise of Motion : wherein are handled, not only the machines or engines, commonly called the mathematical or mechanical powers, but the whole doctrine of motion, derived and demonstrated from its genuine and first principles : the doctrine of percussions, repercussions, springs, and reflexions ; the doctrine of hydrostatics, from the counterpoise of the air ; and many other things newly discovered. In the latter volume are contained, 1. A large Treatise of Algebra, Historical and Practical : siiowing the origin and progress of that art, from time to time, and the steps by which it has attained to its present height. 2. A Treatise of Combinations, Alternations ; and Aliquot Parts and divers Problems relating to the same. 3. A Treatise of Angular Sections, with other things appertain- ing ; as the Canon of Sines, Tangents, and Secants, &c, 4. A Treatise of the Angle of Contact, showing it to be of no magnitude, and not any part of a right lined angle. 5. A Defence of that Treatise, against the Objections of Leotaud and others; with several Discourses concerning Composition of Mag- nitudes, Inceptives of Magnitudes, and Compositions of Motion. 6. A Dis- course concerning Euclid's Fiith Postulate, and his Fifth Definition of his Sixth Book. 7. A Treatise of the Cono-Cuneus, or a Body representing partly a Cone, and partly a Wedge, with the sections thereof made by a Plane; considered in like manner as what are called the Conic Sections. 8. A Geo- metrical Disquisition of Gravity and Gravitation ; wherein the doctrini,e of the counterpoise of the air is defended against that of the ancients' Fuga Vacui. 9. A New Hypothesis concerning Tides, or the Sea's Ebbing and Flowing ; derived from the common centre of gravity of the earth and moon, considered as conjunct bodies. 10. Commercium Epistolicum : being a collection of let- ters whicli passed between Messrs. Fermat and Frenicle on the one part, and Lord Brounker and Dr. Wallis on the other part, by the intervention of Sir KenelmDigby; concerning divers mathematical questions. 11. A Treatise of Trigonometry, plain and spherical, of Mr. John Caswell.* * A ihird, very large volume, was published afterwards, containing many iniscelLmeous plecej. VOL. XIX.] PHILOSOPHICAL TRANSACTIONS. 31 II. Traclatus de Salts Cathartici amari in Aquis Ebshamensibus et hujusmodi aliis contenti Naturd et Usu. ^ut. Nehemia Grew, M. D. Ulriusque Reg. Soc. Soc. Loud. 12mo. N° 21 6, p. 76. This book is divided into 2 parts. The former, on the nature of the purging waters, and of its purging salt. The latter, of their use. The former has 6 chapters. The 1st shows how these waters came to be commonly known and used. The 2d the nature of the waters. The 3d the nature of the purging salt of these waters. Where, among many other experiments, it is observed, that its crystals, when permitted to shoot at a due distance, are most of them rectangular prisms with 4 parallelogram planes. By which, and divers other properties, it is distinguished from all other salts. The 4th shows the difference between this salt and alum : and that the waters are falsely supposed by many to be aluminous. As also between this and common salt; although some quan- tity of common salt be contained in all the purging waters. The 5th demon- strates the difference of this salt, both from nitre, and from the salt of lime ; notwithstanding it has been taken for a calcareous nitre. The 6th contains some further observation on this bitter purging salt,* grounded chiefly on the foregoing experiments. The latter part has 7 chapters. The 1 st shows the use of the waters, and of their purging salt in general. The 2d shows the ways of using the salt; whereof one of the best is to take it dissolved in its own purging water, raw, or first a little boiled : whereby the said water will work, both in a far less quantity, and more kindly and effectually. The 3d, 4th, 5th, and 6lh show in what diseases this medicine is to be used. The last chapter mentions the diseases wherein it is improper and hurtful. ^ Relation of one Hannah Taylor, a very extraordinary Child of about 6 Years of ^ge, who in Face, &c. toas as large as a full grown JVoman ; and of ivhat appeared on the Dissection of her Body : by Dr. Henry Sampson, F. of the Coll. of Physicians. N° 217, p. 80. Hannah Taylor was born in Crutched Friars, June 12, l682. She was, till 3 years old, very sickly, lean, and not able to go alone; but about Bartholomew- tide, l685, she began to grow strong and fat, which increased till the time of her death : she was also a very forward child of understanding, had her pubes * This bitter purging salt consists of vitriolic or sulphuric acid and magnesia ; being the sulphate of magnesia of tfie French nomenclature, and the magnesia vitriolata of Bergman and the London Pharmacopoeia. S3t PHILOSOPHICAL TRANSACTIONS. [aNNO l6g5 . grown thick and long, as also hair under her arm-pits, and a downiness on her chin, unusual with those of her sex, except in some aged persons. About i a year before she died she began to complain of pains, especially on her left side, and voided gravel often by urine, and with pain. Her breath was short, as is usual to fat people, especially when she went up a pair of stairs : yet on that very evening before she died, she walked abroad, was merry and lively, went to bed, and slept as at other times, but after midnight awaked, cried out of a great pain in her side, and said, Mother, I want breath, 1 shall die; and in less than 4 of an hour was quite dead. The measures and weight of her body were as follow. Round the breast 1 yard and 2 inches, over the hips at the navel 1 yard 5 inches, over the stomach a yard, her height 1 yard wanting an inch, round the thigh 1 foot g inches !., calf of the leg 13 inches, upper part of the arm 14 inches^, the wrist 7 inches, her weight g5lb. She had a face as large and broad as any fat grown woman of 20 years. Her chin and breast were so thick laid with fat, that she was forced to hold up her head, or rather throw it backward, as she walked. These measures were all taken before the dissection. The thickness of the fat on the muscles of the abdomen was 2 inches, and not much less on the sternum: after the fat was removed, the abdomen was still very protuberant and round, and yet the fat contained therein not extraordinary much, neither on the omentum nor mesentery (which was as much as is usually in most fat and grown up persons) these, with the other internal parts, were of the largest size. The guts were all inflamed and thick, the liver large, the left kidney, where was the seat of her misery, exceedingly large, and double the size of that on the right side ; on dissectmg it, there issued out a vast quantity of blood, both from all the vessels of it, and out of its pelvis; and although it was several times sponged from it, yet it came flowing in from theemulgent artery ; a certain argument of a great plenitude in the descending trunk. Here was also some small gravel, which possibly had choked up the ureter, though that was not examined ; but because there was no blood in the bladder I justly make this conjecture. The uterine parts were not larger than in others of her age. The ovaria were large, but smooth and white, without protuberances or show of eggs. The bladder had a purulent matter in it. When the breast was denuded of its fat, it showed no larger tlian of another child of her age. The cavity was totally tilled with the lungs and heart. The heart was natural. But the lungs, besides that they were extended to fill up the whole cavity, were annexed strongly to several parts of the pleura, and had several protuberances as large as nutmegs filled with a pulp like an atheroma, and were in divers places rotten VOL. XIX.] fHILOSOFHICAL TRANSACTIONS. 33 and corrupted. Quaere, Why one with so bad lungs was so fat ? Why had she not rather a consumption ? Tne evident cause of her death lay in the inflammation of the lower parts, though the parts concerned in respiration were also disordered. Her face and head were miserably coloured with redness of stagnant blood. The head was not opened. Account ofTadmoT, or Palmyra, in Syria ; and a Journey from Aleppo to that Place. By the Rev. Mr. JVilliam Halifax. N° 217, p. 83. We left Aleppo on Michaelmas-day, 1691, and in 6 easy days' travel over a desert country, nearly in a south direction, but a little inclining to the east, came to Tadmor. As we rode into the town we observed a castle, about half an hour's distance from it, and so situated as to command both the pass into the hills, by which the town is entered, and the city too. But we could easily perceive it was no old building, showing no traces of the exquisite workman- ship and ingenuity of the ancients. We were informed it was built by Man- Ogle, a prince of the Druces, in the reign of Amurath tiie third, anno 1585. But it does not appear that either Man-Ogle, or any Drucian prince, was ever powerful in these parts, their strength lying on Mount Libanus, and along the coast of Sidon, Berytus, &c. It is a work of more labour than art, and the very situation alone is sufficient to render it almost impregnable ; standing on the top of a very high hill, enclosed with a deep ditch, cut out of the very rock, over which there is only one passage, by a draw-bridge, which however is now broken down ; so that there is no entrance remaining, unless you will be at the pains to clamber up the rock, wliich may be done in one place, but so difficult and hazardous, that a small slip may endanger a man's life. Nor is there any thing within to be seen sufficient to recompense the trouble of getting up to it, the building being confused, and the rooms very ill con- trived. Upon the top of the hill there is a well of a prodigious depth. This castle stands on the north side of the town, and from hence you have the best prospect of the country all about. You see Tadmor below, inclosed on three sides with long ridges of mountains, which open gradually towards the east to the distance of about an'hour's riding ; but to the south stretches a vast plain beyond the reach of the eye; in which is a large valley of salt, which is more probably the valley of salt mentioned 2 Sam. viii. 13, where David smote the Syrians, and slew 18000 men, than another, which lies but 4 hours from Aleppo, and has sometimes passed for it. The air is good, but the soil exceedingly barren; nothing green is to be seen, save some few palm-trees in the gardens, and here and there about the town ; and from them it probably had TOL. IV. F 34 PHILOSOPHICAL TRANSACTIONS. [aNNO l6g5. its name, both in Hebrew, Tadmor, which signifies a palm-tree, and in Latin, Palmyra ; and the whole country is thence denominated Syria Palmyrena ; and sometimes Solitudines Palmyrenae : so that the Latins did not change but only translate the old name, which therefore still obtains in these eastern parts, and the more modern is wholly unknown. The city itself appears to have been of a large extent, by the space now oc- cupied by the ruins ; but there are no foot-steps of any walls remaining, nor is it possible to judge of the ancient figures of the place. The present inhabi- tants, as they are a poor, miserable, dirty people, so they have shut themselves up, to the number of about 30 or 40 families, in little huts made of dirt, within the walls of a spacious court, which inclosed a most magnificent heathen temple. Certainly the world itself cannot afford the like mixture of re- mains of the greatest state and magnificence, with the extremity of filth and poverty. The whole inclosed space is a square of 220 yards each side, encompassed with a high and stately wall, built of large square stone, and adorned with pilasters within and without, to the number of 62 on a side. And had not the barbarity of the Turks, enemies to every thing that is splendid and noble, out of a vain superstition, purposely beat down those beautiful cornishes both here and in other places, we had seen the most curious and exquisite carvings in stone which perhaps the world could ever boast of: as here and there a small remainder, which has escaped their fury, abundantly evidences. The west side, where is the entrance, is most of it broken down, and near the middle of the square, another higher wall erected out of the ruins, which seems to have been a strong but rude castle. Within were to be seen the foundations of another wall, which probably might answer the front ; and probably the Mamalukes, whose workmanship it seems to have been, built the castle here for the security of the place. Before the whole length of this new front, except a narrow pas- sage which is left for an entrance, is cut a deep ditcii, the ascent whereof on the inner side is faced with stone to the very foot of the wall, which must have rendered it very difficult to be assaulted. The passage and the door are very narrow, not wider than to receive a loaded camel, or that two men may well walk a-breast. And as soon as you are within the first door, you make a short turn to the right, and pass on to another of the like size ; which leads into the court. This outward wall quite shrouds that magnificent entrance, which belonged to the first fabric ; of the stateliness of which we may judge by the two stones which support the sides of the great gate, each of which is 35 feet in length, and artificially carved with vines and clusters of grapes, ex- ceedingly bold, and to the life. They are both standing and in their places, and VOL. XIX.] PHILOSOPHICAL TRANSACTIOK3. 53 the distance between them, which gives us the width of the gate, 15 feet. But all this is now walled up to the narrow door before-mentioned. On entering the court are seen the remains of two rows of marble pillars, 37 feet high, with their capitals of most exquisite carved work; of these only 58 remaining entire ; but there must have been a great many more, for they appear to have gone quite round the whole court, and to have supported a most spa- cious double piazza or cloister. Of this piazza the walks on the west side, which is opposite to the front of the temple, seem to have exceeded the other in beauty and spaciousness, and at each end are two niches for statues, at their full length, with their pedestals, borders, supporters, and canopies, carved with the greatest art and exactness. The space within this once beautiful enclosure, which is now filled with nothing but the dirty huts of the inhabitants, seems to have been an open court, in the midst of which stands the temple, encompassed with another row of pillars of a different order, and much higher than the former, being above 50 feet high, of which only l6 remain. The whole space, contained within these pillars, is 59 yards in length, and near 28 in breadth. In the middle of which space is the temple, extending in length more than 33 yards, and in breadth 13 or 14. It points north and south, with a most magnificent entrance on the west, exactly in the middle of the building, which by the small remains yet to be seen, seems to have been one of the most glorious structures in the world. I never saw vines and clusters of grapes cut in stone so bold, so lively, and so natural, in any place. Just over the door we could just discern part of the wings of a large spread eagle, extending its whole width. Of this temple there is nothing at present but the outward walls standing, in which it is observable, that as the windows were not large, so they were made narrower towards the top than they were below ; but all adorned with excellent carvings. Within the walls the Turks, or more probably the Mama- lukes, have built a roof, which is supported by small pillars and arches, but a great deal lower, as well as in all other respects disproportionate and inferior to what the ancient covering must have been. And they have converted the place into a mosque, having added to the south end new ornaments after their man- ner, with Arabic inscriptions and sentences out of the Alcoran, written in flou- rishes and wreaths, not without art. But at the north end of the building, which is shut out of the mosque, are relics of much greater art and beauty. Thev are beautified with the most curious fret-work and carvings; in the middle of which is a dome or cupola, above 6 feet diameter, and above is of one piece, either hewn out of one entire rock, or made of some artificial composition, strongly hardened by time; in a word, it is a most exquisite piece of workman- ship. F 2 3^ PHILOSOPHICAL TRANSACTIONS. [aNNO IGqS. Presently after we were struck with an amazing sight of a multitude of mar- ble pillars standing scattered up and down, for the space of near a mile of ground, but so disposed as to afford no foundation to judge what sort of structures they formerly formed. Passing the remains of a handsome mosque, we had the prospects of such magnificent ruins, that if we may frame a conjecture of the original beauty of the place, by what is still remaining, it may be questionetl whether any city in the world could have vied with it in magnificence. Towards the north is a stately obelisk, consisting of/ large stones, besides its capital and the wreathed work above it ; the carvings are exceedingly fine ; its height is above 50 feet , and upon it probably stood a statue, which the Turks may have thrown down and broken in pieces. It is in compnss, just above the pedestal, 12i feet. On each side of this, towards the east and west, are seen two other large pillars, each a quarter of a mile distant, and part of another standing neat that of the east, which would incline one to think there was once a continued row of them. The height of that to the east is more than 42 feet, and the circumference in proportion. Upon the body of it is an inscription, in ancient Greek capitals, from which it seems evident they were a free state, governed by a senate and people, though perhaps under the protection of greater empires, the Parthians, and afterwards the Romans, who for a long time contended for the mastery ; and this government might continue among them till about the time of Aurelian, who demolished the place, and led Zenobia, wife of Ode- natus, captive to Rome ; who, though she be called queen, yet we find not that ever her husband had the title of king : but was only one of the chief inhabi- tants, a leading man in the senate ; who, while the Romans were busied in Europe, made himself great here, and by his own force repelled the Parthians ; who, having mastered whatever was held by the Romans on the other side of the Euphrates, made an incursion into Syria, but were driven back beyond the river by Odenatus. In the course of these wars Odenatus was slain ; but his wife Zenobia, being a woman of a masculine spirit, not only maintained her ground against her enemies abroad, but preserved her authority at home, keep- ing the government in her hands. Afterwards, out of a desire to cast off" the Roman yoke, she caused the whole garrison, which was left there by Aurelian, to be barbarously cut off": which bringing Aurelian back with his army, he quicklv took the city and destroyed it, putting the inhabitants to the sword, and carrying Zenobia captive to Rome, which was the flital period of the glory of the place. The other pillar towards the west in height and circumference answers this, and has upon the side a similar inscription engraved. Proceeding forward, directly from the obelisk, about 100 paces, you come to a magnificent entrance, very large and lofty, and for the exquisiteness of the VOL. XIX.3 PHILOSOPHICAL TRANSACTIONS. 37 workmanship not inferior to any thing before described. This entrance leads to a noble piazza of more than half a mile in length, and 40 feet in breadth, en- closed with two rows of stately marble pillars, 26 feet high, and 8 or 9 in com- pass. Of these 129 remain standing and entire ; but originally there could not have been less than 50o. On almost all the pillars are found inscriptions, both in Greek and the unknown language, of which we had time to take but few, and those not very instructive. And what we may collect from both, and several others of a like import, is, that as the state, the senate, and people, sometimes honoured those that had been in public trust with inscriptions on these })illars, so when this was not done by them, private persons had the liberty to do the same for their friends- The upper end of this spacious piazza was shut in by a row of pillars, stand- ing somewhat closer than those on each side ; and perhaps there might have been a kind of banqueting house above, but now no certain footsteps remain ; but a little farther to the left hand are the ruins of a very stately building, which I am apt to believe might have been for such a use ; it is built of better marble, and has an air of delicacy and exquisiteness in the work beyond what is dis- cernible in the piazza. The pillars which supported it are of one entire stone ; and one of them that is fallen down, but so firm and strong that it has received no injury thereby, measured 22 feet in length, and in compass 8 feet and 9 inches. In the west side of the great piazza are several openings for gates, leading into the court of the palace : two of which when they were in their perfection were perhaps the most magnificent in the world, both for the ele- gance of the work in general, and particularly for those stately porphyry pillars with which they were adorned. Each gate had four, not standing in a line with the others of the wall, but placed by couples in the front of the gate, facing the palace, two on one hand and two on the other. Of these only two remain en- tire, and but one standing in its place. They are about 30 feet in length, and Q in circumference, of a substance so hard, that it was with great difficulty we broke off a few shivers ; the art of making which I think is quite lost. The palace itself is so entirely ruined, that no judgment can be made what it was in its ancient splendour, either for its figure or workmanship. Hot sulphureous .baths are frequent in this country, and hence it obtained the name of Syria Salutifera. The scent of the waters here is much like those of Bath in England, but not so strong, neither is the taste so offensive. On the contrary, when they have run so far from the fountain, as to become cold, they are potable, and are the only waters the inhabitants use. But we, during our stay there, sent to a fountain of very excellent water, about an hour's journey from the city. 38 PHILOSOPHICAL, TRANSACTIONS. [aNNO 16Q5. On the east side of the long piazza stands a vast number of marble pillars, some perfect and others deprived of their beautiful capitals ; but so scattered and confused, that it is not possible to reduce them into any order, so as to con- jecture to what they anciently served. In one place are ] 1 ranged together in a square, paved at the bottom with broad flat stones, but without any roof or covering. And at a little distance from that are the ruins of a small temple, of very curious workmanship, but the roof is wholly gone, and the walls much defaced. Before the entrance, which looks to the south, is a piazza supported by six pillars, two on one hand of the door, and two on the other, and one at each end ; the pedestals of those in the front have been filled with inscriptions, both in Greek and the other language ; but they are become unintelligible. Their sepulchres are very curious, being square towers, four or five stories high, and standing on both sides of a hollow way, towards the north part of the city. They stretch out in length the space of a mile, and perhaps formerly might extend a great way further. They were all of the same form, but of different splendour and size, according to the circumstances of their founders. There were two which stood almost opposite to each other, and seemed most perfect of any, though not without marks of the Turkish fury. They are two square towers, rather larger than ordinary steeples, and five stories high, the outside being of common stone, but the partitions and floors within of good marble ; and beautified with admirable carvings and paintings, and figures both of men and women, as far as the breast and shoulders, but miserably defaced and broken. We entered one of these by a door on the south side, from which was a walk across the whole building just in the middle. But the floor was broken up, and so gave us a view of a vault below, divided after the same manner. The spaces on each side were subdivided into six partitions by thick walls, each partition being capable of receiving the largest corpse, and piling them one above another, as their way appears to have been, each of those spaces might contain at least 6 or 7 bodies. For the lowest, second, and third stories, those partitions were uniform, and altogether the same, except that from the second floor, which answered the main entrance, one partition was reserved for a stair-case. Higher than this, the building, being somewhat con- tracted towards the top, would not afford space for the continuation of the same method : therefore the two uppermost rooms were not so parted, nor per- haps ever had any bodies laid in them. Unless it was that of the founder alone, whose statue wrapped up in funeral apparel, and in a lying posture, is placed in a nich, or rather window in the front of the monument, so, as to be visible both within and without. The other monument opposite was much like this, only the front and entrance VOL. XIX.] PHILOSOPHICAL TRANSACTIONS. 3Q are towards the north, and it is not altogether so smooth nor so well painted. But the carvings areas good, and it appears altogether as stately and magnificent as the former. Besides it has the advantage in age of a whole century of years, as appears from the date of the inscription. It is placed above a nich in the front adorned with handsome borders and cornices, the place, doubtless, of some statue, and probably that of the founder. On the Cycloidal Spaces which are perfectly Quadrable. By Dr. JVallis. N°217, p. 111. It is generally supposed that no part of the semicycloid figure, adjacent to the curve, is capable of being geometrically squared but these two, viz. 1. The segment a /; v, fig. 6, pi. 1, taking a v = -i-Aa, (which was first observed by Sir Christopher Wren, and after him by Huygens and others,) and it is = 4« r = 4 r'^ V" 3. — 2. The trilinear A