TRANSACTIONS ROYAL SOCIETY OF EDINBURGH. VOL. VI: ee a] Mt | til HI i 2 <7; OPO GEC WEL AM. ‘EDINBURGH, PRINTED FOR CADELL AND DAVIES, LONDON, 2 AND - ARCHIBALD CONSTABLE & CO. AND BELL & BRADFUTE, EDINBURGH. . 1812 re eee wy e “es | be =o AN Sey Sha “4 c - wa pee ‘ 0 ‘ V ne ' A - ; ‘ \ -) a ot , site m * \ . st — ” 7 * te of ate , - ’ ce Mite — oa - - f 7 > aoe, eer ee Sk Raa : : MOmaaTe waaa ane f Sn ‘parva avant Ls 9 WOR GAT aagnt i aes 3 ame reioo angatnna ee: >. arpa ve yh Jab ‘ake 0 SAN Mie ahr: dade 1 ae » CONTENTS SDXT-BeVOLU ME, HISTORY of the SOCIETY. Carta Nova Erectionis Societatis Regahs- Edinburgi, 1808, - - - - . Page iit Laws of the Royal Society of Edinburgh, enacted 23d May 1811, te . . ix Presents made to the Royal Society of Edinburgh since the Year 1809, # 2 =f See I. A Description of the Strata which ve ur in ascending Srom the Plains of Kincardineshire te the summit of Mount Battoc, one of the most elevated points in the Eastern District of the Grampian Mountains. By Lieutenant-Colonel Imrie. - - 3 I. A Geometrical Investigation of some curious and inte- resting Properties of the Circle, &c. By James Glenie, Esq. = = a 91 vi | CONTENTS. Ill. Account of a Series of Experiments, shewing the Ef- fects of Compression in modifying the Action 6 Heat. By Sir James Hall, Bart,....... - were’ IV. Of the Solids of Gacatect Attraction, or those which, among all the Solids that have certain Properties, Attract with they greatest Force in a given Direc- tion. By Mr Playfair. - - Page 187 V. An Account of a very extraordinary Effect of Rare- faction, observed at Ramsgate, by the Reverend S. Vince. Communicated by Patrick Wilson, Esq. 245 VI. Some Account of the large Snake Alea-azagur, (Boa Constrictor of Linneus), found in the Province of Tipperah. Communicated by Mr James Russell. 249 VII. Chemical Anah ysis of a Black Sand, from. the River Dee in Aberdeenshire ; and of a Copper Ore, from Arthrey in Mali codes By Thomas. Thomson, Vio ee 0 Ran eee = - - 253 VIII. New Series for the Ee es of the Conic Sections, and the Computation, of Feoepoe ithms.. By Mr Wal- lace. ~ - - (06 & 109 1269 IX. Remarks on a Mineral from Greenland, supposed to be Crystallised Gadolinite, By. Thomas Allan, “Esq. = : = tn aan “SOyiarh S104 345 X. On the Praghss aia Heat when ‘communicated to Spherical Bodies ae their ag acai ‘By Mr Play- fair. a0 AMOUNT INOS FQ SOV OgVesowTd Inorrssete 358 ia XI. CONTENTS. vib XI. Experiments. on Allanite, a new Mineral from Greenland. By Thomas ‘Thomson, M. D. 371 XII. A Chemical Analysis of Sodalite, a new Mineral from Greenland. By'Thomas Thomson, M.D. 387 XIII. Demonstration of the Fundamental Property of the Lever. By David Brewster, LL. D. - 397 XIV. On the Rocks in the vicinity of Edinburgh. By Tho- mas Allan, Esq. - a’ 4OS HISTORY OF THE SOCILET Y.. ee N the original charter of the Royal Society, it was provided that the collections of the Society should be deposited, if belonging to Natural History, in the Museum of the Univer- sity, and if to Antiquities, in the Library of the Faculty of Ad- - vocates. Much inconvenience, however, could hardly fail to result from this arrangement, especially wh@n the researches of the Society, having, as of late, been much turned to Geolo- gy, it became an object to collect together the specimens which served to illustrate the subjects under discussion, and to - have them at hand when reference should be necessary. In a Museum arranged with a view to public lectures, (like that of the University), such an order as was required for this purpose could not easily be preserved; the Professor of | Natural History must feel himself interrupted by the exami- nations which the Members of the Royal Society might wish Vou. VI.—P. IT. a to 2 HISTORY of the SOCIETY. ‘to make ; and it would often be a point of delicacy, not to give him the trouble that such examinations would require. TueEse considerations induced the Society to apply for a new charter, under which its collections should remain in its own possession, so as to be at all times accessible to its Mem- bers. As the interest of the two bodies just mentioned, might be somewhat affected by these alterations, the first step taken was to give them information of the intentions of the Society, and to request their concurrence in a measure of such mani- fest justice and utility. The Faculty of Advocates readily as- sented to this proposal ; and the University, though at first in doubt whether it were not bound in duty to resist the al- teration, on more mature deliberation, resolved to withdraw all opposition. As it was not meant that the new charter should have any retrospect, the Huttonian Collection, with a great number of other articles, the property of the Society, still remain in the University Museum. ‘The foundation of a new collection, in the Society’s apartments, has been laid, by a cabinet presented by Mr Atay, containing specimens of the rocks round Edin- ~ burgh; a collection by Colonel Imrrzg, illustrating the section of the Grampians which he has given in the 5th volume of the Transactions of the Society; and a collection of specimens from Sir Grorce Mackenzie, illustrating the Mineralogy of Iceland. Tux New Charter, which follows, hardly differs in any thing from that contained in the first volume of the Transactions of the Society, except in what respects the two restrictions that have just been mentioned. ‘CARTA CARTA NOV ERECTIONIS. SOCIETATIS REGALIS EDINBURGI.. 1808... GrorGIUS TERTIUS, Det gratia; Britannarum Rea; Fidei. Defensor ; omnibus probis hominibus ad quos prasentes h- tere nostre pervenerint, salutem: Quandoquidem Nos conside= _rantes, quod petitio humilis nobis -oblata fuerit a Regali Socie- tate Edinburgi, et prefideli nostro et» predilecto: consanguineo Henrico Duce de Buccleuch, ejusdem preside, in nomine et vice Societatis, et omnium ejusdem Sociorum ; im qua petitione enar- ratur, quod per regiam nostram cartam, dutam vigesimo nono die mensis Marti anno Domini millestmo septingentesimo et oc- togesimo tertio, Nobis benigné placuisset constituere, erigere et mecorporare quosdam wbi nominatos in corpus politicum et corpo- rate ratum,; 4 HISTORY of the SOCIETY. ratum, nomine titulogue Recawis Socirratis Eptnpurcy, ad promovendas literas et scientiam utilem, cum facultatibus et. privilegus abidem concessis, et speciatim,. ut potens et capax sit tenendi proprietatem realem et personalem, causasque agendi et defendendi, Presidem et Socios eligendi, canones ordinandi, et perpetuam successionem sub tali nomine habendi : quod, virtute predicte carta, Regalis Soctetas Edinburgi, ita creata, substi- tuerit, suisque offictis a prima institutione, rite functa sit: quod carta predicta ordinatum fuerit, cunctas res antiguas, tabulas publicas, librosque manuscriptos, quos acquisiverit Societas, in Bibliotheca Facultatis Juridice deponi ; atque universas res ad historian naturalem pertinentes, quasque Societas acquisiverit, in Museo Academie Edinensis deponi: quod, ab hac constitu- tione incommodum haud parvum ortum fuerit ; cim Regalis Societas, nullum jus in Bibliothecarios Facultatis Juridica, nec in Custodes Musai Academie Edinensis, habeat, nec horas eorum ministeru regulasve admissionis ad ea repositoria pra- scribere possit, nec Societati licitum sit congressus suos in eo- rum alterutro tenere; qué cum ita sint, hactenus Societati non licuit suas collectiones ita disponere, ut Soctorum aliorumve studio et disquisitiont apté subjiciantur, unde et alia dona ex- pectanda essent : Quod predicta Societas, causd hec incommo- da amovendi, nostraque bona proposita in hac institutione ad effectum perducendi, sapientie nostre regia humiliter subjiciat, ut detur Societati jus collectiones suas cujuscunque generis uno in loco deponendi, quo sibi ordine placuerit, sub custodibus a Societate eligendis ejusque potestati subjectis; itaque ut car- tam, cum privilegus idoneis humilibus nostris petitoribus: conce- dere dignemur ; ut et in hac petitione oratum sit, ut Nobis benigne placeret de novo Cartam Nostram Regiam concedere dicta Regali Societati Edinburgi, ejusque Sociis, qua iterum darentur jura, facultates, et privvilegia, in carta regia per quam HISTORY of the SOCIETY. 5 quam corpus istud creatum fuerat concessa, et qua insuper provideretur, uti nobis in regia nostra sapientia adoneum vi- deatur, ut Societati potestas daretur collectiones suas anted memoratas in uno edificio deponendi, eas legibus, et ers mini- stris, qui Societati placerent, hosque subi subjectos haberet : Et nos certiores factt hanc petitionem justam esse rationique consentaneam, et .certis conditionibus et modis, in prasentibus eapressis, concedi debere: IcituR, constituimus, erigimus et éncorporavimus, sicutt Nos regid nostra prerogativd, et gratid speciali, pro Nobis notrisque regiis successoribus, per has pre- sentes, constituimus, erigimus, et incorporamus, predictum Hen- ricum Ducem de Buccleuch, Sociosque dicte Regalis Societa- tis, atgue alios qui postea eligentur Soci, in unum corpus po- liticum et corporatum, vel legalem incorporationem, nomine et titulo Recatis SocretatTis Kpinpurei, ad promovendas literas et scientiam utilem, utque talis existens, et tali nomine, perpetutatem habeat et successionem ; DECLARANTES, Quod dicta Societas capax sit capere, tenere, et frui proprietate reals. seu personal, et petere, causas agere, defendere et re- spondere, et convemri, in jus trahi, defendi et responderi, in omnibus seu ullis nostris Curis Judicature ; et declarantés quod dicte Societats fas sit, sigallo, tanguam Societatis sigillo, uti; dantes potestatem dicte Societati, per majorem suffragio- rum numerum eorum qui aderunt, eligendi Presidem aliosque officiartos pro negotiorum administratione ; necnon ordinandi canones, ad quos Soci sint eligendi et res Societatis sint admi- nistrande, conditionibus hujus carte sive donationis haud in- congruenles, nec legibus et praxi nostri regni Scotie contra- rios ; et declarantes, quod hajusmodi canones sanciri nequeant, nisi rité proposits fuerint in congressu habito saltem uno men- st ante illum congressum quo sanciendi sint : dantes etiam po- testatem Societata ordinandi et administrandi collectiones re- TUM 6. HISTORY of the SOCIETY. y rum antiquarum, tabularum publicarum, librorum manuscrip- torum, et rerum ad historiam naturalem pertinentium, quas Societas posted acquisiverit, easque in Musco et Bibliotheca, tali ordine et modo ut Societati placuerit, deponendi: satvis tamen conditionibus, in hac nostra carta provisis ; declarantes insuper hanc cartam nostram concessam esse sub his conditione- bus sequentibus, videlicet, Quod jura, facultates, et privilegia, per praesentes in dictam Societatem collata, nullo. modo detra- hent de ullo jure domini quod competit Academie Edinensi- in collectiones antehac depositas in Museo Academia, virtute car- te nostra Societatr. Regali data, pradicto vigesimo nono die mensis Martw mallesomo septingentesimo et octogesimo tertio ; antedicta Societate quantum in se est astricta, omne jus, ad collectiones antehac factas et in Museo. pradicto depositas, in dictam Academiam transferre ; et quod Historie Naturalis Pro- fessori copia introitis in Museum et Bibliothecam Societatis Regalis detur equé ac Sociis ipsius Societatis ; et quod dicte Societati non sit licttum constituere Professorem, praltectorem seu Doctorem Mineralogia, Geologie, aut Historie Naturalis, nec swis collectionibus uti ad talem institutionem promovendam, nisi que vel nunc sit, vel posthac fuerit, in Academia Edinensi. —IN cUJUS REI TESTIMONIUM, sigillum nostrum per Unionis Tractatum. custodiend., et in Scotia vice et loco Magni Sigilli ejusdem utend., ordinat., presentibus append: mandavimus ; Apud Aulam nostram apud St James’s, vigesimo septimo die mensis Decembris anno Domini millesimo octingentesimo et octa- 0, regnique nostri anno guadragesimo nono. Per HISTORY of the SOCIETY. 7 Per signaturam mantis D. N. Regis supra script. Written to the Seal, and registered the thirtieth day of August 1811. James Dunpas, Dep’. Sealed at Edinburgh, the thirtieth of August, One thousand. eight hundred and eleven years. James RoBERTSON, Sub‘. £80 Scots. _ Tuts charter, as well as the former, having left the Society in possession of the power of making By-laws for the regula- tion of its affairs, it was proposed to revise the whole of those ‘laws, and to make such alterations as, after the experience of thirty years, might appear to be necessary. Tue Society, therefore, having at several General Meet- angs taken this subject into consideration, after mature deli- beration, 8 HISTORY of the SOCIETY. beration, and with due attention to the clause in the charter that respects. the enactment of such laws, did, at a General Meeting, on the 23d of December 1811, sanction the Laws that follow, and declare them to be the rules by which the Society is to be governed, till all, or any of them are regularly repeal- ed. LAWS. LAWS ROYAL SOCIETY OF EDINBURGH, ENACTED 23d May 1811. I. HE Roya Society oF EpinpurcH shall be composed of Ordi- nary and Honorary Members. ' II. Every Ordinary Member, within three months after his election, shall pay as fees of admission Three Guineas, and shall further be bound. to pay annually the sum of Two Guineas, into the hands of the Trea- surer. Til. Members shall be at liberty to compound for their annual subscrip- tion, each paying according to the value of an annuity on his life, deter- . mined as in the ordinary insurance on lives. The power of raising the annual subscription shall remain with the Society. IV. Ordinary Members, not residing in Edinburgh, and not compound- ing for annual subscription, shall appoint some person residing in Edin- burgh, by whom the payment of the said subscription is to be. made. and shall signify the same to: the Treasurer. b ¥. ‘ xO HISTORY of the SOCIETY. V. Members failing to pay their subscriptions for three successive years, due application having been made to them by the Treasurer, shall cease to be Members of the Society, and the legal means for recovering such arrears shall be employed. Vit None but Ordinary Members are to bear any office in the Society, or to vote in the choice of Members or Office-bearers, nor to interfere in the patrimonial interest of the Society. Vil. The number of Ordinary Members shall be unlimited. VIII. The Ordinary Members shall receive the volumes or parts of the So- ciety’s Transactions, when published, at the booksellers price, or the price at which they are sold to the trade. This regulation to continue in force for five years from the date of its enactment ; and it is left to the Society then to consider, whether the volumes cannot be afforded gr atis to the Members. IX. The Society having formerly admitted as Non-resident Members, gentlemen residing at such a distance from Edinburgh as to be unable regularly to attend the Meetings of the Society, with power to such Non-resident Members, when occasionally in Edinburgh, to be present at the Society Meetings, and to take a part in all their inquiries and proceedings, without being subjected to any contribution for defraying the expences of the Society ; it is hereby provided, that the privileges of such Non-resident Members already elected shall remain as before ; but no Ordinary Members shall be chosen in future under the title and with the privileges of Non-resident Members. ‘The Members at present called Non-resident shall have an option of becoming Ordinary Members; if they decline this, they shall continue Non-resident as for- merly. pe HISTORY of the SOCIETY. rr s a The Honorary Members of the Society shall not be subject to the annual contributions. They shall be limited to Twenty-one, and shall consist of men distinguished for literature and science, not residing in Scotland, pe See The election of Members, whether Ordinary or Honorary, shall be by ballot; it shall require the presence of Twenty-four Members at least to make a quorum, and the election shall be determined by the majority of votes. D.C: Se The election of Members shall be made at one General Meeting an- nually, on the fourth Monday of January. XIII. No person shall be proposed as an-Ordinary Member, without a re- commendation presented by a Member of the Society, and subscribed by Three, to the purport mentioned below * ; which recommendation shall be hung up in the Rooms of the Society, at least during Three Ordinary Meetings (of the Classes) previous to the day of election. XIV. In order to carry on with facility and success those-ymprovements in science and literature, which. are the objects of the institution, the So- ciety shall be divided into two Classes, the Physical and the Literary Class ; the former having for its department the sciences of Mathematics, Natural Philosophy, Chemistry, Medicine, Natural History, and what relates to the improvement of Arts.and Manufactures ; the latter having b 2 for * « A.B. a gentleman well skilled in many. branches of Philosophy and Polite Learning, (Mathematics, Chemistry, Natural History, &c.) being to our knowledge desirous of be- coming a Member of the Royal Society of Edinburgh, we whose names are subscribed, do recommend him as deserving of that honour, and as likely to prove an useful and valux able Member.” Election of Mem~ bers. Office-bearers. 12 HISTORY of th SOCIETY. for its department the inquiries relative to Speculative Philosophy, An- tiquities, Literature and Philology. KV. ‘The Classes shall meet alternately on the first and third Mondays of ‘every month, from November to June inclusive. It shall be competent, ‘however, to bring matters of a Physical or Literary kind, before either lass of the Society indiscriminately. To facilitate this, one Minute- book shall be kept for both Classes; the Secretaries of the respective. Classes either doing the duty alternately, or according to such agreement as they may find it convenient to make. XVI. The Society shall from time to time make a publication of its Trans- actions and Proceedings. For this purpose, the Council shall select and arrange the papers which they shall deem worthy of publication in the Transactions of the Society, and shall’ superintend the printing of the same. The Transactions shall be published in Parts or Fasciculi, and the ex- pence shall be defrayed by the Society. XVII. There shall be elected annually for conducting the publications and regulating the private business of the Society, a Council, consisting of a President ; Two Vice-Presidents ; a President for each Class of the So- ciety ; Six Counsellors for each Class; one Secretary for each; a Trea- surer; a General Secretary; and a Keeper of the Museum and Li- brary. XVIII. The election of the Office-bearers shall be on the fourth Monday of November. XIX. Four Counsellors, Two from each Class, shall go out annually. They ate to be taken according to the order in which they presently stand on the list 4 the Council. XX. HISTORY of the SOCIETY. 1 XX, The Treasurer shall receive and disburse the money belonging to the Society, granting the necessary receipts, and pia the money when due. He shall keep regular accounts of all the cash received and expended, which shall be made up and balanced annually; and at the General Meeting in January, he shall present the accounts for the preceding year to be audited. At this Meeting the Treasurer shall also lay before the Society a list of all arrears due above twelve months, and the Socie- ty shall thereupon give such directions as they may find necessary for recovery thereof. XXI. At the General Meeting in November, a.Committee of Three Mem- bers shall be chosen to audite the Treasurer’s accounts, and give the necessary discharge of his intromissions. —_- The report of the examination and discharge shall be laid before the Society at the General Meeting in January, and inserted in the records. XXil. The General Secretary shall take down minutes of the proceedings of the General Meetings of the Society and of the Council, and shall enter them in two separate books. He shall keep a list of the Donations made to the Society, and take care that an account of such Donations be pu- blished in the Transactions of the Society. He shall, as directed by the Council, and with the assistance of the other Secretaries, superintend the publications of the Society. : XXIII. A Register shall be kept by the Secretary, in which copies shall be inserted of all the Papers read in the Society, or abstracts of those Pa- pers, as the Authors shall prefer; no abstract or paper, however, to be published without the consent of the Author. It shall be understood, nevertheless, that a person choosing to read a paper, but not wishing to put Treasurer. Secretary. 14 HISTORY of ea SOcre TT: put it into the hands of the Secretary, shall be at liberty to withdraw it, if he has beforehand signified his intention of doing sO. For the above purpose, the Secretary shall be empowered to employ a Clerk, to be paid by the Society. XXIV. Another register shall be kept, in which the names of the Membexs shall be enrolled at their admission, with the date. XXV. A Seal shall be prepared and used, as the Seal of the Society. ~ XXVI. The Librarian shall have the custody and charge of all the Books, Manuscripts, objects of Natural History, Scientific Productions, and other articles of a similar description belonging to the Society ; he shall take an account of these when received, and keep a regular catalogue of the whole, which shall lie in the Hall, for the inspection of the Mem- bers. . XXVII. All articles of the above description shall be open to the inspection of the Members, at the Hall of the Society, at such times, and under such regulations, as the Council from time to time shall appoint, PRESENTS > OO PRESENT'S made to the Royat Socrety or Epinzsurcu since the Year 1809. The Sixth Volume of the Scriptores Logarithmici—From Mr Baron Maseres. Treatise on the Gout, by the late Dr Hamiiron of Lynn-Regis.—From the Au- thor. Traité de Mineralogie, par M. tz Comprtr ve Bournon, 3 vols. 4to.—From-the Au- thor. An Engraving, representing all the Mountains of the World, by R. Riddel, Esq; together with the History of Mountains, by Joseph Wilson, Esq; vols. 1st and 2d.—From Messrs Rippex and Wuson. Recueil de Quelques Antiquités trouvées sur les Bords de la Mer Noire, par M. Leon pe Wexer.—From the Author. ‘Nova Acta Petropolitana, tom. 14.—From the Imrenta. Acapemy of St Pe- tersburgh. Philosophical Essays, by Tuomas Gorpon, Esq; 2 vols. 4to.—From the Author. Transactions of the Linnean Society, vol. 8th and 9th.—From the Linnean So- CIETY. ‘ Asiatic Researches, vol. 10th and 11th.—From the Benean SociEry. Philosophical Transactions, for 1809, 1810, 1811.—-From the Rovat Society or Lonpon. Memoirs of the American Academy, vols. Ist, 2d, and 3d.—From the Amer- RICAN ACADEMY. Transactions of the American Philosophical Society, vol. 6th, part 2d.—From the American PuitosopuicaL Sociery. “Observations on the Hydrargyria, by GEorcE AutEy, M. D.—From the Author. ‘Transactions of the Geological Society of London, vol. 1st.—From the Groto- e1caL SociETy. Travels 16. - HISTORY of the SOCIETY. Travels in Iceland, by Sir George Mackenzie, Baronet, Annals of Iceland, from 1796 to 1804. Manuscript copy of the Sturlinga Saga. History of Iceland during the 18th century. A compendium of Anatomy, translated into Ice- landic, from the Works of Martinet. Pope’s Essay on Man, in Icelandic -verse.—From Sir Georce Mackenziz. Essay on the Natural History of se Salt District in ar by Dr Hottanp.— From the Author. Collection of Specimens, illustrating the Mineralogy of the Peaer round Edin- burgh.—From Tuomas pee Esq;. Collection of Specimens, Rusteatiag the Section of the Grampians, at the begin- ning of this volume, with a descriptive Catalogue.—From Lieutenant-Colonel Imrie. Model in Relief, representing the Granite Veins at the Windy Shoulder in Gal- loway.—From Sir James Haut, Baronet. Collection of Specimens, illustrating the Mineralogy of Iceland.—From Sir Grorce Macwenzirz, Baronet. I, 4 DescripTion of the StRATA which occur in afcending from the PLAINS of KINCARDINESHIRE fo ‘he summit of Mount Batroc, one of the moft elevated points in the Eaftern Diftrict of the GRAMPIAN MounTAINs. By Lieutenant-Colonel Imrir,. F.R.S. EpIn.. [Read 5th March 1804.] HE moft mountainous parts of Scotland are fituated in its - -weftern and north-weft diftricts.. From thofe parts of the country, feveral chains of mountains branch off, and continue their courfes in various directions, and to various extent. The moft extended of thofe chains is that of the Grampians. This chain takes its rife from nearly about the centre of the above al- pine diftrict, and continues its courfe in a direction almoft due: eaft, or perhaps a little to the fouth of that point, until it difap- pears in the German Ocean, betwixt the towns of Aberdeen and Stonehaven. Tuis chain, in its eaftern diftrict, confifts of three ranges, run- ning nearly parallel to each other; the two lateral ranges being. confiderably lower than the central one. To the lateral moun- tains are attached a range of lower hills, that flope down into undulated grounds, which fkirt the adjacent plains. Tue general fhape of the individual mountains compofing thofe three ranges, is oblong, rounded, and fometimes flattifh on. the tops; their length is always in the direction of the A 2. chain; .. (DESCRIPTION of the “a chain, that is to fay, from weft to eaft: and I have obferved, not unfrequently, that the weftern ends of thofe oblong moun- tains are more bulky than their eaftern extremities, and that they . flope and taper in fome degree towards this quarter. Their ge- neral covering is that of a coarfe gravelly foil, produced by their own decompofition ; and. the produce of this foil is heath. But upon fome of the heights in the central range, I have found beds or layers of that f{pecies of turf called Peat, from fifteen to twen- ty feet in thicknefs, which repofe upon the gravelly foil that there covers the native rock. At this eaftern part of the Grampians, where I am now about to endeavour to give a defcription of the {tratification, the moun- tains feldom thow any confiderable extent of naked rock. In their courfe to the eaftward, as they approach the fea, they begin to contract in breadth, and cover much lefs {pace of coun- try; and where they finifh their courfe at the fea, their height will fearcely entitle them to the appellation of hills : but although - they become fo diminutive in height and in breadth, yet the ma- terials of which they are formed continue the fame as thofe which compofe the ranges where they are in their greateft alti- tude, and their exterior characters, as to form and figure, alfo continue the fame. AmonG the rivers which have their fource in the Grampians, that of the North Efe is not the firft in rank as to fize, nor is it the moft diminutive. At a confiderable diftance from the plains in the interior of the mountains, a {mall lake called Loch Lee is formed, in a rocky bafon, by a rivulet, and fome fprings and rills flowing from marfhy grounds. From this lake the North Efk iffues, not in a very confiderable flow, but, being foon joined by other {treams and alpine torrents, it {wells to a confiderable fize, and continues a courfe from this lake almoft due eaft, betwixt the central and fouth lateral ranges of the mountains, for an ex- tent of about feven miles: it then fkirts Mount-Battoc, and be- ing STRATA of the GRAMPIANS. 2 ing there impeded, in its eaftern direction, by fome of the hills forming the bafis of that mountain, it then changes its courfe, almoft at a right angle, and from thence flows in a due fouth di- rection. In this laft direction, it opens a way for itfelf through the fouth lateral range, and enters the plains of Kincardine, and - Forfar fhires, where it immediately becomes the line of divifion of thofe two counties. It leaves thofe plains by a hollow betwixt the two low hills of Garvoke and Pert, and after a courfe of near- ly thirty miles from its fource, it joins the fea fomewhat to the eaftward of the town of Montrofe. It is in the bed of this river that I have examined the ftrata of the Grampians of which I am now to give a defcription. The fection extends about fix miles, from the horizontal grit or fandftone in the plain, to the granite of Mount Battoc, which is one of the mountains in the central range, and one of the higheft of the chain in that part of the country. My direction, in this examination, is about due north, piercing through, almoft at right angles, the ftrata of the moun- tains, which are here nearly in a vertical pofition. In this fhort f{tretch of fix miles, a great deal of matter highly interefting to geology prefents itfelf. In it, we pafs from the fe- condary horizontal {trata of the neweft formation, to the verti- cal, contorted, primary {trata of the oldeft date, and terminate with granite, the primitive rock in the conception of many geo- logifts. Thus, it embraces a complete range of the foflil objects, which in this part of Scotland intervene between that which is deemed the oldeft and what is accounted the moft recent in point of formation. From the various ftrata ftanding in a pofition vertical, or nearly fo, and the river North Efk, cutting acrofs thefe ftrata, at right angles, the fucceflion is uncommonly well exhibited to view, and a fair difplay of the ftru@ure of this country, and of the materials compofing it to a great _ depth, is open to the attentive obferver. In addition to this fine difplay of the fucceflion of ftrata, the arrangement of them will be 6 DESCRIPTION of the be found to offer fome very curious and important facts, parti- cularly the gradual elevation, and the final perfect vertical pofi- tion of the fand{tone and puddingftone, as well as the rather un- ufual manner in which the fecondary es the older ftrata meet each other. In the-feries here to be defcribed, the repeated occurrence of rocks of the whiz and of the porphyry formation, refpecting the origin of which opinions are fo much divided, adds confiderable intereft ; efpecially when the form and fituation in which they occur, and the condition of the contiguous rocks, are taken into confideration. In the account which I am now about to give, I fhall endea- vour to lay down a fair reprefentation of the facts as Nature pre- fents them, unbiafled by any of the prevailing theories of cof- mogony. I fhall avoid every geological difcuflion whatever, leaving it to others to draw thofe conclufions, in relation to their own fpeculations, which they fhall imagine the — to: warrant. In that part of the plains of Kincardinefhire from which I take my departure, the native rock confifts of Siliceous Grit or Sandftone, which is here divided into an immenfe number of beds or layers, of various thicknefles, from one inch to four feet,. folidftone. In many places, gravel of various fizes is found im- bedded in this grit ; which gravel confifts moftly of water-worn quartz, and fmall-grained granites. The colour of the general mafs of this grit is a dark-reddifh brown, and in fome few pla- ces it fhows narrow lines and dots of a pearl-grey colour. The component parts of this grit confift of {mall particles of quartz, and {till more minute particles of filvery-luftred mica: thefe owe their cohefion in mafs to a martial argillaceous cement, to which this rock alfo owes its colour. Thofe lines and dots of pearl- grey colour, generally occur in the moft folid and thickeft beds. of. STRATA of ihe GRAMPIANS. 4] of the rock: they are formed of the fame materials with the other parts of the ftone; but into them the ferruginous ftaining matter has not apparently been able to penetrate, and they de- rive their prefent greyifh appearance from the natural -colour of its particles of quartz, which are here per fe of a bluifh-white tint. This rock, in the plain, is perfectly horizontal in its pofi- tion; but upon its approach towards the undulated grounds, which here form the loweft bafis of the Grampians, it begins to rife from its horizontal bed, and, gradually increafing in its ac- clivity tewards the mountains, it at laft arrives at a pofition per- fectly vertical. | For the firft quarter of a mile from where this grit begins to leave its horizontal pofition, the rife is very gradual; but after that diftance, it becomes more rapid, and in a mile it gains its vertical pofition. WueEre this grit or fandftone rock is in its moft folid ftate, and where its pofition is perfectly vertical, betwixt two beds or layers of it, there occurs a bed of Whinftone forty feet broad. THE main body of this bed of whin interfects none of the layers of grit, but ftands upright betwixt two of them, to both of which it is clofely joined. The river, at this place, has, in its paflage, worn down this bed of whin equally with that of the adjoining grit, and a perpendicular face of it can be examined upon each fide of the river, from fifty to fixty feet in height. Upon examining the fection of this bed, I found upon the eaft fide of the river two branches, which fprung from the main body of the whin, nearly where the water of the river at prefent wafhes the bafe of its perpendicular furface. One of thofe branches {prings from the right fide of the trunk, and the other fprings from the left fide; they at firft diverge from the trunk as they ‘afcend, and where they push out laterally, they interfect the con- tiguous ftrata, and penetrate them in a zig-zag manner; but at laft, in a pofition betwixt two of the layers of the grit, they con- tmue 8 DESCRIPTION of the tinue their direction upwards, decreafing in their diameters as they afcend, until they finifh their courfe near to the fuperficial foil which here covers the rock. The grit contiguous upon both fides to the bed of whin, is confiderably harder and more com- pact than it is in any other part of the ftratification; and that angle of the grit which lies between the body of the whin and its branches, is more indurated than the ftrata of the grit upon each fide. iy Te {MM Grit in | Grit in vertical Mb vertical ftrata. ftrata. The Tuts fpecies of whin is not very compadt in its texture. Its fracture is fomewhat earthy, and is of a brownifh-black colour ; but it has a confiderable degree of induration, and has fome fpecks of luftre in it. Having paffed this bed of whin, the grit continues in the fame pofition as immediately before the whin occurred ; but, foon after, the gravel, which I have mentioned to be in fome places imbedded in the grit, increafes in quantity, and at laft the {trata are formed of a rock compofed entirely of that fpecies of gravel, and which may be called Gravel-{tone or Plum-pudding-rock. This aggregate conftitutes a ftratum four hundred yards thick. Its ftretch is nearly from weft to eaft, and it is vertical in its pofition. Its compofition confilts of quartz, porphyries, and fome fmall-grained granites, all of which have evidently been rounded by attrition in water: they are STRATA of the GRAMPIANS. “9 dre of a vaft variety of fize, from that of a pea to the bulk of an oftrich egg. Thefe are all firmly combined by an argillaceous ferruginous cement. In fome parts of this gravel rock, are to be feen thin lines of a fine-grained grit, ftretching through -it from weft to eaft; it is by thofe lines alone that the vertica- lity and the ftretch of this mafs is difcoverable. ts general co- lour, in mafs, is that of a ferruginous red. THis plum-pudding rock is immediately followed by a he ceffion of ftrata of fine-grained grit, in thin layers: it has a very confiderable degree of induration, and is of a dark ferruginous brown colour. This deviates a little from the vertical pofition, and inclines to the fouth : the ftretch is from weft to eaft, and its ex- tent towards the north is two hundred and fixty yards. To this rock immediately fucceeds a fpecies of Porphyry, the principal mafs of which confifts of an-indurated argil. Its colour is of a purple or lilac brown: its induration is very confiderable, and its fracture is rough and earthy. The materials which are im- bedded in its mafs, confaft of fmall particles of quartz, felfpar, blackifh-brown mica, and fpecks of iron ochre; all of thefe are but thinly fcattered. ‘The fpace in the courfe of the river occu- pied by this porphyry is two hundred and twenty yards: its ftretch is nearly-from weft to eaft, and it inclines in a {mall de- gree to the fouth. The rock which fucceeds to this porphyry, and which is in contact with it, is difficult to defcribe ; and this difficulty arifes from the great diforder of the ftratification, and the variety of materials compofing it. The ftrata of this bed do not fucceed each other in a regular manner. Portions of them of various dimenfions lie together, but very varioufly difpofed : fome are vertical, fome horizontal, fome dip to the fouth, one only to the north, affording a folitary inftance of a northern in- ~ clination of the ftrata in this field of examination. Tue materials of this mafs of confufed ftratification, are of very different defcriptions. In one place, a quartzofe ftone VoL. VI.—P. I. B abounds, 10 .\ DESCRIP TION of ‘the abounds, of a granular texture: it here, in general, refembles a fine-grained, highly indurated, and compact quartz fandftone : fometimes, however, it approaches to hornftone, and even fome- times to quartz in mafs. Much of it has a white colour: the reft is tinted of an ochery brown, of different fades. In other places, the ftratified matter confifts of a ftone of a-laminated tex- ture, with undulating lamellz of a ferruginous tint, looking like an indurated fhale ; and various gradations of both kinds prefent themfelves. ‘This jumble is 1n thicknefs three hundred yards; and to it immediately fucceeds a very narrow ftratum of Argil= ~ lite, which is of a greenifh-grey colour, and very thinly lamel- lated. Tuis argillite is fucceeded by a bed of Whin, thirty-three feet. broad. This whin is of a dark blackifh-brown colour, and is of a more compact texture, than the whin which I have defcribed occurring in the grit, and is poflefled of more induration: the materials of compofition are nearly the fame in both. Irs general ftretch is nearly from weft to eaft; but in this ftretch, where it has been expofed to the eye by the river, it is fomewhat curved, and’prefents its convex fide to the mountains. To this bed of whin.fucceeds a narrow ftratum of Argillite, per- fedtly fimilar to that which I have juft now. defcribed upon its fouthern fide. To this fucceeds a feam of Limeftone, fix feet broad: This limeftone is of\a pale blue colour, and is much in- terfected by {mall veins of quartz trending through it in all de. rections: In this limeftone, I was unable to trace the remains of any animal or vegetable production. Its pofition is vertical, and it is immediately fucceeded by another-narrow ftratum of argillite, thinly lamellated. To this narrow ftratum of argillite fucceeds-a bed of Whin, feventy-five feet broad. This whin is, in its texture, more com-- pact ;: and its fracture. difplays a fmoother furface than either of STRATA of the GRAMPIANS. “a of the two former whins which I have had occafion to mention. Its colour is of a dark-bluifh black. In tracing, with my eye, its vertical cracks-and fiffures, I thought I could perceive a rude tendency to prifmatic forms. It is vertical in its pofition; and its ftretch is from weft to eaft. Tus bed of whin is fucceeded by an Argillite of fhivery tex- ture, and confufed ftratification; but as it recedes from the whin, and approaches the mountains, it becomes regularly ftra- tified. This ftratum of flate is of great extended thicknefs ; and it contains a vaft variety of colour and of tint. The colours are, pale greyifh-blue, yellowifh-green, reddifh-brown, purple and black, with a great variety of tints of all thofe colours; but the predominant colours are the greyifh-blue and the yellowifh- green; of which two there are two forts; the one foft, and the other much indurated. The foft is thinly laminated,.and fre- quently paffes over into the highly indurated fert, in which the appearance of the laminated texture is almoft loft. In this long fucceflion of argillite ftrata, fome fubftances oc- cur that are heterogeneous to its rock, fuch as jafpers, limeftone, &c. | THE jafpers are in general of a blood-red colour, and are much veined with white quartz: they occur in large amorphous mafles, and in nefts, of eliptic forms, of great variety of fize. One of thofe bodies of jafper, in the eliptic form, has been cut through by the river, and is now to be feen in the face of the perpendicular rock, upon each fide of the ftream. Its fize is thir- ty feet long, by ten broad: the points of its tranfverfe axis are fharp ; and it ftands upright in the argillite. The maffes.of this matter which occur amorphous in the argillite, are of great mag- nitude. I have traced one of thofe for thirty yards in extent. All of thofe jafpers are of great induration, and take a high po- lifh. Both the amorphous and the eliptical formed maffes are found imbedded, where the argillite is of a greenifh-grey colour, B2 thinly 12 DESCRIPTION of the > thinly lamellated, of a filky, luftre, and faponaceous to the feel : it clings round thofe maffes in all their variety of direction, and of courfe its texture is there much twifted. When the argillite {tratification has extended its thicknefs to near three quarters of a mile, the limeftone which I have mentioned above then occurs, in. a bed of twelve feet thick. Its colour is bluifh-black; and it is much pervaded by veins of quartz, and of calcareous {par ; the laft of thofe are, in many places, of confiderable breadth, and are of a pale flefh colour. Where this limeftone has been wrought, I obferved it forked ; that is to fay, the bed is there {plit or divided into two, by the intervention of an argillaceous ~ body. Upon each fide of this bed of limeftone the argillite oc- curs of two colours. That which is next to, and in contaét with the limeftone, is black, of a fhaly texture, foils the hand, and has veins of ferruginous-coloured quartz trending through it. The argillite which is more remote from the limeftone is of a dark purple colour. IMMEDIATELY after this narrow bed of fhale, the ar gillite re- affumes its greenifh-blue colour, and flaty texture, and becomes highly indurated : here fome fpecks of granulated quartz begin to appear, thinly {cattered in its mafs, and, foon after, it is feen to pafs over into an aggregate rock, chiefly compofed of grains ef quartz, felfpar, and minute particles of mica. The particles of quartz and of felfpar feldom. occur in this aggregate larger than the eight of an inch: thefe have very little the appearance of having fuffered attrition: they are much mixed, and are fre- quently feen to take lineal directions ; and in thofe lines the par- ticles of felfpar have frequently a comprefled appearance, and an. eye- -like form. ‘This rock; in mafs, has a greyifh-blue colour = it is of great induration, and although lamellous or flaty in its texture, a crofs fracture is often more eafily obtained’ than one with the lamelle. Its crofs fracture 1s pretty even, but appears more granular than foliaceous. ‘This rock occurs frequently in the STRATA of the GRAMPIANS. 13 the diftriG of blue:clay flate, and may almoft be faid to.alternate with it. I have been perhaps more minute in the defcription of this rock than it’‘deferved ; but I have been fo, becaufe doubts have arifen relative to what name ought to be given to this aggre- gate. In all my geological refearches, I have found this rock only twice ; once, where I have here defcribed it; and, again; near to Banff, on the Moray Frith. In both of thofe fituations, the ag- gregates are of the fame compofition, and fimilar in pofition :. they both lie among blue clay flate. In this long alternation, two fubftances occur which are he- terogeneous to the rocks among which they lie. The firft of thofe, is a bed of compact Felfpar, of great induration. ‘This bed is ten feet broad: its ftretch is nearly from weft to eaft: its pofition is vertical; and it ftands between two of the layers of the blue clay flate. Its colour is of a reddifh-brown, with a fmall admixture of purple; and its general fracture is conchoi- dal, fomewhat rough, but not earthy. Nor far diftant from this bed, an appearance occurs wor- thy of fome notice. Where the aggregate and the blue clay flate are alternating, a furface of confiderable extent of the aggregate rock is expofed to view, parallel to its ftratifica- tion. This furface is regularly undulated in fmall undula- tions, bearing a very ftrong refemblance to thofe that may be feen upon the fand of the fea-beach, when recently left by the tide. After paffling the bed of compact felfpar, the blue clay flate and aggregate rock again alternate ; but here the blue clay flate predominates. Near to this, the fecond fubftance hetero- geneous to thofe alternating rocks occurs. It is a bed of Whin, the form of which is fomewhat fingular. It confifts of a prin- cipal trunk, which the river, here cuts nearly at right angles. Upon the eaft fide of the river, this principal trunk is feen to fplit into three branches ; and thofe three take an eaftern direc- tion, between the ftrata of the aggregate rock and the blue clay flate, where thofe two rocks are of great induration. The breadth: ay a DESCRIPTION of the ~ breadth of this bed of whin is thirteen feet; and where it {plits, its three branches are, fix, four, and three feet in diameter. The trend or ftretch of this bed is from weft to eaft; but upon the weit fide of the river, 1t curves fomewhat to the fouth-weft. Its compofition is nearly the fame with the three other beds of whin which I have before mentioned. It is of a brownifh-black co- - lour, and, when placed in certain directions, it fhows fpecks of luftre. It is vertical in its pofition, has a great degree of indu- ration, and its general fracture is roughly conchoidal. Soil <== Soil Aggregate Blue rock, Clay flate The river. - ~ Upon pafling this bed of whin, the river ceafes to be deeply imbedded in the rocks ; but the aggregate rock and the clay flate {till continue to be feen for a fhort diftance, in a fhelvy acclivi- ty, where they are loft to view in a long narrow plain, deeply covered with a bed of gravel, compofed of the debris of the in- terior mountains. The river here flows over this bed of gra- vel for a confiderable {pace ; and upon this narrow flat, we pafs through between two of the moft elevated points in the fouth la- teral range of this part of the Grampians. Although the obtru- fion of this mafs of gravel cuts off from infpe@tion the continui- ty of the laft-mentioned rocks, yet the broken and abrupt fides of the mountains, clofe upon each hand, clearly points out, that this part of the fouth lateral range is entirely compofed of mica- ceous fhiftus. Here, we are deprived of the junction of the mi- | caceous fhiftus with the two former rocks; and the lofs of all fuch STRATA of the GRAMPIANS. 15 fuch junctions are always to be much regretted in mineralogical refearch. Havine paffed over this narrow plain, I advanced towards a fecond range of hills, which here form the bafis of the centrak and higheft chain. It is at this place where the river fo fud- denly changes its courfe from eaft to fouth, and where I was un- der the neceflity of leaving its bed, to continue my northern di- rection towards Mount-Battoc. This, however, I was enabled to do to great advantage, by following up the deep cut bed of a winter torrent, which led me into the direction which I withed to follow. Upon entering the bed of this torrent, I found that the bafis~ of the hills here entirely confifted of micaceous fhiftus, much veined with quartz, and much twifted in its texture. ~The ftretch of this rock is here nearly from weit to eaft; and it has a foutherly dip of 45 degrees. In pafling through among thofe hills towards the central range, I found in feveral of the beds of the torrents large blocks of reddifh-brown porphyry, with fcattered mafles of micaceous fhiftus and granite. In tracing up one of thofe torrents, I faw the micaceous- fhif- tus rock and the porphyry both expofed to view, near to each other; and, foon after, in the bed of the fame torrent, I came to a cafcade which had laid bare both thofe rocks at a point where they are in contact; and near thofe a fecond bed of porphyry "made its appearance, in the front of a near hill. From my firft view of thofe, and from their relative pofitions, I was led to ima- gine, that they might here alternate in vertical pofition ; but up- on more minute in{fpedction, I found that the porphyry conftituted vertical dikes, ftretching nearly from fouth to north ; which courfe » cuts the line of direction of the Grampians-here almoft at right angles : and, on the contrary, I found that the. micaceous fhiftus which. flanked thofe dikes of porphyry, had a regular ftretch: from: 16 - DESCRIPTION of the from weft to eaft, and a foutherly dip. To endeavour to have thefe appearances more fully explained to me, I directed my f{teps to the brow of that hill, where I had obferved the rock laid bare; and in pafling along the fronts of the hills from eaft to weit, I foon came to a dike of porphyry fimilar to thofe which I had immediately left. This dike is fixty feet broad, ftretching nearly from fouth to north, and flanked upon both fides by mi- caceous fhiftus, ftretching and dipping as before defcribed. In proceeding farther along the faces of thofe hills, I found feveral other dikes of porphyry, of various breadths, and at various di- {tances from each other; but all of them fimilar in their lines of direction, and the micaceous fhiftus always interpofing between them, through which they feemed to rife. The porphyries of thofe dikes are generally of a ferruginous colour, tending fome- times to an orange-red, and of various tints of thofe colours. They have great induration, are coarfe-grained, and produce a rough fracture. The particles of quartz which, are fcattered in their principal mafles, are {mall, amorphous, and are of a ferru- ginous colour. The particles of felfpar are of a light tint of the fame colour, and are moftly cryftallized. The furface of thofe dikes are in many places bare, and expofed to the eye for long extents, in their lines of direction; and in all thofe lines of di- reCtion which I have traced, I have never found any of them al- ter in their breadths, in their verticality, nor in their directions. Their furfaces, in general, confift of oblong fquare blocks, now loofe and unconnected with each other; and, in many places, the lines of fra€ture of thofe blocks are fo ftraight, that one might almoft fuppofe that they had been disjoined by the hand of art. I HAveE often obferved, in this diftriG, and in other parts of the Grampians, that the loofe and outlying blocks of both gra- nite and ‘of porphyry, (which have not been worn down by at- i Xv trition), STRATA of the GRAMPIANS. 17 trition), confift, in general, of oblong fquare fhapes. This ob- fervation, when I firft made it, led me to imagine, that thofe rocks here were perhaps ftratified. I have, however, as yet, not been able to trace real ftratification of thofe rocks in this diftridt of the Grampians. Uron fome of the fummits of thofe hills which here form the bafis of the central range, I firft difcovered the granite in folid rock. In thofe fituations, the granite is only feen in patches, where the fuperincumbent rocks have worn off it. Thefe fu- perincumbent rocks, which I here found in contaét with the granite, are of two different compofitions, and occur on the fum- mits of different hills) The one of thofe rocks, and the moft prevalent one, is the micaceous fhiftus ; the other is the granitelle,. or a mixture of quartz and fhorl. In fome parts of this laft- mentioned rock, I perceived a {mall admixture of hornblende: where this appears in the compofition, it perhaps ought to receive the appellation of granitine. In thofe elevated fituations, I found both of thofe rocks, (efpecially the micaceous fhiftus), in a ftate of decompofition, and faft leaving the granite expofed to the eye. . From thofe appearances, it is to be inferred, that the interior by thofe hills is compofed of granite, which i is but thinly coated. by the fuperincumbent rocks. Upon leaving thefe hills, hicks I have already faid, form the bafis of the central chain of the Grampians, I regretted very much, that all my endeavours proved abortive to trace out the whole extent, in line, of any one of thofe dikes of porphyry which interfect their fides. I conftantly loft them whder peat or other foils, before I could trace them to their contaét with the granite. It was my anxious wifh to fee how thofe two rocks of porphyry and granite connected with each other at their junc- tion. Vou. VL—P. 2. 6 IN. 18 DESCRIPTION of the In purfuing my refearches towards the fummit of Mount Ba- toc, I proceeded up the bed of a torrent, which, after heavy rains, dafhes down the immediate fide of that mountain. In this bed, the blocks of micaceous {chiftus and of porphyry, (which I had feen fo abundantly fcattered among the hills that I had juft left), totally difappeared, and no outlyers of any kind were to be feen, excepting fome granites, which were fcattered in large maffes ; and in every part, where the torrent had carried off the fuperincumbent foil, the granite was to be feen in folid rock. In my progrefs towards the fummit of this mountain, I fell in with a large face of the native granite rock expofed to the eye. By the cracks in this face being in long-extended horizontal lines, it had at firft the appearance of being ftratified ; but upon a nearer and more minute examination, I found that it was not {tratified, and that the cracks which gave it that appearance were only fuperficial. ArounpD this face were fcattered large blocks of granite, which were moftly in oblong fquare fhapes. Soon after pafling this precipice, I gained the fummit of the mountain, which, though not very highly elevated, is in this part of the chain the higheft of the central range. It is about 3465 feet above the level of the fea; and is entirely compofed of a coarfe-grained granite, in which fhorl fometimes occurs; and its felfpar is very generally cryftallized. Havinc here finifhed the extent of my intended inveftiga- tion, I beggto be permitted to add, that the line which I have here given the defcription of, has been traced with much atten- tion, and the true pofition of each foflil has been moft {crupu- loufly attended to, and is correctly placed in the annexed plate. I UY MOT. = SDS fa sel eciungy papi) Beige CPL yates 2s Gy ay snaydiouy” sndsuy- inp sos 9 poseyiny 4 fens Wanye JOTUOEUaY Ue Layy Mp u nuoppUn gH Fy H LYS, PUN [ y) fi uaa UNS (UpbdznLYy ] \ unos ppp) sua Pines p49 ey, pe= OEetDy tiny SFE MA. tee ABBY IIS PY esnadongy fe -§TRATA of thee GRAMPIANS. 19 I wisH that fome. more able pen than mine, would take up the further defcription of this extended field of geology, fo worthy of inveftigation ; but if none will come forward for that purpofe, I may at fome future period prefume to give to this Society more extended, and more general lines of defcription of the Grampians, than that which I have now had the honour of fubmitting to their examination. IT. 1, 4 GEomETRICAL INVESTIGATION of /ome curious and intereft- ing Properties of the CirciE, ce. By FAMES GLENIE, Efq; A.M. F.R.S. Lonp. & Eoin. [Read April 1. 1805.] DEAR Sir, Edinburgh, 22d March 1805. A S the following paper refers in a great meafure to the gene- ral theorems publifhed by your father, I now commit it to your care, and that of my friend Mr Puayrair, Profeffor of Na- tural Philofophy. I with it to be communicated to the Royal So- ciety of Edinburgh, and, if approved of, to be inferted in their Tranfactions as foon as poflible. Indeed, I truft, that even fimple as it is, it will not be altogether unacceptable to that learned body. I am, Dear Sir, Moft fincerely your, &c. Ja’ GLENIE, Dugald Stewart: E/qs t Profeffor of Moral Philofophy. bo 2 INVESTIGATION of fome Tuat truly elegant and inventive geometer the late Dr Mattuew STEwart, publifhed at Edinburgh, in 1746, without demonftrations, a number of general theorems, of great ufe in the higher parts of mathematics, and much calculated for impro- ving and extending geometry. Such of them as refer to the circle, and to regular figures infcribed in, and circum{cribed about it, have not, as far as I can underftand, been yet demon- (trated. Thefe, with an endlefs variety of other theorems, are derivable, as corollaries, from the following general though fimple geometrical mveftigation, that otcurred. to me fifteen years ago, and which, I fuppofe, has remained fo long unknown. and unattended to chiefly on account of its fimplicity. Let A,B,C, &c. (Pl. I. Fig. 1.) be any numberof points in the circumference of a circle, and let that number be denoted by z. Let RA, RS, ST, &c. be tangents to the circle, in the points A, B, C, &c.; and let POQ be any diameter. Let Qc, Qd, cy, &c. be perpendiculars from the point Q to the diameters paf- fing through the points A, B, C, &c., and Pa, P-d, Pe, &c. per- pendiculars from the point Pto the fame diameters. ‘Tuen it is evident, that PQ = AP +AQ — BP +BQ'= CP +CQ,= &c. Wherefore PQ’ X= AP + BP +CP +, &e. + AQ’ + BQ'+CQ +, &c. But AP’>=AG X Aa=PQ_ x Aa, BP =PO'x B¢, CP: =PO) x Co, Se: and AQ’ + BQ'+ €Q’+, &. = PQXAC+Bf+Cd+, &c. Now Aa, Be, Cb, &c. are refpectively equal to perpendiculars drawn from P to the tangents RA, RS, ST, &c., as are Ac, Bf, Cd, &c. equal to perpendiculars drawn from Q to the fame tangents. Con- fequently the fum of all the perpendiculars drawn from the points P and Q to lines touching; the circle in the points, A, B; C, &c. is equal to PQ X %, or a multiple of the diameter by 2. Tue fame may be proved othewife ; for fince O02 = Oc, Aa — Gc, Aa+Ac=the diameter. In like manner, Be-+Bf= the diameter, and C)-+Cd= diameter, &c. In PROPERTIES of the CIRCLE. 23 In the fame way, it is demonftrated, that if from any,two points /, g, in the diameter PQ, equally diftant from the centre O, perpendiculars be drawn to the lines touching the circle in the points A, B, C, &c. their fum is equal to a multiple of the diameter by 7. But if from any two points V, W, in PQ_ produced, equally diftant from the centre O, lines drawn perpendicular to any dia- meter Br, pafling through any point of contact B, fall beyond its extremities B, 7, the difference of the perpendiculars drawn from W, V, to the line touching the gircles in B, is equal to the diameter, and fo on. So alfo, when perpendiculars from the points V, W in PQ, produced to the diameters pafling through the points of contact A, B, C, &c. do not fall beyond the extremities of any of thefe diameters, perpendiculars from V and W to right lines touching the circle in the points A, B, C, &c. are taken together equal to a multiple of the diameter by the number of the faid points. Cor. 1. Perpendiculars drawn from P and Q, or fp and g, to lines touching the circle in the points A, B, C, &c. are toge- ther equal to a multiple of the radius by 22. Cor. 2. The fum of perpendiculars drawn from P, Q, or p, q) to the fides of any regular figure circumfcribed about the circle, is equal to twice the fum of perpendiculars drawn to the fides of a regular figure of the fame number of fides circumfcribing the circle from any point within the fame regular figure. Cor. 3. a = fum of the perpendiculars drawn from P to right lines touching the circle in the points A, B, C, &c. d denoting the diameter. Or athird proportional to the diameter and the chord AP, to- gether with a third proportional to the diameter and the chord BP, together with a third proportional to the diameter and the chord 24 INVESTIGATION of fome chord CP, &c. is equal to the fum of the perpendiculars drawn: from the point P to right lines touching the circle in the points A, Byic, &e. AQ + BO + CU +, &e. AND C = fum of perpendiculars drawn from Q to the fame lines. AGAIN, fince by a well known property of the circle, AP* + AQ’ = BP* + BQ’ = CP + cQ"'= &e. = 277+ 20>, r denoting radius, the fum of the fquares of lines drawn from the points A, B, C, &c. to any two points /, 9, in the diameter equally diftant from the centre, is = 2277+ 2X Op’ = amul- tiple of 7?, by twice the number of the points A, B, C, &c. to-. gether with the fame multiple of the {quare of Op or O q- In like manner, AV ++ AW +BV*+BW’-+CV +CW7+, &c. =—2nr+2nxXOV =a multiple of 7? by twice the number of the points A, B, C, &c., together with the fame multiple of OY" or OW Anp fince the fquares of the chords AP, BP, CP, &c. are to- gether equal to the fum. of the fquares of the: perpendiculars. drawn from P to the right lines touching the circle in the points A, B, C, &c. together with the fum of the fquares of the per- pendicular diftances of P from the diameters pafling through thefe points, the fum of the fquares of Ap, Bp, Cp, &c. is in like manner equal to the fum of the fquares of perpendiculars from /p to thefe lines, together with the fum of the fquares of the perpendicular diftances from fp to the faid diameters. In like manner, Ag’ +Bq +Cq’ + &c. = fum of fquares of perpendiculars from. ¢ to the lines touching the circle in A, B, C, &c. together with the fum of the fquares of the perpendi- cular diftances of 7 from the diameters paffing through A, B, GC. &c, WHEREFORE PROPERTIES of the CIRCLE. 25 WuereErore the fquares of the perpendicular diftances of ei- ther P or Q, from diameters pafling through the points of con- tact A, B, C, &c., are, taken together, equal to the excefs of the rectangle under half the diameter PQ, and the fum of perpendi- culars:- from P and Q to right lines touching the circle in the points A, B, GC, &c. above half the fum of the fquares of faid perpendiculars = 27? —7s, (s being equal to the fum of per- pendiculars from O, as, in what follows, to right lines touch- ing the circle, of which OQ. is the diameter, in the points c, d, f, &c.). And the fum of the fquares of thefe perpendicular diftances from both P and Q, is =22r?—ars. This is alfo evident, from all angles in a femicircle being equal to right ones. For AP FAQ 4BP BQ 4OP poG + &e. =ax PO =4nr;and 427’ —a2nr'—2rsmanr-—ars. 7 ConsEQUENTLY, when the whole circle is divided into equal parts, in the points A, B, C, &c. Ap + Bp +Cp +&. = Ag +Bg +g + &c =art+ nx Op ; and A Wah Bick CV + &. =AW +BW +CW 4t&e.=ar*+nxOV. For the fum of perpendiculars drawn from / to the fides of any regular figure circum{cribing the circle, is then equal to the fum of the perpendiculars drawn from g to the fides of the fame figure. The fame obfervation holds with regard to perpendicu- lars drawn from the points V, W. From the foregoing general inveftigation, when the circle is fuppofed to be equally divided in the points A, B, C, &c. Dr STEWaRtT’s firft, fecond, third, and eleventh theorems can be im- mediately derived. I sax, however, proceed regularly with the inveftigation ; and, in the firft place, take the fquares of the perpendiculars from P and Q to the right lines touching the circle in the points A, B,C, &c. which perpendiculars are re{pectively equal to A a, Mes BY, Bes Cd, Cbs &e VoL. VI.—P. I. D Now 26 INVESTIGATION of fome Now:Ac + Aa mrt 6 tr=c0 sar 42x00 Bf + Be =r}0f +7r—Of = ar 4 ax OF Gd +Cb p+ Od ee =2r42xOd Sre6) fo) &e. ; &e. WuereErorE the fum of the fquares of perpendiculars from P, Q to lines touching the circle in the points A, B, C, &c. is ECO TOT 2K Oc’ + Of + Od" 4. &c. But the points c, d, f, are in the circumference of a circle, of which the diameter is OQ_ or r, and by Cor. 3. the fum of Oc +Of +Od 4. &c. = 00;% into the fum of perpendiculars drawn from O to lines touch- ing the circle, of which OQ is the diameter, in the points c, d, f, &c. Call the fum of thefe perpendiculars s. Then we have the fum of the fquares of perpendiculars drawn from P, Q to lines touching the circle APQ in the points A, B, C, &c, = 2 2.7% — + tO as —4 at +2rs = (Cor 3.) AP 4BP 4CP + &e. + AQV4+BQ+CQ +&e. rs When the circumference is divided into equal parts by the points A, B, C, &c. or the angles at O are equals = x 0Q, 1 2 or> xrand amr -ars= 307, Ir a regular figure be infcribed in the circle, having its angles at the points A, B, C, &c. or a regular figure be cir- cumfcribed about the circle, having its. fides tangents to it in the points A, B, C, &c. we get from the general expreflion Ap +BP + cP + &c. or AQ Yea Stas CQ'+ &e. &e. ae +4rsaz4nr--a2anrs=6 7, or third proportionals to ra- dius, the chords drawn from either P or Q to the points A, B, _ C, &c. and the cubes of thefe. chords equal, when taken toge- ther, to fix times a multiple of the cube of radius by the num- ber PROPERTIES of the CIRCLE. 27 ber of the fides of the infcribed or circumfcribed figure; or to {peak algebraically, the fum of the fourth powers of thé chords is equal to fix times a multiple of the fourth power of the femi- diameter of the circle, by the number of the fides of the figure. This is Dr SrEwart’s 23d theorem. s In like manner, Ac + Aa =r4+Oc +r—Oc =2 r3-+O6rx Or Bf +Be eof &esor = 2rit+6rxOf (Cd 40h =rF0d +r—Od = 2 46rxO"d &ce. &c. &e. And the cubes of perpendiculars from P and Q to right lines touching the circle in the points A, B, C, &c. are taken together S224 67x Oc +0f +0d 4+ &. = (by Corollary 3.) ceerG 86 a6 ae ee pees aa AP + BP +CP +&c. + AQ +BQ +CQ + &. ARS ae a ae a a nr aaeomee oF + Od + &e = ae, when the circumference is equally divided in the points A, B, C, &c. or when a regular figure is circumfcribed about the circle, with its fides touching the fame in faid points. Wherefore the cubes of perpendiculars from P and Q to the fides of a_ regular figure of a greater num- ber of fides than three circumfcribed about the circle, are taken together = 5273. This is Dr STEWaRT’s 19th theorem. AND if a regular figure of a greater number of fides than three be infcribed in the circle, having its angles in the points A, B, C, &c. third proportionals to the cube of the diameter and the cubes of chords drawn from P and Q to the points A, B, G, &c. will, taken together, be equal to 5 #75; or third proportion- als to the cube of the diameter and chords drawn from either P 52Pr > or Q to the faid angular points, will taken together, be = D2 or, 28 | INVESTIGATION of fome — or, to {peak algebraically, the fum of the fixth power of chords drawn from either P or Q to the faid points, will be equal to twenty times a multiple of the fixth power of radius, by the number of the fides of the infcribed figure. In like manner, a —4 ——4 4, ——— a ME ey tea = okra COE oe, et a ee ee eee alee B am oY oe On earn aee Hes st |) ee a —7+Od +r—0d psec = 273 4+ 127 x Od + Ral | &e. &c. - &e. And third proportionals to radius, perpendiculars from P and Q_ to right lines touching the circle in the points A, B, C, &c. and the cubes of faid perpendiculars are, taken oe equal to 2urierarx Oc + Of 40d + &e. he +07 oe + &c. = (by Cor. 3.) AP'4BP +CP + &c. ACHE &e. d+*Xr axr But Oc + Of +04 + &e = 7" | 2 is equally divided in the points A, B, C, &c. or when a regular figure is circumfcribed about the circle, with its fides touching when the circumference nr? the fame in faid points, and 127 X= 6473. Alfo 4ar is then = Wherefore thefe 5 Oc + Of +04 + &e. ‘i 3 third proportionals are taken together equal to 8"73 + 3 a == 350 r3 4 ; and four times their aggregate is equal to 35.773. Or, te PROPERTIES of the CIRCLE. 29 to fpeak algebraically, eight times the fum of the fourth powers of perpendiculars from either P or Q_ to the fides of a Tegular figure of a greater number of fides than four circum{fcribed about the circle, and_ touching it in the points A, B, C, &c. are equal to thirty-five times the multiple of the fourth power of radius by the number of the fides of the figure. This is Dr STEWART’s 25th theorem. | ’ Anp if a regular figure of a greater number of fides than four be infcribed in the circle, having its angles in the points A, B, C, &c. the eighth powers of the chords drawn from either P or Q_ to the points A, B, C, &c. (to fpeak algebraically) is equal to 7onrs =x = ay oe 2+,r° = feventy times a multiple of the eighth power * radius by the number of the fides of the fi- gure. - In like manner, a er r+Oc +r=O08" —2r3taorxOc +a Oc" b r a aon x Of + 10 xs 5 5 uey, r+Od +r—Od _ ar Od rp? = 2734207 x Od #10 xi &e. &ec WHEREFORE ge Eisai Sis —Of 47400 +r—0a° acne + &c. Sant ronr4 0% 2B sake 4 equal (by Cor, 3.) to AP +BP +P. + &c. + AQ +BQ + CQ. + &c. dir the circle is equally divided in the points A, B, C, &c, , when AND 30 -INVESTIGATION of ome AND generally when m is any integer whatfoever, we have r+Oc +700" 7 +Of' +7—OF r+Od 4r—oOd r"—3 v3 + a mr I m— m + &c. equal to meiiees > : e eR NO RE © 2h eed y Oct OF 1 Ob4 ier ae eee I I 3 4 i I I 3 6 aver RANG Lee SOT ea, Ee re Ore Of 4 Od + Ken &c. = (Cor. 3.) 4 5 6 r3 AP +BP +CP + &e. | AQ + BQ. +CQ 4+ &e. a” r—3 TSS 0 eo, ee ar which, when the circle is equally divided in the points A, B, CG, &c. by the circumfcription or infcription of a regular figure, coincides with the 36th and 38th of Dr Srewart’s general theorems. Anp ‘univerfally if m have to / any ratio whatfoever, m1 m m m r4Oc™ + r—Oc ae: + Of “+r—Of ‘2 7 r+Od! 4.7—Od" a ma + &e. re 3 x + ae is ant rx Oc + Of + Od + Be, +4.) mal 5 bins 2 a4 m—3l, Oc + Of +Od + &e. 4 Bf! ted m3! mad Se wer) gee TO ee ET ee = pee thas) m—s! Oc 4+ Of +Od XKC. Be. Ke, 6/ Pa . Turs laft theorem, or expreflion, is more general than any of Dr Stewart’s theorems, and will furnifh an endlefs num- ber of new and curious infinite feries, with their fumma- tions. It may alfo be extended to the chords AP, BP, &c. and expreffed in terms of them. And as to the truth of the bi- nomial PROPERTIES of the CIRCLE. — 31 nomial and refidual theorems, when m has to / the ratio of any two homogeneous magnitudes whatfoever, I muft refer the reader to my general demonftration of both in Baron MasEREs’s Scrip- tores Logarithmici, vol. 5. and to fome of the geometrical formu- le in my Univerfal Compari/on. , In like manner, if pg, p 5, pi, &c. be perpendiculars refpec- tively to BO, CO, AO, &c. we have 7 +O i + r—Oi +74 OF - r—OF + 7405 4+ r—Ob + &c. = 2ur* + 2K Oi + Os + Oh +c. 2097 a. Op , when the circle is equal- ly divided in the points A, B, CG, &c. or when a regular figure is circumfcribed about it, with its fides touching it in thefe points. This is Dr Srzwart’s third theorem, of which he gives. a demonftration of confiderable length. In like manner, 740i pr—O1 47408 £r—OF £7408 +r—O8' + &c. are equal to2"7r? +67 ~ Oi + Og 4-Ob +. &c. 207? +3 rxOp when the circle is equally divided.in the points, A, B, C, &c. or when a regular figure circumfcribing it touches it in thefe points. This is Dr STEwart’s 20th theorem. In like manner, x + Of +7507 4740s +7—OF , 7 EON +r—08 i r r + &c. is equal to 2773+ 12 rxOi +Og +05 + &e. + 2x: —* ald —+4 : —-4 ee Oe nek St Re ear eerie at 31-0P when the circle-is equally divided in the points: A, B, C, &c. or when-a regular figure, circum{cribing it, touches it in thefe points.- Anda multiple of this. by four, or eight times the aggregate of third: 32 INVESTIGATION of fome third proportionals to 7, the perpendiculars from either p or 7 to the fides of the regular circumfcribing figure, and the cubes of thefe perpendiculars is equal to 8ar? + 24urx Op 4.32% 24 , Pe ; or, {peaking algebraically, eight times the fum of the fourth ; powers of perpendiculars from either / or g are equal to 8274, together with 24 times a multiple by z of the fourth power of the line whofe f{quare is equal to r xO g, together with thrice a multiple by 2 of Op. This is Dr SrEwart’s 26th theorem. In like manner, r40i $7071 , 7 FOR 270g 4 FEO +705 r* r? r? 4+. &e. Sn gv, OF FOF OF 4 Bee. are = 2nr*-+-20rKOi +O¢ +0/ st Beg te re | a die =2nr?+10nr xOp swe: when the circle is equally divided in the points A, B, GC, &c. or when a regular figure cir- cumfcribing it touches it in thefe points. Anp generally when m is any integer whatfoever, r-Oi tr—Or ii 7+Oe +r—Og- if r4+Ob 4+r—Oh- a ui—3 , ae 3 773 ml Mmi—=t —=z_— 2 tt—~—S m= + — «——-~., + &c. Ae eee wrx OF Og + Ob + &e. +7 a Ss um m—2 m—2 x OnOr + Ob + Se. | m eae ee nF Se A aes ‘2 8 4 6 5 bane 5 ages t OE oh +Og +0b + & , &c. &c.; which, WSs rege xinto — = . when the circle is equally divided in the points A, B,C, eg when PROPERTIES of the CIRCLE. 33 when a regular figure circumfcribing it touches it in faid points, gives Dr STEwArT’s 37th theorem, fince the fame reafoning and mode of demonftration holds good in regard to half the amount of this expreflion, whether the points mt and q be in PQ; or in PQ _ produced. Anp univerfally if m have to / any ratio whatfoever, — ” m —_——_" —— —_——,7 ara >a pe BP Orr—_O;7 art 08! +r—Og 7 4 7t06 4 eR + 8c. tH an iit eee eS ae -—=3 -_—3 ——=3 Ban iy FRO + OF + Ob +&c. + a m—al 3¢ m—=3l OF. +07 +05. + &c. mmol m—21 m—3l m—z n—al aight PEND agh Ml REE ae Pie ()z. 4. Og wi 4. Ste; 4 Bec. Ke, r _ Tus laft theorem or expreflion is more general than any of Dr Srewarvt’s theorems, and from it may eafily be derived an endlefs number of new and curious infinite feries, with their fummations. Ir is almoft needlefs for me to obferve, that befides thefe theo- rems of Dr MaTrTHEW STEWART, an unlimited number of other theorems, refpecting figures both regular and irregular, circum- feribing and infcribed in the circle, may eafily be derived from the foregoing inveftigation, as well as a great number of geo- metrical infinite feries, with their fummations. And as to theo- rems refpecting given points, right lines and figures either re- gular or irregular, given by pofition, and right lines interfecting each other either in one point or in different points in angles either equal or unequal, that are deducible from it, they are in- numerable. ; VoL. VI.—P. I. KE Now, 34 INV ESTIGATION of fome Now, let a circle (Fig. 2.) be divided into an uneven number- of equal parts, by the points A, B, C, D, E, &c. and let PQ be any diameter; from P let Pa, P4, Pc, Pd, Pk, &c. be drawn perpendicular to the diameters pafling through the points A, B, C, &c. and from Q let Qe, Q f, Qg, Qh, Qi, &c. be perpen- dicular to the fame diameters. THEN it is evident, that Aa, Ae are refpectively equal. to perpendiculars drawn from P, Q; to a tangent to the circle in the point A; and fince Oa = Oc, their fum Aa + Ae = s—Oa+r+Oa, In like manner, the fum of the perpendicu- lars from P, Q to the tangent at B is = r—Oc +7 + Oc, to the tangent at C is = — r—OF + r+ 0h, to the tangent at D is = r+0Ob6+7—O4, and to the tangent at E is = r+Od + r—Od. But r—Oa+7—Oc 4r—Ok +r +O0b+r+O0d= r+Oat+r+O0c+rt+O0kh+r—0b+r—Od; 2x0b+0d= 2xOatOc+OF and O+0d=02440c+0 8, and fince 7—Oa + r—Oc + r—Ok + r+Ob +r+0d =rtOa 4 ¢—Oc -+r+Ok +7—Ob + r—Od , we have this equation 4x7rxObtrxOd = 4xrxOatrxOct+rxOk, or OS+ Od=Oa+Oc+Ohk. : Burt if from a point in the circumference of a circle, perpen- diculars be drawn to the alternate fides of a regular figure of an even number of fides circum{cribing the circle, or, which comés to the fame thing, beginning with any one fide, perpendiculars be drawn to the Ift, 3d, 5th, 7th, &c. fides, the fum of thefe perpendiculars, the fum of their fquares, the fum of their cubes, n—2'b 7 ae &c. to the fum of their Pash: Hea powers, is refpectively equal to the fum of the perpendiculars drawn from the fame point PROPERTIES of the CIRCLE. 35 point to the other fides, viz. the 2d, 4th, 6th, 8th, &c. the fum of their {quares, the fum of their cubes, &c. to the fum of their N—2 powers, but not in powers ate 2 being the num- ber of the fides). Tuus for inftance, if a regular hexagon circum{cribe a circle, and from any point in the circumference perpendiculars be ~ drawn to the alternate fides, that is, to the fides of an equilate- ral triangle circumfcribing it, the fum of thefe perpendiculars, | and the fum ef their fquares, are refpeCtively equal to the fum ef the perpendiculars drawn to the other three fides, and the fum of their fquares. For the fum of the perpendiculars to the three fides of an equilateral triangle, 1s equal to -half the fum of the perpendiculars to the fides of the hexagon, and the fum of their fquares in the one, equal to half the fum of their fquares in the other. But this does not hold in regard to the fum of - their cubes, as the fum of the cubes of perpendiculars to the fides of the triangle is not invariable. In like manner, if perpendiculars be ec from a point in the circumference to any four fides of a regular circum{cribing octagon, taking them alternately, that is, to the fides of a cir- _ cumfcribing fquare, their fum, the fum of their fquares, and the fum of their cubes, are refpectively equal to the fum of perpen- diculars to the other four fides, the fum of their {quares and the fum of their cubes. But this does not hold in regard of the fum of their fourth powers, which to the fides of a {quare are not invariable. In like manner, the fum of perpendiculars to the receonie fides of a regular circumfcribing decagon, that is, to the fides of a pentagon, the fum of their fquares, the fum of their cubes, and the fum of their fourth powers, are refpectively equal to the E 2 fun, 36 INVESTIGATION of fome fum, the fum of the fquares, the fum of the cubes, and the fum of the fourth powers of perpendiculars to the other five fides. But this equality does not hold in the fifth powers, which to the . te 1o—2 | fides of a pentagon are not invariable. For 3 4 And fo on. | ; N. B. Tue fame holds true if the perpendiculars be drawn from any point within the figure for odd powers, and either within or without, in even ones, ; Bur as it was obferved in the preceding page, that the equa- lity between the fum of the powers of perpendiculars, drawn from any point in the circumference of a circle, to the alternate fides of any regular figure of an even number of fides, and the fum of the powers of perpendiculars drawn from the fame point ae ath : 2 = power ; fo the equa- lity between the fum of the powers of perpendiculars drawn from the extremities P and Q of any diameter to the fides of a regular figure of an odd number of fides circumfcribing the circle, and the fum of perpendiculars from either of thefe, or any point in the circumference, to the fides of a regular circumferi- bing figure of double the number of fides, exits only to the to the other fides, exifted only to the u— 2" power. A WIDE field is here opened for the geometrical folution of both determinate and indeterminate problems. } For inftance, having two equal right lines given, to cut one mto two parts, and the other into three, fo that the fum of the fquares on the two parts, into which the one js cut, fhall be equal PROPERTIES of the CIRCLE. ifs equal to the fum of the fquares on the three parts, into which the other is cut. SOLUTION. With radius equal to one-third part of either of the given: lines defcribe a circle. If a regular hexagon circumfcribe it, per- pendiculars drawn from the point where any fide of the hexagon touches the circle, to the other five fides, are refpectively equal: to the parts into which the two given equal right lines are requi- red to be divided. Calling the fide, from a point in which the perpendiculars are drawn, the ft, the perpendiculars drawn to the 3d and sth are the parts, into which one of the two equal gi-. ven right lines is cut, and thofe drawn to the 2d, ath, and 6th fides, the three parts into which the other given line is cut. N. B. If the perpendiculars be.drawn from any point in the circumference, that is not one of the points of contact, three of them taken alternately, are together equal to the other three, and equal to either of the given lines, and the fum of their fquares equal to the fum of the fquares of the other three. And if they be drawn from a point in the circumference equally diftant from two points of contact, the 1ft = the 6th, the 2d = the sth, and 3d = the 4th. AGAIN, let it be required to divide each of two equal given: right lines into four unequal parts, fo that none of the parts of the one fhall be equal to any of the parts of the other, but the fum of the fquares.of the parts of the one fhall be equal to the fum of the fquares of the parts ofthe other, and alfo the fum of the cubes of the parts of the one equal to the fum of the cubes of the parts of the other. | SOLUTION. 38 INVESTIGATION of jome _ SOLUTION. Wits a fourth part of either of the equal given right lines as radius defcribe ‘a circle. If a regular decagon circumfcribe the circle, and from any point in the circumference, that is nei- ther one of the points, where the fides of the figure touch the circle, nor at an equal diftance between the points of contadt, perpendiculars be drawn to the fides of the oGtagon, thefe taken alternately are the parts into Wace the given right lines are re- quired to be divided. IF the point coincide with one of the eae of contact, one of the given lines is cut into three parts, and the other into four. | Ir the point be equally diftant from two points of contact, the ift perpendicular is = the 8th, the 2d = 7th, the 3d = 6th, and . the 4th = sth. <2 = = 3 the higheft power. Witu fuch problems one might proceed without end. Sincz (fig. 1.) AP +BP +CP + &c.AQ +BQ. YEG: &e. are equal to the-fquares of lines drawn to P and Q from the angles of a regular infcribed figure of the fame number of fides with the irregular circum{cribing figure, or from the points where the fides of a regular circumfcribing figure touch the circle, it is evident, that the fum of the fquares of perpendicu- lars drawn from P and Q to the fides of any circumfcribing fi- gure, regular or irregular, of a given number 2 of fides, together with the fquares of the perpendicular-diftances of Pand Q from | the diameters pafing: oe the points of contact A, B, C, &e. ViZe Pa + PB ee: + Se. 4 OF + Ord ENcty +, Be. = 2x Pa+Bo+Pe + &c. is an invariable quantity. For Pa +aO = Pb +450 = Pe +20 = PO whether the angles PROPERTIES of the CIRGLE. 30) angles at O formed by the diameters. paffing through the points: of contact A, B, C, &c. be equal or unequal, or whether the cir cumference of the circle, of which the diameter is PO, be equal- ly or unequally divided in the points a, 4, ¢, &c.; and the fum of perpendiculars from P and O, to the fides of a circumfcribing figure, touching this circle in the points 2, b, e; &c. is the fame whether the figure’ be regular or irregular. , ; OTHERWISE, ; Aa ee seer ar = Aa + Qe (2G) aig 2AayxaG (2 x Pa). In like manner, 477 = Gi. +. Cd. (52) + 2 Cc b x bb (2X Pb) and fo.on: g2r'= Aa + Ac +:Ce + Cd + Be + Bf + &. + 2xPa 4 PS + Pe + &e. whether the angles AOB, BOC, &c. at O be equal or unequal. In like manner, the {um of the cubes of perpendiculars drawm - fron’ Prand Q to the fides of any circum{cribing figure, regular or irregular, of a given number n of fides, together with thrice the folid, which has the {quares of perpendiculars from P and eg to the diameters pafling through A, B, C, &c. for its bafe, and r for its altitude, is an invariable quantity ; that is, 2773 + 67 x Oa + Ob-+ O02 + &c. + Pa + Pb + Pe + &c. is an inva- riable quantity, being = 2773+ 67r xX PO = 8277 =nx Gr. For Oa +06 +e + &c.4+Pa +Ph + Pe + &e. =nr. Confequently, when AB, BC, &c. are unequal among themfelves, the fum of the {quares, cubes, &c. of perpendiculars drawn from P and Q to lines touching the circle in. oe points: A, B, C, &c. is not invariable. AP + BP + OP + &c. + AQ +BQ + CQ + &. be- ing = 477", hisarnanees AB, AC, &c. be equal or unequal, is in-. variables. 40 INVESTIGATION of fome variable. But their 4th, 6th, 8th, &c. are not invariable, when AB, BC, &c. are unequal among themfelves. . It is manifeft, that when AB, BC, &c. are equal among them- felves, whatever be the number of the points A, B, C, &c. or whatever be the part of the circumference they take in or extend to, the fum of the fquares, cubes, &c. of perpendiculars drawn from P and Q to lines touching the circle in thefe points, is the fame with the fum of the fquares, cubes, &c. of perpendiculars drawn from P and Q to the fides of a regular circum{fcribing figure having the fame number of fides as there are points, A, B, C, &c. Thus, if the number of the points be five, and thefe be comprehended in a femicircle, a quadrant, or any fector, the fum of the fquares, cubes, and fourth powers of perpendiculars drawn from P and Q to lines touching the circle in thefe points, is the fame with the fum of the fquares, cubes, and fourth powers of perpendiculars drawn from P and Q to the fides of a regular pentagon circumfcribing the circle. And fo on. PERPENDICULARS drawn from P, (fig. 2.), one extremity of the diameter PQ; to the fides of the figure of an uneven or © odd number of fides circumferibing the circle, and touching it in the points A, B, C, D, E, &c. are refpectively equal to per- pendiculars drawn from Q; the other extremity of the diameter, to the fides of a circumfcribing figure of double the number of fides, which touch the circle in the points H, I, K, L, G, &c..or Hee Be Ges Aa hd = E d Kh =D, eee and perpendiculars drawn from Q to the fides of the figure of an odd number of fides, are re{peCtively equal to perpendiculars drawn from P.to the fides of a figure of double the number of fides, which touch the circle in the points H, I, K, L, G, &c. or Df Ke, Cialh, Bee Ht, AepenGianEs sly, eee Wherefore, the fam of the m powers of perpendiculars, drawn from P and Q to the fides of any circumfcribing figure of an odd number of fides, is equal to half the fum of the ™ powers of PROPERTIES of the CIRCLE. gt of peapepidibglatt drawn from P and Q to the fides of a circum- feribing figure of double the number of fides. Tuus, if the figure be a pentagon, we get eae Or = 27° +67x Ga. et SY OOo r+Ok +7—OF = 2r' + Or x OF r4Ob° +00 = = ar +6rx OD ect r+Od On _ = art 6rxOd ae for+6rxOa +0e +08 400 40d Sagrcroxios \ Wuen the diameter PQ ‘bife€ts the arcs BK, DH, or is per- pendicular to one of the diameters pafling through a point of contact, O%, Oi vanifh, and it is then demonftrated exa@tly in the fame way as in figures of an even number of fides, that the fum of the cubes of perpendiculars drawn from either P or Q is ae rs, and confequently that the fum of the cubes of thofe drawn from P, is equal to the fum of thofe drawn from Q. But let the figure be a pentagon, and let the diameter-AG be perpen- dicular to any fide in the point of contact A. Draw Cm, Ba perpendicular to AG. Then Gm is equal to each of the per- pendiculars drawn from G to the fides touching the circle in the points C and D; and Am to each of the perpendiculars drawn from A to the fame fides; G1 is equal to each of the perpendi- culars drawn from'G to the fides touching the circle in the points B, E, and A2, to each of the perpendiculars drawn from A to the fame fides.’ Wherefore 2G m+2Ga2+GA (2r) = 2An+2Am, orr—Om+r+On4r—=2An+Ama=r—On VoL. VI.—P. I. ae ee 42 INVESTIGATION of the 4+ r+Om,andr=20m—20z2, ThereforeOm = pit Aad a, r—2O0n gr+20n oe wa 2 7—O m = — Gam, andr+Om = —An. Confequently we have 2x7 + Oma 27 sar x On sae" xOn +8xOz3, —— "3 —rz ——3 and 2xr—On =2r—6rxOn+6rxOn —2xOn, and 3 2 ; ee | thefe added together give aries In rn—6r°xOn+12rxOn—8xOn, 4 pea fits Se: | and 2Xr+On — 27+67rxOn2+67 XOn + 2x On’, ma 3 — dike manner, 2x7—-Om = f 3 8r7>x O ae po thefe added together give gre erxertsersts is which, if GA’ — 873, the cube of the perpendicular to the Gide touching the circle in the point A, be added, we get 41 P+18rxOn+367xXOn pt Se REL 4 Bur the fum of the cubes of perpendiculars, drawn from A and G to the fides of the pentagon is 257%, as- has been demonftrated, when PQ _ coincides with AG. Wherefore pnt gsr xort 6rx On or tg -i2r XOn+ 247 OF = agri and 4rxOa + 2rxOxv=1. Now, if for gir$187°xOn-+367x00,, this value of 73, there be fubftituted 73 in ; we PROPERTIES of the CIRCLE, gg 3 3 : We get Avie ax a ; and if 75 be fubftituted for its equal in a5 +30PxOn+ 6orx O07 ha eet 35r? isn ad 257% S Ga: 2 Wherefore the fum of the cubes of perpendiculars drawn from the point G to the fides of the pentagon, is equal to the fum of the cubes of perpendiculars drawn from the point A to the fame. Since 2X7+Om +2X7r—On =4ri+ 37° 20m—20n L2G +37xX2xOm +2 On +2x0m'—2xOn? = and 3 Se ee r a 0 ee a 2xOm + On = 3, we have 37 X2XOm—2x On + Fs 2xOm —Onv =4r’.. But 20"—20n=r7 therefore 2X Om —On=7, or Om’ —On'’ — as | 23 Ir P, inftead of bife@ting the arc BK, be any point between B and K. the fum of the cubes of perpendiculars drawn from it to the fides of the circumf{cribing pentagon, is equal to the fum of the cubes of perpendiculars drawn from Qo the fame. For fince Oc + O2 +O =Ob+ Odand Oc — 04, O0d—Oz and O8, begin together, and become maxima together, Oc_O4 has to O£ a given ratio. Let that be the ratio of » to 1. Then Oc—O8= mxOk, and Od—Oa = Oc—O3 +Okh— m+1XO8. Oc=O0b+mxO0k, Oa=Od—mFti xO}. =—— 3 3 net =? , } | 2 Oc = O06 +3mxXOd xX Ok+ 3m*xObx OR -+- mx hyp EE anata | > Oa = Od —3xm+1XOd XOk+3xm+1%0dx F 2 ——a Oz 44 INVESTIGATION of fome Of — OF. Wherefore Oct-+ Oa +0f =0s +042 +3mx Ob -x Ob— 3x mF x Od x OF + 3m! x O x OF +3x oe ae x Od x OF + OF. Now, let this be = 00 +04. = V3. Then 90H magn i 8 + 3m * Ob x OF + cael “x Odx OF+OF ae mx Ob =m+1 Vv? sg Oe K ohh eG yikeee x Od a = when Ok = 0. But mx Ob = m+1 X Od eat: Va=o. For when O P — 0, O4 is the fine of 72°, AN O d the fine of 36°. When O £ is a maximum, it is the fine of 18°, Oc is =+r, O84 is the cofine of 362, and Oc —Od the verfed fine: of 36°. © Wherefore,’ -++ 1: m = Os -Od =O Yer fed fine of 36° + fine of 18°: verfed fine.of 36°.>~ ~~ Ler BD (Pl. ILfig. 3) = BH = fide of an infcribed pene ; bife@t BD.in F, and draw OFC, AC, BC and'DH. Then, fincé the angle FOB is 36°, CF is the verfed fine of 36°, OG is the fine: of 18°. But fince the triangles CFB, DGO, are fimilar OG :CF= DG: FB — DH’: DB, and OG + CF:CF= DH+DB: DB = DH: DB =DG:FB = {quare of the fine of 72° : {quare of the fine of 36°. For when DH 1 is cut in extreme and’ mean ratio, the greater part is equal to the fide of the pentagon. DH is cut in extreme and mean ‘ratio in the point L, and LH = BD; the triangle CDP is fimilar’ to the triangle DOB ; and. the Hiatigh! MDN to the triangle BOC. = - Tuis demonftration, however, was unneceflary.. For if the fam of the cubes of perpendiculars drawn from P to the fides of the) pentagon, be equal to the fum of the cubes of perpendiculars drawn PROPERTIES of the CIRCLE. 45 drawn from Q to the fame, both when O é is = 9, and when it is a maximum, this equality muft exift whatever be the magni- tude of O& between thefe limits. -AND in a fimilar manner is it demonftrated, that the fum of the cubes of perpendiculars drawn from P to the fides of any other regular figure of an odd number of fides circum{fcribing the circle, is equal to the fum of the cubes of perpendiculars _ drawn from Q to the fame. For if Om, &c. or fuch parts of. perpendiculars drawn from A to the fides of any regular cir- cumfcribing figure of an odd number, 2, of fides as lie between: O and G, be denoted by A, B, C, &c.; and Ou, &c. or fuch parts of perpendiculars drawn from G to the fame as lie between O and A, be denoted by 2,4, c, &c. A+ B+ C+ &e, to — n—I ake terms, —a — b — &e. to terms, = 5, if 7—1 be a multiple. of 2 by aneven number. Alfo.A-+-B “+C “4 Bec. to” SES tar tet oe + &c. ‘to A 2 ems, = Pe, and‘ A*+-B3' +O ie. to 0X53: a= atin! — 4 —}3—c3— &e. to. terms; a = 2 Bue 2 n—TI be a multiple of 2 by an odd number; A -+-- B+C-+ &c. to nt = teks > a+ Kei to, “Ses, 4 - pe EBC Be. to a terms, +f a’ + b+ &e._to ajassks terms ae 7, Pose Age: Bets eee, ett | 4 terms. 46 INVESTIGATION of fome q 10! bh ore : terms, = >- Thus, in the terms, — a) — b3 — &c. to saa heptagon A+ B—az= , A+B4+a= 7a, and A}-+ Bs — “3 2 a= ~ In the enneagon A+ B—a—b= =; MV+tB+a ys A aid Ast pO Ihe pen - dad fo baal the enneagon, A is the cofine of 20°, or the fine of 70°, B is the fine of 30° = s a is the fine of 10°, and & the fine of 50°. Wherefore the fine of 70° = fine of 10° + the fine of 50°; the {quare of the fine of 70° + fquare of the fine of 10° + fquare of the fine of sor ae ; and the cube of the fine of 70° — ~ __ eube of ae fine of 10° — cube of the fine of 50° = ra viz. thrice the cube of half the radius. And fo on. 7—O0" = =rt+—4r xOe+ 67 xO — 47 x Oc + Oc, - 7—Oa = 40 me Oe x Oa 47X00 +02; 7—OF =r —4PxOk+6r x OF —4rxOF +f; aan ey = r++ arn x OF+ 67° Oo +4r x Os +05" “ r+0d =rt+4rxOd+6rx Od +47rxOd +0d, Bur PROPERTIES of the CIRCLE. 47 Bur fince 57 ( = fum of the perpendiculars drawn from P or Q to the fides of the pentagon) = . ee vi = AQ +00 +CQ+3Q +1Q, and fince the fum of the nite {quares of perpendiculars drawn from P to the fides of die pen- tagon, has been demonttrated equal to the fum of the fquares of AP’ +BP +CP +DP +EP_ 4r _ Ag + uO’ + ca +DQ ta: fom rete And as the cir- —_—d, from Q to the fame, Ree hdc of the rade which fis PO for its diameter, is di- ‘vided into five equal parts, in the points a, 4, c, d; &, Oc a Oa +0Ok +06 +0d =r x SOF = a and Oc + Oa +08 OP (1g OP Te ee hh Se oe 4 < 4 grt 6° x Oc + Oa + OF + OF + Od +02 + Oa" 4 OF + Os + Od = 4x Wherefore, +O) + Od =A 4 Org SE = Mix = sX5 Mire half the fum of the fourth powers of perpendiculars drawn from both P and Q to the fides of the pentagon. In the fame way is it demonftrated, that the fum of the fourth powers of perpendiculars drawn from P to the fides of any regu- Jar circumfcribing figure of an odd number, 2, of fides, is equal to the fum of the fourth powers of perpendiculars drawn from Q@ 48 INVESTIGATION of fome Q to the fame, = 2 Mort = 9 a oak ae X74. When P coin- I cides with B, Oc is =7, and A*+Bt+Ci+ &e. to = I ‘ ° terms, (when 2 — 1 is a terms, + at + b+ + c+ cae &e. to 2 multiple of 2 by an even spelt is = 37 gor 8 xX rt; and a. 3 : terms, -+ a++ 5++ &e. to id = terms, Bt + Crt: Bc. tom (when 2—1 is a multiple of 2 by an odd number) i. n—8 : ; if ee 7 In like manner is it demonftrated, that the fum of the fifth powers of perpendiculars drawn from P to the fides of any regu- lar circumfcribing figure of a greater number 2 of fides than 5, is equal to the fum of the fifth powers of perpendiculars drawn —— FS from Q to the fame; and that AS+ B5+Cs+ &c. to —— ; terms, (when —1 isa terms, — @5 — 65 —cs — &c. to C rs : multiple of 2 by an even number) is = oe and “AS + B: + Cs n+ y1 + &c. to rig? terms — a3 — Js — &c. to i rr 3 terms, (when A rs m—t11is a multiple of 2 by an odd number) is = wi PROPERTIES of th CIRCLE. 49 , B is the cofine of 3x $y ‘Cis = at 1s the cofine of 2 A cofine of 5 X = , and fo on; and when 2—1 is a multiple of 2 by an even eae the laft of the terms A, B, C, &c. is the _cofine eee 3 xX Se ; ", and a, the firft of the terms a, 4, c, &c. ‘is the cofine ce ues if oe aie b, the fecond, is the cofine of OTS. 3g x ae 3 c, the third, is the cofine of m9 x ae 3 and the laft of the terms 4, 4, c, &c. is the cofine of + x oor 180° But when “—1 is a multiple of 2 by an odd number, the laft of the terms A, B, ro} &c. is the cofine a2——1 ie of - ae ee the ‘firtt of the terms 4, b, &c. is the cofine of “43 Xx 1 and the laft of the terms @, 4, &c. is the co- 180° Univerfally, if m be any odd number fine of n»—2 X lefs than 2, we have A" + B" + C” - &c. to ea terms, 1 —TI r ; terme, = a when 7—TI is Ae: eo BC — &e. to * a om of 2 by an even number; and A” + B”+ C+ &c., r * terms, — qe — b® — &ce. to lara terms, = —, when n to Vou. VIL—P. Te ce n—T 50 « ENVESTIGATEON of omer 2——1 18 a multiple of 2 by an odd number. Thus, in the en- neagon, or figure of nine fides, A is the cofine of'26° or'the fine of 70°, B is the cofine of 60° or the fine of 30°, a is the cofine of Loo° or the fine of 10°, and 4 is the cofine of 140° or the 27's pa 1, And if p-be fine of 50, and A"— 4” i" =. any even number lefs than 2, A? +B’ +. Ce + ee. to° Daa —$—$___ N-+- tT terms or a i+ tae j aS terms, (according as#——1 is a multiple of 2 by an even . #42 Fe Je we} —TI ¢ P Pp er odd number) iss= x 113-5 Zis2 Pat x Fas hy Fo. Tighe: — 2 T.2.3+4e 2 2 SS Se EEE | fequently, A? + qe 4 A” — q” + B? ot b 4 B“ — $2 - &c. to 4—TI 2 We 2h cape. ¢ aa pe ye? rm ; terms, 1S ae Hips eee We uo ~ a= = | T.2.3-f eee 3. aoe ior ¢ hits one DY ge Pies 7P ad 773 ie 2 = Xo — -4+ 4 (when L aS < 2 #—1 is a multiple of 2. by an even number) = : 2 . 7— x Pea When r=1; 2+ An 6a to, 23 and when n—I is a.multiple of 2 by an odd number A?-+ A” + Bat BY + OPO" + &e, to = terms, -+ a?—a”™ +. I 4 teeliee neal oo PROPERTIES of the CIRCLE. st o? — b™ 4. ch —¢™ -+ &e. to 4 3 terms, is equal to 7 X 1.3-5-7 +++ to Pp—1 ie 1:2-3.4.+. tof ie 2 rp Lat: . —— + —, which when 7=1 2 2 . pte pt6 ptio. | 2p—2 e . to 2 2 is) 4 2 ee a p BOSS Bre 06 +++ to % 2 PE2PFO...t020=2 ig 2 ais A < —p4q- Hence the fummation of £ j a-(Te lie, @ 2 an endlefs variety of feries, of which the terms are powers of the fines and.cofines of angles ; and though they do not confift of an infinite number of terms, they may confift of any number of na——T -terms whatever, fince may be equal to any given number + as well as and forse expreffion nx EB5-7 th 4SOP——t Pp 1.2.3-4,-.'. to — 4 ae pays hee peg 5 2 2 2 T ‘ X oer is =aX 3 p X 377», when Mae We RAs Aas s tor 22 oe : is equal to an even number, or f is a multiple of 2 by an even number. But when . is an odd number, it is equal to 7 X pts pt8 --to2p—2 I = = ae 5 % ez» which muft then be ufed. 22 2 ; 2 e 4. CO) to G 2 THE 52: INVESTIGATION of fome THE fums of thefe feries, however, vary with the variations. 2. 5 : ¥ TP - in the magnitude of 7. For when ee 4, A, Stra sit yu . re. « Pp. 3 wee does not. vanifh, and 3¥z becomes. refpectively | en. west 2. 4 Led 2 : . 4 b+>27 Ke. 2 2 oor —_.. iJ At ay gt & Bb" =p? CG? om ae pm er &tc. to terms — my Fl a when #:—1 isa multiple of 2 by an even. number, and-the fum of. the- feries is conftant. or- invariable when m js. siven; let the number of the terms ~—— be great or fmall. This feries, when 2 is infinitely great, or 7—1 has-to 4 a-ratio greater than. any given or affignable ratio, may be confidered as infinite. . * tA, A Ts a7 Wi wT. s tee ~ terms, * A"—a a” 4 Bh" b + —e’ -+ &c orm pls ee a Sa = 1— A—a +B—64C—c+4 &c. to. T terms. 2 2. Av — a” + BY — Be — pn + &e, to =a pia age Kx A—a+ ae b pnb lel la oe a lle + &e. an ~ terms, ap be BEET ae 7m r + &c. to eam. terms = —>- = I, when rie ——<——$$<$_<$<—<— —_—— ——> ¥ A” —A—a"—a+ B”—B—b"—} + &c. to terms, n 4. 7"—Fr — —— oO, When? © a PROPERTIES of the CIRCLE. 53 Ir may not be unaceeptable to geometers to fee the foregoing conclufions in regard to regular figures circum{cribed about and infcribed in a circle, derived by making ufe of one point only, inftead of two, either in or not in. the -circumference, which is: ~ eafily effected in the following manner. Ler the fides of any regular figure of an even number:of fides: touch the circle BRETCQES (Pl. ff. fig..4s) in the points B, R, E, T, C, Q; L, S, and. let DN, DH, DM, DV, be perpendiculars from the point) D to the.diameters joining the points of contact ; and from the points of contact let chords be drawn to any point A in the circumference. Ir GE, or the-radius of the circle, be denoted by 7; and ‘A a, Ab, Ac, Ad, be perpendiculars to the diameters joining the points.of contadt, 4C, aB, T OSs Te oe Bic, Qd, dR, are re- aes equal to the game from the point A to the files of the fur and are aie refpectively equal to ad AB, 26) ae aT. as aE AE exe aE But if N denote the num-- By 2%) )2ar Br? ar 2 | ber’ of fides of the figure;- the fum of ‘the perpendiculars i aS a N Xvrvl “Wherefote~ AC + AB. +.AT. + &. =2N 7% nae is Prop. 4. Dr STEWART’ s Theor.: “AG AN, the fur’ “OF the fquares ‘of the: two! perpenditulats Pama, parallel to BC, or Ba eae = oF pes 2% Ga; and the » a of? the’ two: perpendicular roms A \ parallel to LE, or Ee: POE'S PPRG SEP PTT si 2a 29 xX'GV ald Rd 4+7Q = 27°42xGd. Wherefore the fam of the {quares of ‘the perpendiculars drawn from the .point A to the. fides. 54 INVESTIGATION of ome fides of the figure, is = Nx7*+2x yk: Gi # Ge +Gd’. But fince the angles GaA, Gb A, Ge A, GGA, are right ones, a circle pafles through the points A, a, J, G, c, d. having GA for its diameter. And becaufe the angles CGT, QGC, LGQ: are equal, this circle is equally divided by the points a, 4, ¢, d. Confequently the fquares of the chords drawn from thefe points to the point A, are together = N X at that is, Ac (Ga ) + Aa (Gc) + Ad (Gb) + Ab (Ga) = wx Shin N X =) aioe ae the fum of the fquares of the perpendicu- lars drawn from A to the fides of the figure, is = N Xr*>+2N x 5 — Nx 3. Hee the fum of the fotidets of thefe Big AT AL | AE, AQ’ 4r SAL Tar ag diculars is = oak ee = ——é4 a SS ee = eas _ "Therefore AC’ + AB’ +, &. = NX674+= NX 27% | 4r ars i Wad + AB +, &c. X 37°... Whence this propofition: Ir a circle be divided into any-even number of parts, and from the points of divifion chords be drawn to any point in the circumference, the fum of the fourth powers of thefe chords is equal to the fum of their {quares, maultiplied by.thrice the {quare of radius. : WHEREFORE PROPERTIES of the CIRCLE. 55 WHEREFORE Aa + Ab Ac +Ad = Aa + Ab + Ac Ad Cb ee Fe AS ft p aR) Seer" ran: a ere =.N.Xx, 37; and DM + DV + DH + DN’ tea Seo SE pw x SxS 2 aXNY GL = DM 4.DV + DH + DN ine ee s. eu. De® ; bee 246 Now, it is evident, that perpendiculars drawn from the point D to'the fides of the figures, are’ refpectively 7 + DM, 7 — DM, r+ DV, r—DV,;7+ DH, r— DH, ¢ + DN, r — DN. Bur r-DM + 7—DM = ar 4 2x DM, | 7A DV + 7—DV = or + 2xDvV, 7. DH -.r—DH = 27° + 2x DH, * | PDN + 7— De = 27° -+ 2 x DN. WHEREFORE the fum of their fquares is equal:'to Nx 1’ + 2XDM + DV'+ DH + DN a Oe GD fnee the circle which paffes through the points D, M, V, G, H, N, is: equally divided by the points M, V,H,N. This is Prop. 5. ef Dr STEWART’s Theorems. 7 DM: + r—DM = 273 + 6rXDM> POU ef OV oe ere DN 7+ DH +7—DH! = 27 4 6rx DH» r+DN +r—DN = 27 + 6rX DN. WHEREFORE, 56 INVESTIGATION of fome WHEREFORE, the fum of the cubes of the perpendiculars, drawn from the point D to the fides of the figure, is = Nx 734 67 X DM’ + DV’ + DH’ + DN =Nxr73 + 67 X oS This is Prop. 23. SrEwaRt’s Theor. When DG =;, the fum of the cubes of the perpendiculars is = N X Sx rs =NxX 95 73, This is Prop. 22. Dr STEWART’s Theor. WHEN DG = = #65, 1D coincides als A, ae fum. of the ca AC” A Sr ee -of the perpendicular is equal to + ae = east, Bc. ; and, confequently, we get AG. > AB + AT +, &c. = 5 MIN MGS icc ae oe wee Tg BEN X 20 x= Nx 107+x AG + AB + AT 4, &ci Ir, therefore, ws circumference of a Saale be divided into an — even number of equal parts, and from the points of divifion chords be drawn to any point in the circumference, the fum of the fixth powers of thefe chords is equal to the fum of their fquares, multiplied by ten times the fourth power of radius. ~ 7 DM' 7 DM! = art +1artx DM + 2x DM, FED der DV, =: 2 +127 x DV +2xDV' ai 5 ye i “7 DH = artbiar x DH +2x DH’, r= DN‘ +r—DN = 2art+i2r x DN. + 2x DN. WHEREFORE PROPERTIES of the CIRCLE. | 57 WHEREFORE, thé fum of the fourth powers of the perpendicu- lars drawn from thé point D to the fides of the figure, is = N x rt+NX3r x GD +NxX : x GD’ ; and eight times this fam =N x 8rt + 247. GD + 3:GD*. This is Prop. 29. of Dr SrEwaRT’s Theorems. ? Wuen GD =7, the fum of thefe fourth powers is NX ae ; PUSORN ar a N ue = ak oa x r+, which is Prop. 28. of Dr Srewart’s eects ——3 —— AC Anp fince ——; 4 AB pan! “+f, Se. =N x 357 , we get 2+ 24 r+ 24 r+ 2+ r+ AG +AB LAT 4+, &. = Nx 33. oe = ila Weare atr’, ) Anpb by proceeding in this way, (the law of continuation be- ing evident), we get Propofitions 39, 40, 41, &c. of Dr SrEwaRT’s- Theorems, ‘fince the powers of DM, DV, DH, DN, &c. however high, may always be expreffed by thofe of DG andr. The fame reafoning ‘holds in all even powers, when the point D is without the figure, by taking the powers of DH, +7, &c., when ‘DH, &c.' 1s greater’ than ‘than 7; inftead of the poets of r+ DH, &c. Ler any regular figure of an odd number of fides, (Pl. III. Fig. 5.), circumfcribe the circle, and touch it in the points B, BE, C, QL; and from any point D, let perpendiculars DP, DR, DS, DO, DT, ’be drawn ‘to the ‘fides of the figure; and DF, DM, DN, DH, DV, perpendiculars to the diameters pafling through the points of contact: Vou. VI—P. I. H THEN 58 INVESTIGATION of fome THEN, if radius be denoted by 7, it is.evident, that DP is — y—GN, DR =r—GF, DS=r+GH, DO =r + GM, and: ‘DT =r-+GV; and calling N the number of the fides of the figure, the fum of the fquares of thefe lines is NX 7? 4+ 27 & GH + GM+GV—GN—GF + GH +GM +Gv + GN +GF. But fince the angles HGN, NGM, MGF, FGY, are. equal, and the angles at H, N, M, F, V, right ones, a-circle, ha- ving its diameter = GD, pafles through the points G, H, N, D, M, F, V, and its circumference is divided into equal parts at the points H, N, M, F, V. Wherefore GH + GN +GM + GF + GV = ax Nx 52 aNxee But DP + DR’ 4 ae GD. } DS +DO +DI =Nxr+Nx-—>: (SrewARt?’s Theors. Prop. 5.). Therefore 27 X GH + GM + GV — GN — GF- — o,or GN + GF = GH + GM.+ GV.. Whence this propo- ' Gition: If, from any point, perpendiculars be drawn to the fides: of any regular figure of an odd number of fides, circumfcribing: a circle, the fum of the parts by which thofe perpendiculars,, which are greater than radius, exceed it, is equal.to the fum of, thofe parts by which the perpendiculars, which are lefs than ra; dius, fall fhort of it. And this propofition is alfo.true with re-. gard to any regular figure, of which the number of its fides is a, multiple of any odd number by 2, fince the perpendiculars DF, DM, DN, DH, DV, &c. are the fame both in number and mag= nitude, in any regular figure of an.odd number of fides, and a; regular figure of double the’ number, of fides. Confequently, in a, hexagon, one of the three perpendiculars drawn from any point; D to the diameters joining the oppofite, points of contact, is. equal PROPERTIES of the CIRCLE. 59 equal to the fum of the other two, and fo on; and if, in the hexagon, the point D be taken in one of the ies diameters, the perpendiculars drawn from it to the other two are equal. Acain, DP’ + DR + DS’ +D0 +Dr=Nxr+4+ 3r er. OM +0G ie GN. —' GE oar x - 6H 4 GM FOV FON TOF 4 Oy OH Lav GN —GF =Nx?? Nx gr 4 Gi CM. + Gv’ — GN’ —GF. But finceN X73 -++N X 37 i = DP + DR’ +DS +DO'+DT GH + GM’ 4+GV = GN’ + GF. | Ir D be in a line perpendicular from G the centre, to a dia- meter drawn from any point of contaét L, the odd chord GV svanifhes, (V coinciding with G), and GN = GM, GH = GF; and the expreffion for the fum of the cubes of the perpendicu- lars, drawn from D to the fides of the circum{cribing figure, is aah fimply N x r? +N x 37x SO Ir the figure circumfcribing the circle be a pentagon, a line drawn from G, bifecting the angle QG d nearer to G, is perpendi- cular to LG ; alfo, if D be in the line Gd, the point M coincides with D, GN = GF, GH = GV, and GM coincides with GD, and twice the cube on GF or GN is equal to the three cubes on GD, GH, GV, or to the cube on GD with twice the cube on GH or ‘GY; and the difference of the cubes on GF, GV, or on GN, GH, H 2 ; is 60 INVESTIGATION of fome is then equal to half the cube on GD, or 3GF —aGV = GD. Hence an eafy folution of this problem. HavinG two equal right lines given, it is required to cut one of them into two parts, and the other into three parts ; fo that the cubes on the two parts, into which the one of thefe lines is cut, fhall, together, be equal to the cubes on the three parts, in- ‘to which the other is cut, taken together. HENCE, alfo, an eafy conftruction for this problem: On a gi- ven right line, to conftitute a triangle, fuch that twice the dif- ference of the cubes on the other two fides, fhall be equal to the cube on the given line. Ler AC be the given line, (PI. IIL Fig. 6. ). With A as radius, defcribe an arc AB. Take the angle ACB = 36°. _ Draw AG perpendicular to CB, and join AB. From A and C as centres, defcribe arcs with the radii a and CG, interfe@ing in the point F. Then CFA is the triangle required; and 2-CF —2-AF =CA. - DEMONSTRATION. SINCE the angle ACB is 36°, AB is the fide of a decagon in- {cribed in the circle, which has AC for its radius ; and CG is the perpendiculat to the fide of an infcribed pentagon. Butitis ~ = sets well known, that CG is = , and ‘AG = — AB ~ ACXAB. Confequently 3AC’ = 3AC?x AB+3ACXAB; add AG’ to both, and we have 4AC* = AC’ + 3 AC x AB 4. 3AC © PROPERTIES of the CIRCLE. 61 KC + 3AC x AB+3AC x AB ant 3 AC x AB’, and AC’ = ri a “RF AB + AB? 8 the cube of radius is equal to twice tlie difference between the cubes on the perpendicular to the fide of the infcribed pentagon, and half the fide of the infcribed decagon. 3 = 2XCG — AF. Ehus, in any circle, Proposition. Let any: regular figure of an odd number of fides, be circumfcribed about a circle, and let (7) be any odd number, lefs than the number of the fides of the figure; and from any point within the figure let perpendiculars be drawn to the fides of the circumfcribing figure ; then the fum of the (z) powers of the parts by which thofe pérpendiculars, which are greater than radius, exceed it, is equal to the fum of the (7) pow- ers of thofe parts by which the perpendiculars, which are lefs than radius, fall fhort of it. _ Hence thefe problems. _Havine two equal given right lines, to cut one of them into two parts, and the other into three, fo that the cubes on the two parts, into which one of them is cut, fhall, together, be equal to the cubes on the three parts, into which the other is cut, taken ~ together. _ AnD having two equal right lines given, to cut one of them into feven parts, and the other into eight, fo that the cubes, the 5th powers, the 7th, gth, 11th and 13th powers, of the feven parts, into which the one is cut, fhall, together, be refpectively equal to the cubes, the 5th, the 7th, the gth, the 11th, and the 13th powers, of the eight parts, into which the other is cut. TuHE firft of thefe two problems is effected by a pentagon, inf{cribed in a circle; and the fecond, by a quindecagon infcri- bed. | Ir 62 INVESTIGATION of Jome Ir V be as much on the other fide of the centre G, towards L, as it is towards C, the lines GN, GM, exchange their values or _ magnitudes, as alfo do the lines GH, GF; and the perpendicu- lars to the fides of the circumfcribing figure then become _r—GM, r — GH, 7 +GN, 7+ GF, r— GV; and the fum of their cubes N X73 -+ NX 37. ae + GN 4Giee ae a GH? — GV*; which added toNx +N x 37.22 4 Ga GH 4G 24 GN. .=) GF viche Mum! of ee bee before found, and the aggregate divided by 2, gives N X73-+N x 37. —y) GD : Led ; : me the fum of their cubes, when D is in the line drawn from the centre G perpendicular to LG, LET a circle, (Pl. II. Fig. 7.), be defcribed on BC, with the centre G, and let BF be a {quare on the diameter BC ; draw EGD from E, through the centre G, to meet the circle in D, and join DF. TuEn, fince BG x CS, or CG x CS = GS, GC is cut in ex- treme and mean proportion in the point S, and GS is the fide of a regular decagon, infcribed in the circle. And fince the per- pendicular from G to the fide of a regular infcribed pentagon, is _ BG+GS BG +GS — 2 3 , BS is twice that perpendicular. But 2 3 3 f NG ee | eve ae a = _ Confequently BS’ — GS’, or 7 a Gs’ — GS'= 49. Therefore 37° = 3r° x GS-+ 37x GS» and r3 PROPERTIES of the CIRCLE. — 63 y3 =r xGS+rxGsS. But BS is cut in G, in the fame manner as GC is cutin S. Wherefore, if another circle be de- fcribed, with BS as radius, and a line be drawn from one of the angles of a fquare, defcribed on the diameter, through the centre, to meet the circumference in a point, and if this point, and the other oppofite angle of the fquare be joined, 2r-+,GS.—r will in like manner be say! x7 + GS ,or4XBS, and 3+ i2r’. GS + 67. Gs +GS° Agi ail 27. GS + 127. GS + 4. Gs. . Therefore 373 = 67. GS + 3GS’, and r? = 27.GS + GS’ =r. GS+r. GS. Therefore 27. GS Cae =r +r.Gs, and Beir r. GS 47, andy. =r. GS—~z. Gs. “1dr, therefore, from any point in the circumference of the circle. BDC, perpendiculars be drawn to the fides of ‘any regular figure circumfcribed. about it, the fum of their cubes being = N X 3. r3, (calling N the number of the fides of the figure), is = NX 57. Gs. +N x 2 cs. 3 and twice the fum-of the cubes of thefe perpendiculars is N X 5. Gs. +N XIor. GS ; that is, equal to five times a multiple by the number of the fides.of the figure of the cube on the fide of an infcribed regular decagon, and ten times a multiple, by the fame number. of the folid, which has the fquare of the fide of the, infcribed decagon for its bafe, and radius for its altitude; and if the perpendiculars be drawn from any point P, within the circumfcribed figure, that is, not in the circumference of the circle, . 64. INVESTIGATION of fome circle, twice the fum of their cubes will be equal to 2NX - GS' + 27r.GS +2N x 6rx Se, that is, equal to twice a multiple by the number of the fides of the figure of the cube on the fide of the infcribed decagon, together with four times a mul-. tiple, by the fame number of the folid which has the fquare of the fide of the decagon for its bafe, and 7 for its altitude, toge- ther with thrice a multiple by the fame number of the folid, which has the fquare of GP for its bafe, and 7 for its alti- tude. : In like manner, may the fixth powers of lines drawn from the angles of any regular infcribed figure of a greater number of fides than three, to any point either in, or not in the circumfer- ence, be expreffed in terms of the fide of an infcribed decagon, fince their {um is a multiple of the fum of the cubes of the per- pendiculars, to the fides of the circumfcribing figure, by 8 73. AGAIN, fince 7+ GS:7r::7:GS:: GS:r—GS, we have ar+GS:r+GS::r+GS:r::r:GS::GS:r—GsS. eh a ee ee : WHEREFORE 37-4 2G5 eee GS = 4X 27-+Gs, or 26 73-517. GS + 337. GS + 7GS° = 3273+ 487.GS + 247. GS £46s’, or 37%. GS + 97. GS + 3GS = 67°, or GS + 37r-GS + CS ead 0 WueErerorg, fince four times the fum of the cubes of the perpendiculars drawn from any point in the circumference of the circle to the fides of any regular circumfcribing figure, is Nx.5 X25; four times the. fum of thefe cubes is = NX g7r.GS+ 157. Gs +5GS =-5 NX. Ge F 37: GS +GS; that PROPERTIES of the CIRCLE. oO _ that is, equal to five times.a multiple, by the number of the fides of the figure of the cube on the fide of the infcribed decagon, together with fifteen times a multiple, by the fame number, of the folid, which has the fquare on the fide of the infcribed decagon as its bafe, and 7 for its altitude, together with five times a multiple, by the fame number of the folid, which has 7” for i its bafe, and the fide of the decagon for its altitude. . Ler the circumference of a circle be divided into any number n of equal parts, and from any point in the circumference let chords be drawn to the points of divifion, and let 3m be any number lefs than 7, the fum of the 2 powers of the lines which have refpectively to 27 the diameter, the ratios which the cubes of a, ‘chords have refpectively to 873, the cube of x 113. eS 7h - 6 4 baie: Is De - 3° 45 3 Mm a Let the chords be denoted by A, B, C, D, &c. to 2 terms; aad fet 872: A’ Sor-4, 87°: B= e'7:b, 8G = 2r:e, ™ erecta is equal to n ; A3 ‘ AS” Sea = 27d, &c.. Then ¢—~—, and a” =", — P= ae gay A Bom : Asm [ae oe. ; and a” -- GARE &c. to 2 eer: 15. aes ies ia a Be } ; ; +——s + (&e. to m terms.. If p= 3m, we have a + ag bem & ae zs Bc, ; 59 c= p+m pre ic ee aa, 4 Ce pnt the fum of the 2/ powers of the chords A, B, &c. is” X UR ey Ae ae a aes ; ° 1+2+3+4e oo ? ; VoL. VI.—P. I. I Therefore 66 INVESTIGATION of fome am am pate 1.3.5.7: Sigheet Y, — ‘Faraone y i) A Li Res Fa Myh 1.2.34 e Pp 2% = ng 2n3-Aas 3” in 2, 2 : if 2%, a+b + &e. =a XX” = 5"* ; and the 1.2.3 2 diameter, (or 27) X a + 4° +c’ + &c. = fum of the cubes of perpendiculars drawn from any point in the circumference, to. the fides of a regular circumfcribing polygon of 7 number of fides, and a + 4° + &c. is to the fum of the fquares of thefe perpendiculars as 5 to 6; and if the perpendiculars to the fides of the polygon correfponding to the chords A, B, C, D, &c. and drawn from the fame point in the circumference that thefe chords are drawn from, be denoted by P, Q.R, 5, &c. a+b + Ps AXP 2XQ 4 cx® i DxS wth Ke 24 + &c. 2” 7” x amn+omtom + & = P+ Qs + RI” -+ &c. = the fum Cee snip yg 13-597- te of the 3 2 powers of thefe perpendiculars, = 2 X TeBi3eqe 0 ye THEOREM ®. From any point G, (PI. III. Fig. 8.), let the chord GA be drawn ; let GAF bea tangent tothe circleat A; and let AD be perpendicular to the diameter BC, and CF, BG toGF. The right line which has to BC (27), the ratio.of AC” to BC’, or the triplicate ratio of the chord of the arc AC, to the diameter, is ACx CF ~~ ACXCD | m —Ba 2 8 “BE =a fourth proportional to the diameter, the chord and the perpendicular drawn from one extremity of the PROPERTIES of the CIRCLE. 67 the chord to the tangent to the circle at the other extremity, or a fourth proportional to the diameter, the chord and verfed fine Bes Confe- of the arc AC. For, the angle CAF = the angle ABC = the angle CAD. BC Therefore CD ='CF, and AD = AF. But CD = AeSEb! AE 7% AG quently — = = Bc which has to BC the ratio of AG’ to BC. Q.E.D. Cor. 1. BD = perpendicular BG ; GF = the chord AE of double the arc AC = twice the fine of the arc AC. ee a 2m, has to BC, the ratio of AB’ Bie Ae 8G OR 2 —o— Of —“ to BC. =———5 ee De Cor. 3. GF ae, BG = ita CF’ cafe aed BC BC BG BC = ace Ee and BG = BC AB X BG x BC; and the lines, which have to BC the ratios of eh es fae bawle . AC: BG and AB : BG are to each other as ACXCD to AB x BD, or as AGx CF : ABx BG. 12 SEE 68 INVESTIGATION of fome . . Sze Fig. 1. and Theorem ®. ‘Since’ the part of the tangent at the point A, that would be intercepted between perpendicu- lars drawn to it from P-and Q; is equal to 2 Pa, or 2 Qe, the part of the tangent at the point B, that would be intercepted be- tween perpendiculars drawn to it from P and Q, is = 2 Pe, or 2Q fs; and the part of the tangent at C, that would be inter- cepted between perpendiculars drawn to it from P and Q; is = 2P4, or 2Qd, we have (when AB, BC, &c.. are equal, or when the diameters pafling through A, B, C, &c. make equal angles with one another at the centre O) the fum of the fquares of thefe parts of the tangents, (calling 7 the number of the I : ' points of contact), = 2 x + r’; the fum of their fourth pow- I ers = 72x a x 277+; and the fum of the. 2 ™ powers of thefe parts (# being any integer lefs than 7) = 2x a X 2m7” (r being the radius OP or OQ) = the fum of the 2m powers of the chords drawn from either P or Q, at right angles to the diameters pafling through A, B, C, &c. = the fum of the 2m powers of chords, drawn to any point in the circumference from the angles of a regular infcribed. figure of » number of fides, or from the’ points where a regular infcribed figure of number of fides, touches the circle, = the fum of the 2 7 powers of perpendiculars, drawn from P or Q to # number of right lines pafling through Q _ or P,.and interfeéting each other at equal angles. And the fum of the 2m powers of the halves of thefe parts of the tangents, or of the parts intercepted between the points of contaét and perpendiculars drawn from either P or Q to the fides of the equal fided figure circumfcribing the circle, or fegment, is 1-3.5s006 2M—I ; =) x 1203-000 ie xr” = the fum of the 2m powers of the PROPERTIES of the CIRCLE. 69 the fines of the angles formed at the centre O, by OP or OQ; and the diameters pafling through the points of contact to the radius omoceee 2, 12 OP or OQ; that is, = Pa +Pe +Pd 4,&c.orQe + ay” a Od. +, &c. = the fum of the 2m powers of perpen- diculars drawn from any point in the circumference of a circle defcribed from P as a centre, with PO as radius to 7 number ‘of right lines, interfecting each other in P, and making all the angles equal, = the fum of the m powers of the rectangles A a X aG, Bexer, Cbx dh, &c.; or of the reftangles Ge x c A, rf x fB,bd x dC, &c. when the regular figure circumfcribing the circle has an odd number of fides; but equal to twice the fum of the 2 m powers of faid fines, or to twice the fum of the m powers of faid rectangles, when the regular figure circum{cri- bing the circle has an even number of fides, fince the number of the diameters drawn through the oppofite points of contact, and making equal angles with each other, at their interfection in the centre O, is only half the number of the points of contact or fides of the figure. But thefe reétangles are refpectively equal ' tor—Oa X r+Oa, r—Oe X r+O04, r—ObxX7r+04, &e. —— fee ies | ——2 : or r—Oa, r—Oe, r°—O4b, &c.; and the fum of their m"™ 71 isos), (ages GE | VL a aa powers is 77?" Ho ata eats Oa +O¢ +046 +, &c. to” terms, m—I m Ss pe ay 2a + ary a Se On Oe -f Oo. +, &c. to m terms +, &c &e. + ees Gn Us +, &c. to (7) terms, or to ——— 22 —27 —-2" —Oa —Oe —Od —, &c. to (7) terms, according as m is even or odd. IIT. ie oleae «3 Fein 2983 . “rad aarbaah, is 93 ’ od ie aisha 8 Pi — he ae alae és Deolyas vt shi oda, algaahsn bin toe | aba: to vedectoic beach Ae ed ol yi bie oan ‘A t digwords ct "eth: eorsinsily 9d e y golgens ‘ope Bln art a aoe he RA eet me OR Goh aed aon ott Vrans:RS EdnVol6." P 7e. Fig.l. PLATE LL. Diizars Jculp.- 3 Trans:R.S Edin? Lb P70. Pram TD, : ee 5 in| Fig. 7. DLizarsleupl y et n | ne rh III. Account of a SeR1ES of EXPERIMENTS, /hewing the Er- FECTS of CoMPRESSION in modifying the ACTION of HEAT. By Sir ¥AMES HALL, Bart. ¥.R.S, Evin. ) ' [Read Fune 3. 1805.] a Ancient Revolutions of the Mineral Kingdom.—Vain attempts to explain them.— Dependence of Geology on Chemiftry.—Importance of the Carbo- nate of Lime.—Dr Biacx’s difcovery of Garbonic Acid, fubverted the former theories depending on Fire, but gave birth to that of Dr Hur- ron.—Progrefs of the Author’s Ideas with regard to that Theory. —Enxperiments with Heat and Compreffion, fuggefted to Dr Hurron in 1790.—Undertaken by the Author in 1798.—Speculations on which bis hopes of fuccefs were founded. HOEVER has attended: to the ftruCture of Rocks and - Mountains, muft be convinced, that our Globe has, not always exifted in its prefent flate ; but that every part of its mafs, fo far at leaft as our obfervations reach, has been agitAias - and fubverted by. the moft violent revolutions. Facts leading to fuch ftriking conclufions, however i imper= fectly obferved, could hot fail to, awaken -curiofity, and give rife to a defire. of tracing the hiftory, and) of inveftigating the caufes, of fuch ftupendous events ; and various attempts were made in this ways but with little fuccels ; 3 for while difcoveries . of "2 EFFECTS of HEAT of the utmoft importance and accuracy were made in Aftrono- my and Natural Philofophy, the fyftems produced by the Geo- logifts were fo fanciful and puerile, as fcarcely to deferve a | ferious refutation. ) ONE principal caufe of this failure, feems to have lain in the very imperfect flate of Chemiftry, which has only of late years begun to deferve the name ofa fcience. While Chemiftry was in its infancy, it was impoflible that Geology fhould make any progrefs ; fince feveral of the moft important circumftances to be accounted for by this latter fcience, are admitted on all hands to depend upon principles of the former. The confoli- dation of loofe fand into ftrata of folid rock; the cryftalline arrangement of fubftances accompanying thofe ftrata, and blended with them in various modes, are circumftances of a chemical nature, which all thofe who have attempted to frame theories of the earth have endeavoured hy" chemical reafon- ings to reconcile to their hypothefes. Fire and Warer, the only agents in nature by which ftony fubftances are produced, under our obfervation, were employ- ed by contending fects of geologifts, to explain all the phe- nomena of the mineral kingdom. - But the known properties of ‘Water, are quite repugnant to the belief of its univerfal influence, fince a very great propor- tion of the fubftances under confideration are infoluble, or near- ly fo, in that fluid; and fince, if they were all extremely fo- luble, the quantity of water which is known to exift, or that could poffibly exift in our planet, would be far too {mall to ac- complith the office affigned to it inthe Neptunian theory *. On the other hand, the known properties of Fire are no lefs inade- quate to the purpofe; for, various fubftances which frequently occur in the mineral kingdom, feem, by their prefence, to pre- clude * Whiftrations of the Huttonian Theory, by Mx Profeflor PLAYFAIR, 430- MODIFIED ty COMPRESSION. 73 clude its fuppofed agency ; fince experiment fhews, that, in our fires, they are totally changed or deftroyed. Unpe_r fuch circumftances, the advocates of either element were enabled, very fuccefsfully, to refute the opinions of their adverfaries, though they. could but feebly defend their own: and, owing perhaps to this mutual power of attack, and for want of any alternative to which the opinions of men could lean, both fyftems maintained a certain degree of credit; and writers on geology indulged themfelves, with a fort of im- punity, in a ftyle of unphilofophical reafoning, which would not have been tolerated in other fciences. Or all mineral fubftances, the Carbonate of Lime is unque- ftionably the moft important in a general view. As limeftone or marble, it conftitutes a very confiderable part of the folid mafs of many countries ; and, in the form of veins and no- dules of {fpar, pervades every fpecies of ftone. Its hiftory is | thus interwoven in fuch a manner with that of the mineral kingdom at large, that the fate of any geological theory muft very much depend upon its fuccefsful, application to the va- rious conditions of this fubftance. But, till Dr Back, by his difcovery of Carbonic Acid, explained the chemical nature of the carbonate, no rational theory could be formed, of the che- mical revolutions which it has undoubtedly undergone. _ Tuus difcovery was, in the firft inftance, hoftile to the fup- pofed action of fire ; for the decompofition of limeftone by fire in every common kiln being thus proved, it feemed abfurd to afcribe to that fame agent the formation of limeftone, or of any ma{s containing it. _ Tue contemplation of this difficulty led Dr Hutron to view the action of fire in a manner peculiar to himfelf, and thus to. form a geological theory, by which, in my opinion, he has fur- - nifhed the world with the true folution of one of the moft inte- MptieWlin F's ee K refting 74 . EFFECTS of HEAT refting problems that has ever engaged the attention of men of {cience. | He fuppofed, I. Tuat Heat has acted, at fome remote period, on all rocks. Il. Tuar during the action of heat, all thefe rocks (even fuch as now appear at the furface) lay covered by a fuperin- cumbent ma({s, of great weight and ftrength. III. Tuar in confequence of the combined action of Heat and Preflure, effects were produced different from thofe of heat on common occafions; in particular, that the carbonate of lime was reduced to a ftate of fufion, more or lefs complete, without any calcination. Tue effential and characteriftic principle of his theory is thus comprifed in the word Comprefion; and by one bold hypothefis, founded on this principle, he undertook to meet all the objec- tions to the action of fire, and to account for thofe’ circum- {tances in which minerals are found to differ from the ufual products of our furnaces. - Turs fyftem, however, involves fo many fuppofitions, appa- rently in contradiction to common experience, which meet us on the very threfhold, that moft men have hitherto been deterred from the inveftigation of its principles, and only a few indivi- ' duals have juftly appreciated its merits. It was long before I belonged to the latter clafs; for I muft own, that, on read- ing Dr Hurron’s firft geological publication, I was induced to reject his fyftem entirely, and fhould probably have continued ftill to do fo, with the great majority of the world, but for my habits of intimacy with the author ; the vivacity and perfpicui- ty of whofe converfation, formed a ftriking contraft to the ob- {curity MODIFIED ly COMPRESSION. 75 {curity of his writings. I was induced by that charm, and by _ the numerous original facts which his fyftem had led him to obferve, to liften to his arguments, in favour of opinions which I then looked upon as vifionary. I thus derived from his con- verfation, the fame advantage which the world has lately done from the publication of Mr PLayrair’s Lllufirations ; and, ex- perienced the fame influence which is now exerted by that work, on the minds of our moft eminent men of {cience. Arter three years of almoft daily warfare with Dr Hut- Ton, on the fubje& of his theory, I began to view his funda- mental principles with lefs and lefs repugnance. There is a period, I believe, in all fcientific inveftigations, when the con- jeCtures of genius ceafe to appear extravagant; and when we balance the fertility of a principle, in explaining the phe- nomena of nature, againft its improbability as an hypothefis : The partial view which we then obtain of truth, is perhaps the moft attractive of any, and moft powerfully ftimulates the exertions of an active mind. The mift which obfcured fome objects diffipates by degreee, and allows them to appear in their true colours; at the fame time, a diftant profpect opens to our view, of fcenes unfufpected before. ENTERING now ferioufly into the train of reafoning fol- lowed by Dr Hutron, I conceived that the chemical effects aferibed by him to compreflion, ought, in the firft place, to be inveftigated ; for, unlefs fome good reafon were given us for believing that heat would be modified by preffure, in the man- ner alleged, it would avail us little to know that they had acted together. He refted his belief of this influence on ana- logy; and on the fatisfactory folution of all the phenomena, furnifhed by this fuppofition. It occurred to me, however, that this principle was fufceptible of being eftablithed in a di- rect manner by experiment, and I urged. him to make the at- tempt; but he always rejected this propofal, on account of K 2 ) Pe 8 3 “6 ONES Eo) PR aTe the immenfity of the natural agents, whofe operations he fup- pofed to lie far beyond the reach of our imitation; and he feemed to imagine, that any fuch attempt muft undoubtedly fail, and thus throw difcredit on opinions already fufficiently eftablifhed, as he conceived, on other principles. I was far, however, from being convinced by thefe arguments ; for, with- out being able to prove that any artificial compreflion to which we could expofe the carbonate, would effectually prevent its calcination in our fires, I maintained, that we had as little proof of the contrary, and that the application of a moderate force might poffibly perform all that was hypothetically, af- fumed in the Huttonian Theory. On the other hand, I con- fidered myfelf as bound, in practice, to pay deference to his opinion, in a field which he had already fo nobly occupied, and abftained, during the remainder of his life, from the pro- fecution of fome experiments with compreflion, which I had: begun in 1790. In 1798, I refumed the fubject with eagernefs, being ftill of pinion, that the chemical law which forms the bafis of the Huttonian Theory, ought, in the firft place, to be inveftigated. experimentally ; all my fubfequent reflections and obferva-. tions having tended to confirm my idea of the importance of this purfuit, without in any degree rendermg me more ap-. prehenfive as to the refult. In the arrangement of the following paper, I fhall firft con- fine myfelf to the inveftigation of the chemical effects of Heat and Compreffion, referving to the concluding part, the appli- cation of my refults to Geology. I fhall, then, appeal to the volcanoes, and fhall endeavour to vindicate the laws of ac- tion afflumed in the Huttonian Theory, by fhewing, that lavas, previous to their eruptions, are fubject to fimilar laws; and that the volcanoes, by their fubterranean and fubmarine exer- tions, MODIFIED ly COMPRESSION. a9 tions, muft produce, in our times, refults fimilar to thofe afcri- bed, in that Theory, to the former action of fire. In comparing the Huttonian operations with thofejef,the volcanoes, I fhall avail myfelf of fome facts, brought to light in the courfe of the following inveftigations, by which a pre- cife limit is afligned. to the. intenfity of the heat, and ,to the force of compreilion, required to fulfil the conditions of Dr Hutron’s hypothefis : For, according to him, the power of thofe agents was very great, but quite indefinite ; it was there- fore impoflible to compare their fuppofed effects in any precife “manner with the phenomena of nature. . My attention was almoft exclufively confined to the Carbo- nate of Lime, about which I reafoned as follows: ‘The carbonic acid, when uncombined with any other fubftance, exifts natural- ly ina gafeous form, atthe common temperature of our atmo- {phere ; but when in union with lime, its volatility is repreffed,. in that fame temperature, by the chemical, force of the earthy. fubftance, which retains it ina folid, form. When the tem- perature is raifed, to a full red-heat, the acid acquires a yola- tility by which that force is overcome, it, efcapes from the lime, and aflumes its gafeous form. It is. evident, that were the attractive force of the lime increafed, or the volatility of the acid diminifhed by any means, the compound would be enabled. to bear a higher heat without decompofition, than it can in the prefent ftate of things. Now, preflure muft produce an effe@ of this kind ; for when,a mechanical force oppofes the expan- fion of the acid, its volatility muft, to a certain degree, be di- minifhed. Under. preffure, then, the carbonate may be expedt-- ed'to remain, unchanged in a heat, by which, in the open. air, it would have, been calcined. _ But experiment alone can teach us, What comprefling force is requifite to enable it to refift any given elevation of temperature ; and what is to be the refult of fuch an operation. . Some of the compounds of lime with acids. are 48 EFFECTS of HEAT are fufible, others refractory ; the carbonate, when conftrained by preflure to endure a proper heat, may be as fufible as the muriate. | One ‘circumftance, derived from the Huttonian. Theory, © induced me to hope, that the carbonate was eafily fufible, and indicated a precife point, under which that fufion ought to be expected. Nothing is more common than to meet with nodules of calcareous fpar inclofed in whinftone ; and we fup- pofe, according to the Huttonian Theory, that the whin and the fpar had been liquid together; the two fluids keeping fepa- rate, like oil and water. It is natural, at the junction of thefe two, to look for indications of their relative fufibilities ; and we find, accordingly, that the termination of the {par is generally elobular and {mooth; which feems to prove, that, when the whin became folid, the {par was ftill in a liquid ftate ; for had the fpar congealed firft, the tendency which it fhews, on all oc- cafions of freedom, to fhoot out into prominent cryftals, would have made it dart into the liquid whin, according to the pecu- liar forms of its cryftallization; as has happened with the various fubftances contained in whin, much more refractory than it- felf, namely, augite, felfpar, &c.; all of which having con- gealed in the liquid whin, have affumed their peculiar forms with perfect regularity. From this I concluded, that when the whin congealed, which muft have happened about 28° or 30° of Wepcwoop, the {par was ftill liquid. I therefore expected, if I could compel the carbonate to bear a heat of 28° without decompofition, that it would enter inte fufion. The fequel will fhew, that this conje€ture was not without foundation. I sHALL now enter upon the defcription of thofe experiments, the refult of which I had the honour to lay before this Society. on the 30th of Auguft laft (1804) 3 fully aware how difficult it is, in giving an account of above five hundred experiments, all tend- ing to one point, but differing much from each other in vari- ous Da MODIFIED by COMPRESSION. 79 ous particulars, to fteer between the oppofite faults of prolixity and barrennefs. My object fhall be to defcribe, as fhortly as poffible, all the methods followed, fo as to enable any chemift to repeat the experiments ; and to dwell particularly on fuch circuraftances only, as feem to lead to conclufions of gale ance. Tue refult being already known, I confider the account I am about to give of the execution of thefe experiments, as addref- fed to thofe who take a particular intereft in the progrefs of chemical operations: in the eyes of fuch gentlemen, I truft, that none of the details into which I muft enter, will appear fu- 4 2 as Il. Principle of execution upon which the following Experiments were con- dutted.— Experiments with Gun-Barrels filled with baked Clay, and welded at the muzzle. —Method with the Fufible Metal.—Remarkable ' effects of its expanfi ion. aig) cab ty of introducing Air.—Refults ob- tained. Wuen I firft undertook to make experiments with heat acting under compreflion, I employed myfelf in contriving various devices of fcrews, of bolts, and of lids, fo adjuft- ed, I hoped, as to confine all elaftic fubftances; and per- haps. fome of them might have anfwered. But I laid afide all fuch devices, in favour of one which occurred to me in January 1798; which, by its fimplicity, was of eafy appli- cation in all cafes, and accomplifhed all that could be done by any device, fince it fecured perfect ftrength and tightnefs to the utmoft that the veffels employed could bear, whether form- ¢d of metallic or earthy fubftance. The device depends upon ; the 86 EFFECTS of HEAT the following general view: If, we-take a hollow tube or bar- rel (AD, fig. 1.) clofed at one end, and open at the other, of one foot or more in length; it is evident, that by introducing - one end into a furnace, we can. apply to it as great heat as art can produce, while the other end is kept cool, or, if necef- fary, expofed to extreme cold. If, then, the fubftance which we mean to fubject to the combined action of heat and pref- fure, be introduced into the’breech or clofed end of the barrel (CD), and if the middle part be filled with fome refractory fubftance, leaving a fmall empty {pace at the muzzle (AB), we can apply heat to the muzzle, while the breech containing the fubject of experiment, is kept cool, and thus clofe the barrel by - any of the numerous modes which heat affords, from the weld- ing of iron to the melting of fealing-wax. Things being then reverfed, and the breech put into the furnace, a heat of any required intenfity may be applied to the fubject of experiment, now in a ftate of conftraint. My firft application of this fcheme was carried on with a common gun-barrel, cut off at the touch-hole, and welded very ftrongly at the breech by means of a plug of iron. Into it I introduced the carbonate, previoufly rammed into a cartridge of paper or pafteboard, in order to protect it from the iron, by which, in fome former trials, the fubje@t of experiment . had been contaminated throughout during the action of heat. I then rammed the reft of the barrel full of pounded clay, previouf- ly baked in a ftrong heat, and I had the muzzle clofed like the breech, by a plug of iron welded upon it in a common forge; the reft of the barrel being kept cold during this operation, by means of wet cloths. The breech of the barrel was then introduced horizontally into a common muffle, heated to about 25° of WepGwoop. ‘To the muzzle a rope was fixed, in fuch a manner, that the barrel could be withdrawn with- out MODIFIED ly COMPRESSION. Sr out danger from an explofion*. I likewife, about this time, clofed the muzzle of the barrel, by means of a plug, fixed by folder only ; which method had this peculiar advantage, that I could fhut and open the barrel, without having recourfe to a workman. In thefe trials, though many barrels yielded to the expanfive force, others refifted it, and afforded fome re- fults that were in the higheft degree encouraging, and even fatisfactory, could they have been obtained with certainty on - repetition of the procefs. In many of them, chalk, or com- mon limeftone previoufly pulverifed, was agglutinated into a ftony mafs, which required a {mart blow of a hammer to break it, and felt under the knife like a common limeftone ; at the fame time, the fubftance, when thrown into nitric acid, diffolved entirely with violent effervefcence. In one of thefe experiments, owing to the action of heat on the cartridge of paper, the baked clay, which had been ufed to fill the barrel, was ftained black throughout, to the diftance of two-thirds of the length of the barrel from its breech. This circumftance is of importance, by fthewing, that though all is tight at the muzzle, a protrufion may take place along the barrel, greatly to the detriment of com- plete * On one occafion, the importance of this precaution was ftrongly felt. Having inadvertently introduced a confiderable quantity of moifture into a welded barrel, an explofion took place, before the heat had rifen to rednefs, by which, part of the barrel was {pread out to a flat plate, and the furnace was blown to pieces. Dr KENNEDY, who happened to be prefent on this occafion, obferved, that notwith- ftanding this accident, the time might come when we fhould employ water in thefe experiments to affift the force of compreffion, I have fince made great ufe of this valuable fuggeftion: but he fcarcely lived, alas! to fee its application ; fer my firft fuccefs in this way, took place during his laft illnefs.—I have been expofed to no rifk in any other experiment with iron barrels; matters being fo arranged, that the {train againft them has only commenced in a red heat, in which the metal has been fo far foftened, as to yield by laceration like a piece of leather. Vot. VI.—P. I. L, 82 EFFECTS of HEAT plete compreffion: and, at the fame time, it illuftrates what has happened occafionally in nature, where the bituminous matter feems to have been driven by fuperior local heat, from one part of a coaly bed, though retained in others, under the fame compreflion. The bitumen fo driven off being found, in other cafes, to pervade and tinge beds of flate and of fandftone. I was employed in this purfuit in {pring 1800, when an event of importance interrupted my experiments for about a year. But I refumed them in March 1801, with many new plans of execution, and with confiderable addition to my 23 paratus. In the courfe of my firft trials, the following mode of execu- tion had occurred to me, which I now began to put in practice. It is well known to chemitts, that a certain compofition of differ- ent metals *, produces a fubftance fo fufible, as to melt in the heat of boilimg-water. I conceived that great advantage, both in point of accuracy and difpatch, might be gained in thefe ex- periments, by fubftituting this metal for the baked clay above mentioned: That after introducing the carbonate into the breech of the barrel, the fufible metal, in a liquid ftate, might be poured in, fo as to fill the barrel to its brim: That when the metal had cooled and become folid, the breech might, as before, be introduced into a muffle, and expofed to any required heat, while the muzzle was carefully kept cold. In this manner, no part of the fufible metal being melted, but what lay at the breech, the reft, continuing in a folid ftate, would effectually confine the carbonic acid : That after the ac- tion of ftrong heat had ceafed, and after all had been allowed to cool completely, the fufible metal might be removed entire- ly from the barrel, by means of a heat little above that of boil- ing water, and far too low to occafion any decompofition of the - * Eight parts of bifmuth, five of lead, and three of tin. MODIFIED ty COMPRESSION. 85 the carbonate by calcination, though acting upon it in free- dom ; and then, that the fubject of experiment might, as be> fore, be taken out of the barrel. Tuts {cheme, with various modifications and additions, “which practice has fuggefted, forms the bafis of moft of the following methods. In the firft trial, a ftriking phenomenon occurred, which gave rife to the moft important of thefe modifications. Ha- ’ ying filled a gun-barrel with the fufible metal, without any carbonate ; and having placed the breech in a muffle, I was furprifed to fee, as the heat approached to rednefs, the liquid metal exuding through the iron in innumerable minute drops, difperfed all round the barrel. As the heat advanced, this exudation increafed, till at laft the metal flowed out in continued ftreams, and the barrel was quite deftroyed. On feveral occafions of the fame kind, the fufible metal, being forced through fome very minute aperture in the barrel, {pouted from it to the diftance of feveral yards, depofiting upon any fubftance oppofed to the ftream, a beautiful af- ’ femblage of fine wire, exactly in the form of wool.’ I imme- diately underftood, that the phenomenon was produced by the fuperior expanfion of the liquid over the folid metal, in con- fequence, of which, the fufible metal was driven through the iron as water was driven through filver * by mechanical per- cuffion in the Florentine experiment. It occurred to me, that this might be prevented by confining along with the fufible metal a fmall quantity of air, which, by yielding a little to the expanfion of the liquid, would fave the barrel. This re- La medy * Effays of Natural Experiments made in the Academie del Cimento, tranfla- ted by Wa ter, London, 1684, page 117. The fame in MusscHENBROEK’s Las. tin tranflation, Lugd. Bat. 1731, p. 63. 84 -.. EFFECTS of HEA medy was found to anfwer completely, and was applied, in all the experiments made at this time *. I now propofed, in order to keep the carbonate clean, ta inclofe it in a fmall veffel ; and to obviate the difficulty of removing the refult at the conclufion of the experiment, I further propofed to connect that veflel with an iron ramrod, longer than the barrel, by which it could be introduced or withdrawn at pleafure. A sma. tube of glafs +, or of Raumur’s porcelain, about a quarter of an inch in diameter, and one or two inches in length, (fig. 2. A) was half filled with pounded carbonate of lime, rammed as hard as poflible; the other half of the tube. being * I found it a matter of much difficulty to afcertain the proper quantity of air which ought to be thus inclofed. When the quantity was too great, the refult was injured by diminution of elafticity,as I fhall have occafion fully to fhew here- after. When too {mall, or when, by any accident, the whole of this included air was allowed to efcape, the barrel was deftroyed. I hoped to afcertain the bulk of air neceflary to give liberty to the ex- panfion of the liquid metal, by meafuring the aétual quantity expelled by known heats from an open barrel filled with it. But I was furprifed to find, that the quantity thus difcharged, exceeded in bulk that of the air which, in the fame heats, I had confined along with the carbonate and fufible metal in many fuccefsful experiments. As the expanfion of the liquid does not feem ca- pable of fenfible diminution by an oppofing force, this fact can only be accounted for by a diftention of the barrel. In thefe experiments, then, the expanfive force of the carbonic acid, of the included air, and of the fufible metal, aéted in combi- nation againft the barrel, and were yielded to in part by the diftention of the bar- rel, and by the condenfation of the included air. My objeét was to increafe the force of this mutual action, by diminifhing the quantity of air, and by other de- vices to be mentioned hereafter. Where fo many forces were concerned, the laws of whofe variations were unknown, much precifion could not be expected, nor is it wonderful, that in attempting to carry the compreffing force to the ut- moft, I fhould have deftroyed barrels innumerable. + I have fince conftantly ufed tubes of common porcelain, poacsae glafs much too fufible for this purpofe. MODIFIED ly COMPRESSION. 85: being filled with pounded filex, or with whatever occurred as moft likely to prevent the intrufion of the fufible metal in its liquid and penetrating ftate. This tube fo filled, was: placed in a frame or cradle of iron (df 4, figs. 3, 4, 5, and 6,) fixed to the end (m) of a ram-rod (m n). The cradle was from fix to three inches in length, and: as much in diame- ter as a gun-barrel would admit with eafe. It was compofed of two circular plates of iron, (def g and. 42k /, feen edge- wife in the figures,) placed at right-angles to the ram-rod, one of thefe plates (de fb) being fixed to it by the centre (m). Thefe plates were connected together by four ribs or flattened _ wires of iron (dh, ¢ 7, f #, and.g /,) which formed the cradle into which the tube (A), containing the carbonate, was intro- duced by thrufting the adjacent ribs afunder. Along with the tube juft mentioned, was introduced another: tube (B), of iron or porcelain, filled only: with air. Likewife, in the cradle, a: pyrometer-* piece (C) was placed in contact with (A) the tube containing the carbonate. Thefe articles generally occupied: | . the »\* Tuer pyrometer-pieces ufed inthefe experiments were made under my own. eye. Neceflity compelled me to undertake: this laborious and difficult work, in. which Ihave already fo far fucceeded as to obtain a fet of pieces, which, though far from complete, anf{wer my purpofe tolerably well. I had lately an oppor- tunity of comparing my. fet-with that of Mr WepGwoob, at: various tempera- ‘tures, in-furnaces of great fize and fteadinefs. The refult has proved, that my- pieces agree as well with.each other as.his; though with my fet each tempera- ture is indicated by a different degree of the fcale. J have thus been enabled to conftru& a table, by which my obfervations have been corredted, fo that the temperatures mentioned inthis paper are fuch as would have. been indicated by Mr Wepcwoon’s pieces. By Mr WepDGwoon’s pieces, I mean thofe of the. only fet. which has been fold to the public, and by:which the melting heat of pure filver is indicated at the 22d degree. I am well aware, that the late Mr WenvewooD, in his Table of Fufibilities, has ftated that fufion as taking place at the 28th degree ; but I am convinced that his obfervations muft-have been mede. with fome fet different from that which was afterwards fold, 86 - EFFECTS of HEAT the whole cradle ; when any fpace remained, it was filled up by a piece of chalk dreffed for the purpofe. (Fig. 4. reprefents the cradle filled, as juft defcribed).. TuinGs being thus prepared, the gun-barrel, placed erect. with its muzzle upwards, was half filled with the liquid fufible metal. The cradle was then introduced into the barrel, and plunged to the bottom of the liquid, fo that the carbonate was placed very near the breech, (as reprefented in fig. 5, the fu- fible metal ftanding at 0). The air-tube (B) being placed fo as to enter the’ liquid with its muzzle downwards, retained great part of the air it originally contained, though fome of it might be driven off by the heat, fo as to efcape through the liquid. ‘The metal being now allowed to cool, and to fix round the cradle and ramrod, the air remaining in the air-tube was effectually confined, and all was held faft. The barrel being then filled to the brim with fufible metal, the apparatus was ready for the application of heat to the breech, (as fhewn in fig. 6.) | . In the experiments made at this time, I ufed a fquare brick furnace (figs. 7 and 8), having a muffle (r s) traverfing it ho- rizontally and open at both ends. This muffle being fupported in the middle by a very flender prop, was expofed to fire from below, as well as allround. The barrel was placed in the muffle, with its breech in the hotteft part, and the end next the muzzle projedting beyond the furnace, and furrounded with cloths which were drenched with water from time to time. (This arrangement is fhewn in fig. 7). In this fituation, the fufible. metal furrounding the cradle being melted, the air contained in the air-tube would of courfe feek the highetft po- fition, and its firft place in the air-tube would be occupied by fufible metal. (In fig. 6., the new pofition of the air is fhewn at p-q). ; AT MODIFIED by COMPRESSION.. 87 Arthe ect Aubin of the experiment, the metal was generally removed by placing the barrel in the tranfverfe muffle, with its muzzle pointing a little downwards, and fo that the heat was applied firft’ to the muzzle, and then to the reft of the barrel in fucceflion. (This operation is fhewn in fig.8). In fome of the firft of thefe experiments, I loofened the cradle, by plunging the barrel into heated brine, or a ftrong folution of muriate of lime; which laft bears a temperature of 250° of Fanrenueit before it boils. For this purpofe, I ufed a pan three inches in diameter, and three feet deep, having a flat bafon at top to receive the liquid when it boiled over. The method anfwered, but was troublefome, and I laid it afide. I have had occafion, lately, however, to refume it in fome experiments in which it was of confequence to open the barrel with the leaft. poflible heat *. _ By thefe methods I made a great number of experiments, with refults that were highly interefting in that ftage of the bufi- nefs, though their importance if fo much diminifhed by the fubfequent progrefs of the inveftigation, that I think it proper to mention but very few of them. / Ow the 31ft of March 1801, I rammed forty grains of pound- ed chalk into a tube of green bottle-glafs, and placed it in the cradle as above defcribed.. A pyrometer in the muffle along with the barrel indicated 33°. The barrel was expofed to heat during feventeen or eighteen minutes. On withdrawing the cradle, the carbonate was found in one folid mafs, which had Nah fhrunk i in bulk, the fpace thus left within the tube being accurately * In many of the following experiments, lead was ufed in place of the fafible metal, and often with fuccefs ; but I loft many good refults in this way: for the heat required to liquefy the lead, approaches fo near to rednefs, that it is difficult to difengage the cradle without applying a temperature by which the carbonate is injured. I have found it anfwer well, to furround the cradle and a few inches of the rod, with fufible metal, and to fill the reft of the barrel with lead. 88 EFFECTS of HEAT accurately filled with metal, which plated the carbonate all over without penetrating it in the leaft, fo that the metal was eafily removed. The weight was reduced from forty to thirty- fix grains. The fubftance was very hard, and refifted the knife better than any refult of the kind previoufly obtained ; its frac- ture was cryftalline, bearing a refemblance to white faline marble; and its thin edges had a decided femitranfparency, a circumftance firft obferved in this refult. On the 3d of March of the fame year, I made a fimilar experiment, in which a pyrometer-piece was placed with- in the barrel, and another in the muffle; they agreed in indi- cating 23°. The inner tube, which was of Reaumur’s porce: lain, contained eighty grains of pounded chalk. The carbo- nate was found, after the experiment, to have loft 34 grains. A thin rim, lefs than the 20th of an inch in thicknefs, of whitifh matter, appeared on the outfide of the mafs. In other refpects, the carbonate was in a very perfect ftate ; it was of a yellowifh colour, and had a decided femitranfparency and -faline fracture. But what renders this refult of the greateft value, is, that on breaking the mafs, a fpace of more than the tenth of an inch fquare, was found to be completely cryftal- lized, having acquired the rhomboidal fracture of calcareous fpar. It was white and opaque, and prefented to the view three fets of parallel plates which are feen under three different angles. This fubftance, owing to partial calcination and fub- fequent abforption of moifture, had loft all appearance of its remarkable properties in fome weeks after its production ; but this appearance has fince been reftored, by a freth fradure, and the fpecimen is now well a by being hermetically inclofed. III. MODIFIED ly COMPRESSION. 89 + ABT. Experiments made in Tubes of Porcelain.—Tubes of Wedgwood’s Ware. —WMethods ufed to confine the Carbonic Acid, and to clofe the Pores of the Porcelain in a Horizontal Apparatus.—Tubes made with a view to thefe Experiments.—The Vertical Apparatus adopted.—View of Refults obtained, both in Iron and Porcelain.—The Formation of Lime- ftone and Marble.—Inquiry into the Caufe of the partial Calcinations. —Tubes of Porcelain weighed previous to breaking.——Experiments with Porcelain Tubes proved to be limited. Waite I was carrying on the above-mentioned experiments; I was occafionally occupied with another fet, in tubes of por- celain. So much, indeed, was I prepoflefled in favour of this laft mode, that I laid gun-barrels afide, and adhered to it du- ring more than a year. The methods followed with this fub- ftance, differ widely from thofe already defcribed, though founded on the fame general principles. I procurep from Mr WeEpcwoop’s manufactory at Etruria, in Staffordthire, a fet of tubes for this purpofe, formed of the ‘fame fubftance with the white mortars, in common ufe, made there. Thefe tubes were fourteen inches long, with a bore of half an inch diameter, and thicknefs of 0.2; being clofed at one end (ies: FO Et Eo, Tae) I proposep to ram the carbonate of lime into the breech (Fig. 9. A); then filling the tube to within a fmall diftance of its muzzle with pounded flint (B), to fill that remainder (C) with common borax of the fhops (borat of foda) previoufly res duced to glafs, and then pounded ; to apply heat to the muzzle alone, fo as to convert that borax into folid glafs ; then, re- verfing the operation, to keep the muzzle cold, and apply the requifite heat to the cgrbonate lodged in the breech. ‘Voz. VI.—P.I. M tae 90 EFFECTS of HEAT I THus expected to confine the carbonic acid; but the at- tempt was attended with confiderable difficulty, and has led to the employment of various devices, which I fhall now fhort- ly enumerate, as they occurred in the courfe of practice. The fimple application of the principle was found infufficient, from two caufes: Firft, The carbonic acid being driven from the breech of the tube, towards the muzzle, among the pores of the pounded filex, efcaped from the compreffing force, by lodging itfelf in cavities which were comparatively cold: Secondly, The glafs of borax, on cooling, was always found to crack very much, fo that its tightnefs could not be depended on. To obviate both thefe inconveniences at once, it occurred to me, in addition to the firft arrangement, to. place fome borax (fig. 10. C) fo near the breech of the tube, as to, undergo heat along with the carbonate (A); but interpofing between this borax and the carbonate, a ftratum of filex (B), in order to prevent contamination. I trufted that the borax in a liquid or vifcid ftate, being thruft outwards by the expanfion of the carbonic acid, would prefs againft the filex beyond it (D), and totally prevent the elaftic fubftances from efcaping out of the tube, or even from wandering into its cold parts. In fome refpects, this plan anfwered to expectation. The glafs of borax, which can never be obtained when cold, with- out innumerable cracks, unites into’ one continued vifcid mafs in the loweft red-heat ; and as the ftrefs in thefe experiments, begins only with rednefs, the borax being heated at the fame time with the carbonate, becomes united and impervious, as foon as its action is neceflary. Many good refults were accord- ingly obtained in this way. But I found, in practice, that as the heat rofe, the borax began to enteér into too thin fufion, and was often loft among the pores of the filex, the fpace in which it had lain being found empty on breaking the tube. It was therefore. MODIFIED by COMPRESSION. or therefore found neceflary to oppofe fomething more fubftan- tial and compaét, to the thin and penetrating quality of pure borax. In fearching for fome fuch fubftance, a curious property . bottle-glafs occurred accidentally. Some of this glafs, in powder, having been introduced into a muffle at the tempe- rature of about 20° of WEepDGwoop ; the powder, in the {pace of about a minute, entered into a ftate of vifcid agglutination, like that of honey, and in about a minute more, (the heat al.’ ways continuing unchanged,) confolidated into a firm and com- pact mafs of Reawmur’s porcelain*. It now appeared, that by placing this fubftance immediately behind the borax, the penetrating quality of this laft might be effectually reftrained ; for, Reaumur’s porcelain has the double advantage of being refractory. and of not cracking by.change of temperature. I found, however, that in the act of confolidation, the pounded bottle-glafs fhrunk, fo as to leave an opening between its mafs and the tube, through which the borax, and, along with it, the carbonic acid, was found to efcape, But the object in view was obtained by means of a mixture of pounded bottle-glafs, and pounded flint, in equal parts. This compound ftill agglutinates, not indeed into a mafs fo hard as Reaumur’s porcelain, but fuf- ficiently fo for the purpofe ; and this being done without any fenfible contraction, an effectual barrier was oppofed to the bo- rax ; (this arrangement is fhewn in fig. 11.) ; and thus the me- thod of clofing the tubes was rendered fo complete, as feldom to fail in practice +. A ftill further refinement upon this me- M 2 thod * In the fame temperature, a mafs of the glafs of equal bulk would undergo the fame change ; but it would occupy an hour. t A fubftance equally efficacious in reftraining the penetrating quality of borax, ’ was difcovered by another accident. It confifts.of a mixture of borax and com- mon fand, by which a fubftance is formed, which, in heat, afflumes-the ftate of a very tough pafte, and becomes hard and compaét on cooling. 92 EFFECTS of HEAT thod was found to be of advantage. A fecond feries of powders, like that already defcribed, was introduced towards the muzzle, (as fhewn in fig. 12.). During the firft period of the experi- ment, this laft-mentioned feries-was expofed to heat, with all the outward half af the tube (24); by this means, a folid mafs was produced, which remained cold and firm during the fubfequent action of heat upon the carbonate. I soon found, that notwithftanding all the above-mention~ ed precautions, the carbonic acid made its efcape, and that it pervaded the fubftance of the Wedgwood tubes, where no flaw could be traced. It occurred to me, that this defe& might be. remedied, were borax, in its thin and penetrating ftate of fu- fion, applied to the infide of the tube ; and that the pores of the porcelain might thus be clofed, as thofe of leather are clofed by oil, in an air-pump. In this view, I rammed the carbonate into a fmall tube, and furrounded it with: pounded glafs of borax, which, as foon as the heat was applied, {pread on the infide of the large tube, and' effectually clofed its pores. In this man- ner, many good experiments, were made with barrels lying ho- rizontally in common muffles, (the arrangement jutt hehe. : being reprefented in fig. 13.) I was thus enabled to carry. on experiments with this porcelain, to the utmoft that its ftrength would bear. But I was not fatisfied with the force fo exerted; and, hoping to obtain tubes of a fuperior quality, I fpent much time in expe-_ riments with various porcelain compofitions. In this, I fo: far fucceeded, as to produce tubes by which the carbonic acid: _was in a great meafure retained without any internal glaze. The beft material I found for this purpofe, was the pure por- celain-clay of Cornwall, or a compofition in the proportion of two of this clay to one of what the potters call Corni/b-/tone, which I believe to be a granite in a ftate of decompofition. Thefe tubes were feven or eight inches long, with a bore. tapering ‘ ' See MODIFIED by COMPRESSION. 93 tapering from 1 inch to 0.6... Their thicknefs was about 0.3 at the breech, and tapered towards the muzzle to the thinnefs of a wafer. _I now adopted a new mode of operation, idacing the. tube: yenticilly? and: not horizontally, as before. By obferving the thin-ftate of borax whilft. in fufion, I was convinced, that it ought to be treated as a complete liquid, which being fupport- ed in the courfe of the experiment from below, would fecure perfect tightnefs, and obviate the failure which often happen- ed in the horizontal ppbeon, from the falling of the borax to the lower fide. In this view, (fig. 16. ); I filled the breech in the manner defcribed above, and introduced into the muzzle fome: bo- rax (C) fupported at the middle of the tube by a quantity. of filex mixed with bottle-glafs (B). I placed the tube, fo prepared, with its breech plunged into a crucible filled with fand (E), and. its muzzle: pointing upwards. It was now my objea to apply heat to the muzzle-half, whilft the other re- mained cold. In that view, I conftructed a furnace (fig. 14. and. 1s.),° ‘having a muflle placed vertically (cd), furround- ed on all’ fides with fire-(e¢), and open both above (at c), and ‘below (at'd@). » The crucible juft mentioned, with its. tube, being then placed on a fupport dire@tly below the ver-. tical muffle, (as reprefented in fig. 14. at F), it was raifed, fo that the half of the tube next the muzzle was introduced into, the fire. In confequence of this, the borax was feen from above to melt, and run down in the tube, the air contained in. the powder efcaping in the form of bubbles, till at laft the. borax ftood with a clear and fteady furface like that of wa-. ter. Some of this falt being thrown in from above, by means of. a tube of glafs, the liquid furface was raifed nearly to the muz- _ ale, and, after all had been allowed to become cold, the po-. fition of the tube was reverfed; the muzzle. being now plun-. ced. 94 EFFECTS of HEAT ged into the fand, (as in fig.17.), and the breech introduced into the muffle. In feveral experiments, I found it anfwer well, to occupy great part of the fpace next the muzzle, with a rod of {and and clay previoufly baked, (fig. 19. KK), which was either introduced at firft, along with the pounded borax, or, being made red hot, was plunged into it when in a liquid ftate. In many cafes I aflifted the compactnefs of the tube by means of an internal glaze of borax; the carbonate being pla- ced in a {mall tube, (as fhewn in fig. 18.) TueEsk devices anfwered the end propofed: Three-fourths of the tube next the muzzle was found completely filled with a mafs, having a concave termination at both ends, (f and g figs. 17. 18. 19.), fhewing that it had ftood as a liquid in the two oppofite pofitions in which heat had been applied to it. So great a degree of tightnefs indeed was obtained in this way, that I found myfelf fubjected to an unforefeen fource of failure. A number of the tubes failed, not by explofion, but . by the formation of a minute longitudinal fiffure at the breech, through which the borax and carbonic acid efcaped. I faw that this arofe from the expanfion of the borax when in a li- quid ftate, as happened with the fufible metal in the experi- ments with iron-barrels ; for, the crevice here formed, indi- cated the exertion of fome force acting very powerfully, and to a very fmall diftance. Accordingly, this fource of failure was remedied by the introduction of a very fmall air-tube. This, however, was ufed only in a few experiments. In the courfe of the years 1801, 1802, and 1803, I made a . number of experiments, by the various methods above defcrib- ed, amounting, together with thofe made in gun-barrels, to one hundred and fifty-fix. In an operation fo new, and in which the apparatus was ftrained to the utmoft of its power, conftant fuccefs could not be expected, and in fact many expe- riments failed, wholly or partially. The refults, however, upon the MODIFIED ty COMPRESSION. 95 the whole, were fatisfactory, fince they feemed to eftablith fome of the effential points of this inquiry. _ THEsE experiments prove, that, by mechanical conftraint, the carbonate of lime can be made to undergo ftrong heat, without-calcination, and to retain almoft the whole of its car- bonic acid, which, in an open fire, at the fame temperature, would have been entirely driven off: and that, in thefe circum- ftances, heat produces fome of the identical effects popes to it in the Huttonian Theory. By this joint action of heat and preflure, the carbonate of lime which had been introduced in the ftate of the fineft powder, is agglutinated into a firm mafs, poffefling a degree of | hardnefs, compactnefs, and fpecific gravity *, nearly approach- ing to thefe qualities in a found limeftone; and fome of the refults, by their faline fracture, by their femitranf{parency, and their fufceptibility of polifh, deferve the name of marble. Tue fame trials have been made with all calcareous fub- ftances ; with chalk, common limeftone, marble, fpar, and _ the hells of fifh. All have fhewn the fame general property, with fome varieties as to temperature. Thus, I found, that, in the fame circumftances, chalk was more fufceptible of .ag- glutination than {par ; the latter requiring a heat two degrees higher than the former, to pring it to. the fame pitch of ag-- glutination. Tue chalk ufed in my firft experiments , always afflumed the. ; character of a yellow marble, owing probably to fome flight contamination of iron. When a folid piece of chalk, whofe bulk had been previoufly meafured in the gage of Wedgwood’s. pyrometer was fubmitted to heat under compreffion, its con- traction was remarkable, proving the approach of the particles during their confolidation ; on thefe occafions, it was found. t@ * See Appendix. 96 EFFECTS of HEAT to fhrink three times more than the pyrometer-pieces in the fame temperature. It loft, too, almoft entirely, its power of im- bibing water, and acquired a great additional fpecific gravity. ‘On feveral occafions, I obferved, that mafles of chalk, which, before the experiment, had fhewn one uniform character of whitenefs, aflumed a ftratified appearance, indicated by a feries of parallel layers of a brown colour. This circumftance may hereafter throw light on the pealinon hiftory of this extraor- dinary fubftance. I uave faid, that, by dhealdeaital conftraint, almoft the whole -of the carbonic acid was retained. And, in truth, at — this period, fome lofs of weight had been experienced in all the experiments, both with iron and porcelain. But even this circumftance is valuable, by exhibiting the sires ase of the carbonic acid, as varied by its quantity. WHEN the lofs exceeded 10 or 15 per cent *. of the weight of the carbonate, the refult was always of a friable texture, and without any ftony character; when lefs than 2 or 3 per cent. it was confidered as good, and pofleffed the properties of a natural carbonate. In the intermediate cafes, when the lofs amounted, for inftance, to 6 or 8 per cent., the re- fult was fometimes excellent at firft, the fubftance bearing every appearance of foundnefs, and often poffefling a high cha- racter of cryftallization ; but it was unable to refift the ac+ tion of the air; and, by attracting carbonic acid or moifture, or both, crumbled to duft more or lefs rapidly, according to circumftances.. This feems to prove, that the carbonate of. lime, though not fully faturated with carbonic acid, may pof- fefs the properties of limeftone ; and perhaps a difference of this * T have found, that, in open fire, the entire lofs fuftained by the carbonate va- ries in different kinds from 42 to 45.5 per cent. MODIFIED by COMPRESSION, 97 this kind may exift among natural carbonates, give rife to their different degrees of durability. I nave obferved, in many cafes, that the calcination has ‘reached only to a certain depth imto the mafs ; the internal part remaining ima ftate of complete carbonate, and, in ge- neral, of a very fine quality. The partial calcination feems thus to take place in two different modes. By one, a fmall proportion of carbonic acid is taken from each particle of carbonate ; by the other, a portion of the carbonate is quite calcined, while the reft is left entire. Perhaps one refult is the effect of a feeble calcining caufe, acting during a long time, and the other of a ftrong caufe, acting for a fhort time. Some of the refults which feemed the moft perfe& when firft produced, have been {ubject to decay, owing to partial calcination. It happened, in fome degree, to the beautiful fpecimen produced on the 3d of March 1801, though a freth fracture has reftored it. A sPECIMEN, too, of marble, formed from pounded {par, on 15th May 1801, was fo complete as to deceive the workman | employed to polifh it, who declared, that, were the fubftance a little whiter, the quarry from which it was taken would’ be of great value, if it lay within reach of a market. Yet, ina few weeks after its formation, it fell to duft. NumBERLEss {pecimens, however, have been obtained, which refift the air, and retain their polifh as well as any marble. Some of them continue in a perfect ftate, though they have been kept without any precaution during four or five years. That fet, in particular, remain perfectly entire, which were fhewn laft year in this Society, though fome of them were made in 1799, fome in 1801 and 1802, and though the firft eleven were long foaked in water, in the trials made of their fpecific gravity. Vor, VI.—P.- 1, N : : A 98 | EFFECTS of HEAT A curRtous circumftance occurred m one of thefe experi- ments, which may hereafter lead to important confequences. Some ruft of iron had accidentally found its way into the © tube : 10 grains of carbonate were ufed, and a heat of 28° was — applied. The tube had. no flaw ; but there was a certainty that the carbonic acid had efcaped through its pores. When bro- ken, the place of the carbonate was found occupied, partly by a black flaggy matter, and partly by fphericles of various fizes, from that of a fmall pea downwards, of a white fub- ftance, which proved to be quicklime; the fphericles being interfperfed through the flag, as {par and agates appear in whinftone. The flag had certainly been produced by a mix- ture of the iron with the fubftance of the tube ; and the fphe- rical form of the quicklime feems to fhew, that the carbonate had been in fufion along with the flag, and that suey had feparated onthe efcape of the carbonic acid. Tue fubje& was carried thus far in 1803, when I fhould probably have publifhed my experiments, had I not been in- ~ duced to profecute the inquiry by certain indications, and accidental refults, of amature too irregular and uncertain to meet the public eye, but which convinced me, that it was poflible to eftablith, by experiment, the truth of all that was hypothetically aflumed in the Huttonian Theory. Tue principal objec was now to accomplith the entire fu- fion of the carbonate, and! to obtain fpar as the refult of that fufion, in imitation of what we conceive to have taken place in nature. Ir was likewife important to acquire the power of retaining all the carbonic acid: of the carbonate, both on account of the fact itfelf, and'on account)of its confequences ; the refult be- ing vifibly mmproved by every approach! towards complete fa- turation. . I therefore became anxious to inveftigate the caufe of the partial calcinations which had always. taken place, to a MODIFIED ly COMPRESSION. 99 a greater or a lef degree, in all thefe experiments. The que- ftion naturally fuggefts itfelf, What has become. of the car- bonic acid, feparated in thefe partial calcinations from the earthy ‘bafis? Has it penetrated the veffel, and efcaped en- tirely, or has it been retained within it in a gafeous, but highly compreffed ftate? It occurred to me, that this que- ftion might ‘be eafily refolved, by weighing the veflel before - and after the action of heat upon the carbonate. WirH iron, a conftant and ‘inappreciable fource of irregu- larity exifted in the oxidation of the barrel. But with porce- lain the thing was eafy ; and I put it in practice in all my ex- periments with this material, which were made after the que- ftion had occurredto me. The tube was weighed. as foon as its mauzzle was clofed, and again, after the breech had been expo- fed to the fire; taking care, m both cafes, to allow all to cool. In every cafe, I found fome lofs of weight, proving, that even in the beft experiments, the tubes were penetrated to a certain degree. Inext withed to try if any of the carbonic acid fepara- ted, remained within the tube in a gafeous form; and im that view, I wrapt the tube, which had juft been weighed, in a fheet of paper, and placed it, fo furrounded, on the fcale of the balance. As foon as its weight was afcertained, I broke the tube by a fmart blow, and then replaced upon the fcale the paper containing all the fragments. In thofe experiments, in which entire calcination had taken place, the weight was found not to be changed, for all the carbonic acid had already efcaped during the action of heat. But in the good refults, I always found that a lofs of weight was the confequence of break- ing the tube. : Tuese facts prove, that both caufes of calcination had ope- rated in the porcelain tubes; that, in the cafes of fmall lofs, part of the carbonic acid had efcaped through the veflel, and that part had been retained within it. I had in view methods: N 2 by 100 EFFECTS of HEAT by which the laft could be counteracted ; but I faw no remedy for the firft. I began, therefore, to defpair of ultimate fuccefs with tubes of porcelain*. ANOTHER circumftance confirmed me in this opinion. I found it impracticable to apply a heat above 27° to thefe tubes, when chargedas above with carbonate, without deftroying them, either by explofion, by the formation of a minute rent, or by the actual fwelling of the tube. Sometimes this fwelling took place to the amount of doubling the internal diameter, and yet the porcelain held tight, the carbonate fuftaining but a very fmall lofs. This ductility of the porcelain in a low heat is a curious fact, and fhews what a range of temperature is embraced by the gradual tranfition of fome fubftances from a folid to a liquid ftate: For the fame porcelain, which is thus fufceptible of being ftretched out without breaking in a heat of 27°, ftands the heat of 152°, without injury, when ex- pofed to no violence, the angles of its frature remaining fharp and entire. IV. * Tam neverthelefs of opinion, that, in fome fituations, experiments with com- preflion may be carried on with great eafe and advantage in fuch tubes. — I allude to the fituation of the geologifts of France and Germany, who may eafily procure, from their own manufattories, tubes of a quality far fuperior to any thing made for fale in this country. . MODIFIED ty COMPRESSION. rer IV. Experiments in Gun-Barrels refumed.—The Vertical Apparatus applied to them.—Barrels bored in folid Bars.—Old Sable Iron.—Fufion of the Carbonate of Lime.—lIts action on Porcelain.— Additional appa-- ratus required in confequence of that action.—Good refults ; in par- ticular, four experiments, illustrating the theory of Internal Calcina- tion, and /hewing the efficacy of the Carbonic Acid us a Flux. SincE I found that, with porcelain tubes, I could neither confine the carbonic acid entirely, nor expofe the carbonate in them to ftrong heats; I at laft determined to lay them afide, and return to barrels of iron, with which I had formerly ob- tained fome good refults, favoured, perhaps, by fome acciden- tal circumftances. -On the 12th of February 1803, I began a feries of experi- ments with gun-barrels, refuming my former method of working with the fufible metal, and with lead; but altering the pofition of the barrel from horizontal to vertical; the breech being placed upwards during the action of heat on the carbonate. This very fimple improvement has been productive of advantages no lefs remarkable, than in the cafe of the tubes of porcelain. In this new pofition, the included air, quitting the air-tube on the fu- fion of the metal, and rifing to the breech, is expofed to the greateft heat of the furnace, and muft therefore react with its greateft force; whereas, in the horizontal pofition, that air might go.as far back as the fufion of the metal reached, where its elafticity would be much feebler. The fame difpofition enabled me to keep the muzzle of the barrel plunged, du- ring the action of heat, in a veffel filled with water; which contributed | EFFECTS of HEAT contributed very much both to the convenience and fafety of thefe experiments. In this view, making ufe of the brick-furnace with the vertical muffle, already defcribed in page 93. I ordered a pit (aaa, fig. 20.) to be excavated under it, for the purpofe of receiving a water-veflel. This veflel (reprefented feparately, fig. 21.) was made of caft iron; it was three inches in diame- ter, and three feet deep; and had a pipe (d e) ftriking off from it at right angles, four or five inches below its rim, communi- cating with a cup (¢/) at the diftance of about two feet. The main veflel beng placed in the pit (a4) dire€tly below the vertical muffle, and the cup ftanding clear of the furnace, wa- ter poured into the cup flowed into the veffel, and could thus conveniently be made to ftand at any level. (The whole ar- rangement is reprefented in fig. 20.) The muzzle of the barrel (g) being plunged into the water, and its breech (4) reaching up into the muffle, as far as was found convenient, its pofition was fecured by an iron chain (gf). The heat communi- ‘cated downwards generally kept the furface of the water (at c) in a ftate of ebullition; the wafte thus occafioned being fupplied by means of the cup, into which, if neceffary, a con- ftant ftream could be made to flow. As formerly, I rammed the carbonate into a tube of porce- lain, and placed it in a cradle of iron, along with an air-tube and a pyrometer ; the cradle being “eel to a rod of iron, which rod I now judged proper to make as large as the barrel would admit, in order to exclude as much of the fufible me- tal as poflible ; for the expanfion of the liquid metal being in proportion to the quantity heated, the more that quantity could be reduced, the lefs rifk there was of deftroying the barrels. In the courfe of practice, a fimple mode occurred of remo- ving the metal and withdrawing the cradle: it confifted in pla- - cing MODIFIED ly COMPRESSION. 103 cing the barrel with its muzzle downwards, fo as to keep the breech above the furnace and cold, while its muzzle was expofed to ftrong heat in the muffle. In this manner, the metal was difcharged from the muzzle, and the pofition of the barrel being lowered by degrees, the whole metal was removed in fucceffion, till at laft the cradle and its contents became entirely loofe. As the metal was delivered, it was received in a crucible, filled with water, ftanding on a plate of iron placed over the pit, which had been ufed, during the firft ftage of the experiment, to contain the water- veflel. It was found to be of fervice, efpecially where lead was ufed, to give much more heat to the muzzle than fimply what was required to liquefy the metal it contained ; for when this was not done, the muzzle growing cold as the breech was heating, fome of the metal delivered from the breech was congealed at the muzzle, fo as to ftop the paflage. AccoRDING to this method, many experiments were made. im gun-barrels, by which fome very material fteps were gained. in the inveftization. On the 24th February, I made an experiment with {par and. chalk; the {par being placed neareft to the breech of the bar- rel, and expofed to the greateft heat, fome’ baked. clay inter- vening between the carbonates. On opening the barrel, a long-continued hifling noife was heard. The fpar was in a ftate of entire calcination; the chalk, though crumbling at the outfide, was Sadeghi hard and firm in the heart. The temperature had rifen to 32°. In this experiment, we have the firft clear example, in iron: barrels, of what I call Internal Calcination ; that is to fay, where the carbonic acid feparated from: the earthy bafis, has been. accumulated in cavities within the barrel. For, fablequently to the action of {trong heat, the barrel had been completely cooled; the air therefore introduced by means of the air- tube, muft have » 104 EFFECTS of HEAT ~ have refumed its original bulk, and by itfelf could have no tendency to ruth out ; the heat employed to open the barrel being barely fufficient to foften the metal. Since, then, the opening of the barrel was accompanied. by the difcharge of ela- {tic matter in great abundance, it is evident, that this mutt have proceeded from fomething fuperadded to the air originally in- cluded, which could be nothing but the carbonic acid of the carbonate. It follows, that the calcination had been, in part at leaft, internal ; the feparation of the acid from the earthy matter being complete where the heat was ftrongeft, and only partial where the intenfity was lefs. | Tue chemical principles {tated in a former part of this paper, authorifed us to expect a refult of this kind. As heat, by increafing the volatility of the acid, tended to feparate it from the earth, we had reafon to expect, that, under the fame compreflion, but in different temperatures, - one portion of the carbonate might be calcined, and ano- ther not: And that the leaft heated of the two, would be leaft expofed to a change not only from want of heat, but like- wife in confequence of the calcination of the other mafs; for the carbonic acid difengaged by the calcination of the hot- teft of the two, muft have added to the elafticity of the confined elaftic fluid, fo as to produce an increafe of compreflion. By this means, the calcination of the coldeft of the two might be altogether prevented, and that of the hotteft might be hindered from making any further advancement. This reafoning feemed to explain the partial calcinations which had frequently occur- red where there was no proof of leakage; and it opened fome new practical views in thefe experiments, of which I availed myfelf without lofs of time. If the internal calcination of one part of an inclofed mafs, promotes the compreflion of other mafles included along with it, I conceived that we might for- ward our views very much by placing a {mall quantity of.carbo- | nate, MODIFIED ty COMPRESSION. 105 nate, carefully weighed, in the fame barrel with a large quanti- ty of that fubftance ; and by arranging matters fo that the {mall fiducial part fhould undergo a moderate heat, while a ftronger heat, capable of producing internal calcination, fhould be ap- plied to the reft of the carbonate. In this manner, I made _ many experiments, and obtained refults which feemed to con- firm this reafoning, and which were often very fatisfaCory, though the heat did not always exert its greateft force where I intended it to do fo. On the 28th of February, I introduced fome carbonate, ac- curately weighed, into a {mall porcelain tube, placed within a larger one, the reft of the large tube being filled with pounded chalk ; thefe carbonates, together with fome pieces of chalk, placed along with the large tube in the cradle, weighing in all 95.7 grains. On opening the barrel, air rufhed out with a long-continued hiffing noife. The contents of the little tube were loft by the intrufion of fome borax which had been in- troduced over the filex, in order to exclude the fufible metal. But the reft of the carbonate, contained in the large tube, came out in a fine ftate, being porous and frothy throughout ; fpark- ling every where with facettes, the angular form of which was diftinguifhable in fome of the cavities by help of a lens: in fome parts the fubftance exhibited the rounding of fufion ; in many it was in a high degree tranfparent. It was yellow towards the lower end, and at the other almoft colour- lef. At the upper end, the carbonate feemed to have united with the tube, and at the places of contact to have {pread up- on it; the union having the appearance of a mutual action. The general mafs of carbonate effervefced in acid violently, but the thin ftratum immediately contiguous to the tube, fee- bly, if at all. On the 3d of March, I introduced into a very clean tube of porcelain 36.8 of chalk. The tube was placed in the upper Vout. VI.—P. I. O part 106 : EFFECTS of HEAT part of the cradle, the remaining {pace being filled with two pieces of chalk, cut for the purpofe; the uppermoft of thefe being excavated, fo as to anfwer the purpofe of an air-tube. The pieces thus added, were computed to weigh about 300 grains. There was no pyrometer ufed ; but the heat was guefled to be about 30°. After the barrel had ftood during a few minutes in its delivering pofition, the whole lead, with the rod and cradle, were thrown out with a {mart report, and with con- fiderable force. The lowermoft piece of chalk had fcarcely been a¢ted upon-by heat. The upper part of the other piece was in a ftate of marble, with fome remarkable facettes. The car- bonate, in the little tube, had fhrunk very much during the firft action of heat, and had begun to fink upon itfelf, by a fur- ther advancement towards liquefaction. The mafs was divi- ded into feveral cylinders, lying confufedly upon each other; this divifion arifing from the manner in which the pounded chalk was rammed into the tube in fucceflive portions. In feveral places, particularly at the top, the carbonate was very porous, and full of decided air-holes, which could not have been formed but in a foft fubftance; the globular form ‘and fhining furface of all thefe cavities, clearly indicating fu- . fion. The fubftance was femitranfparent ; in fome places yel- low, and in fome colourlefs. When broken, the folid parts fhewed a faline fracture, compofed’ of innumerable facettes. The carbonate adhered, from end to end, to-the tube, and in- corporated with it, fo as to render it impoflible to afcertain what lofs had been fuftained. In general, the line of conta& was of a brown colour; yet there was no room for fufpecting the prefence of any foreign matter, except, perhaps, from the iron- rod which was ufed in ramming downthe chalk. But, in fubfe- quent experiments, I have obferved the fame brown or black: colour at the union of the carbonate with the porcelain tubes, where the powder had been purpofely rammed with a piece of | wood ;. MODIFIED by COMPRESSION. 107 _ wood ; fo that this colour, which has occurred in almoft every | fimilar cafe, remains to be accounted for. The carbonate ef- fervefced violently with acid ; the fubftance in contact with the tube, doing fo, however, more feebly than in the heart, lea- ving a copious depofite of white fandy matter, which is doubt- lefs a part of the tube, taken up by the carbonate in fufion. - On the 24th of March, I made a fimilar experiment, ma ftout gun-barrel, and took fome care, after the application of heat, to cool the barrel flowly, with a view to cryftallization. The whole mafs was found in a fine ftate, and untouched by the lead; having a femitranfparent and faline flructure, with various facettes. In one part, I found the moft decided cry- ftallization I had obtained, though of a fmall fize: owing to its tranfparency it was not eafily vifible, till the light was made to reflect from the cryftalline furface, which then produced a dazzle, very obfervable by the naked eye : when examined by means of a lens, it was feen to be compofed of feveral plates, bro- ken irregularly in the fraQure of the fpecimen, all of which are parallel to each other, and reflect under the fame angle, fo as to unite in producing the dazzle. This ftructure was obfervable equally well in both parts of the broken fpecimen. Ina for- mer experiment, as large a facette was obtained in a piece of folid chalk; but this refult was of more confequence, as having been produced from chalk previoufly pounded. Tue foregoing experiments proved the fuperior efficacy of iron veflels over thofe of porcelain, even where the thicknefs was not great; and I perfevered in making a great many experi- ments with gun-barrels, by which I occafionally obtained very. fine refults: but I was at laft convinced, that their thicknefs was not fufficient to enfure regular and fteady fuccefs: For this purpofe, it appeared proper to employ veffels of {uch ftrength, as to bear a greater expanfive force than was juft ne- ceffary ; fice, occafionally, (owing to our ignorance of the re- ma 2 lation. 108 EFFECTS of HEAT lation between the various forces of expanfion, affinity, tena- city, &c.), much more ftrain has been given to the veflels than was requifite. In fuch cafes, barrels have been deftroyed, which, as the refults have proved, had acted with fufficient ftrength during the firft ftages of the experiments, though they had been unable to refift the fubfequent overftrain. Thus, my fuccefs with gun-barrels, depended on the good fortune of ha- ving ufed a force no more than fufficient, to conftrain the car- bonic:acid, and enable it to a@ as a flux on the lime. I there- fore determined to have recourfe to iron barrels of much greater ftrength, and tried various modes of conftruction. I wap fome barrels executed by wrapping a thick plate of iron round a mandrel, as is practifed in the formation of gun-bar- rels ; and likewife by bringing the two flat fides together, fo as to unite them by welding. Thefe attempts, however, failed. I next thought of procuring bars of iron, and of having a cavi- ty bored out of the folid, fo as to form a barrel. In this man- ner I fucceeded well. The firft barrel I tried in this way was of {mall bore, only half an inch: Its performance was highly fatisfactory, and fuch as to convince me, that the mode now adopted was the beft of any that I had tried. Owing to the {mallnefs of the bore, a pyrometer could not be ufed internal- ly, but was placed upon the breech of the barrel, as it ftood in the vertical muffle. In this pofition, it was evidently expofed to a much lefs heat than the fiducial part of the apparatus, which was always placed, as nearly as could be gueffed, at the point of greateft heat. On the 4th of April, an experiment was made in this way with fome fpar ; the pyrometer on the breech giving 33°. The fpar came out clean, and free from any contamination, adhering to the infide of the porcelain tube: it was very much fhrunk, ftill re- taining a cylindrical form, though bent by partial adhefions. Its furface bore fcarcely any remains of the impreflion taken by | the MODIFIED ty COMPRESSION. 109 the powder, on ramming it into the tube: it had, to the naked eye, the roughnefs and femitranfparency of the pith of a ruth - ftripped of its outer fkin. By the lens, this fame furface was feen to be glazed all over, though irregularly, fhewing here and there fome air-holes. In fracture, it was femitranfparent, more vitreous than cryftalline, though having a few facettes : the mafs, was feemingly formed of a congeries of parts, in them- felves quite tranfparent: and,at the thin edges, {mall pieces were vifible of perfect tranfparency. Thefe muft have been produ- _ced in the fire; for the fpar had been ground with water, and paffed through fieves, the fame with the fineft of thofe ufed at Etruria, as: defcribed by Mr Wrpcwoop, in his paper on the _conftruction of his Pyrometer. Wira the fame barrel I obtained many interefting refults, giving as ftrong proofs of fufion as in any former experiments ; with this remarkable difference, that, in thefe laft, the fub- ftance was compact, with little or no trace of frothing. In the gun-barrels where fufion had taken place, there had al- ways been a lofs of 4 or 5 per. cent., connected, probably, with ‘the frothing. In thefe experiments, for a reafon foon to be ftated, the circumftance of weight could not be obferved; but appearances led me to fuppofe, that here the lofs had been fmall, if any. 31 On the 6th of April, I made another experiment with the {quare barrel, whofe thicknefs was now much reduced by fuc- ceffive {cales, produced by oxidation, and in which a fmall rent began to appear externally, which did not, however, pe- netrate to the bore. The heat rofe high, a pyrometer on the breech of the barrel giving 37°. On removing the metals,. the cradle was found to be fixed, and was broken in the at- tempts made to withdraw it. The rent was much widened externally : but it was evident, that the barrel had not been -laid open, for part of the carbonate was in a ftate of faline marble ; 110 EFFECTS of HEAT marble ; another was hard and white, without any faline grains, and fearcely effervefced in acid. It was probably quicklime, formed by internal calcination, but in a ftate that has not oc- curred in any other experiment. Tue workman whom I employed to take out sad remains of the cradle, had cut off a piece from the breech of the barrel, three or four inches m length. As I was examining the crack which was feen in this piece, I was furprifed to fee the infide of the barrel lined with a fet of tranfparent and well-defined cry- ftals, of {mall fize, yet vifible by the naked eye. They lay to- gether in fome places, fo as to cover the furface of the iron with a tranfparent coat ; in others they were detached, and {cattered over the furface. Unfortunately, the quantity of this fubftance was too {mall to admit of much chemical examination; but I immediately afcertained, that it did not in the leaft effervefce in acid, nor did it feem to diflolve in it. The cryftals were in general tranfparent and colourlefs, though a few of them were tinged feemingly with iron. Their form was very well defined, being flat, with oblique angles, and bearing a ftrong refemblance to the cryftals of the Lamellated Stylbite of Haiiyv. Though made above two years ago, they ftill retain their form and tranfparency unchanged. Whatever this fubftance may be, its appearance, in this experiment, is in the higheft degree interefting, as it feems to afford an ex- ample of the mode in which Dr Hurron fuppofes many in- ternal cavities to have been lined, by the fublimation of fub- ftances in a ftate of vapour ; or, held in folution, by matters in a gafeous form. For, as the cryftals adhered to a part of the barrel, which muft have been occupied by air during the ac- — tion of heat, it feems next to certain that they were produced by fublimation. Tue very powerful effe@s produced by this laft barrel, the ~ fize of which (reduced, indeed, by repeated oxidation) was not above - MODIFIED ly COMPRESSION. IfT above an inch fquare, made me very anxious to obtain barrels of the fame fubftance, which being made of greater fize, ought ‘to afford refults of extreme intereft. J found upon inquiry, that this barrel was not made of Swedifh iron, as I at firft fup- pofed, but of what is known by the name of O/d Sable, from the figure of a Sable ftamped upon the bars; that being the armorial badge of the place in Siberia where this iron is made *. A worKMAN explained to me fome of the properties of diffe- rent kinds of irons, moft interefting in my prefent purfuit ; and he illuftrated what he faid by actual trial. All iron, when expo- fed to a certain heat, crufhes and crumbles under the hammer ; ‘but the temperature in which this happens, varies with every different fpecies. Thus, as he fhewed me, caift iron cruthes ma dull-red heat, or perhaps about 15° of WepcGwoop ; fteel, in a heat perhaps of 30°; Swedifh iron, in a bright - white heat, perhaps of 50° or 60°; old fable, itfelf, likewife yields, but in a much higher heat, perhaps of 100°. I merely guefled at thefe temperatures ; but Iam certain of this, that ina heat fimilar to that in which Swedifh iron crumbled under the _ hammer, the old fable withftood a ftrong blow, and feemed to poffefs confiderable firmnefs. It is from a knowledge of this quality, that the blackfmith, when he firft takes his iron from the forge, and lays it on the anvil, begins by very gentle blows, till the temperature has funk to the degree in which the iron can bear the hammer. I obferved, as the ftrong heat of the forge acted on the Swedith iron, that it began to boil at the furface, clearly indicating the difcharge of fome gafeous mat- ter ; whereas, the old chit in the fame circumftances, acqui- red the fhining furface of a liquid, and melted away without any effervefcence. I pr ocured, at this time, a confiderable . . number ~* IT was favoured with this account of it by the late Profeffor Rozison. 112 EFFECTS of HEAT number of bars of that iron, which fully anfwered my expec- tations. 3 . By the experiments laft mentioned, a very important point was gained in this inveftigation ; the complete fufibility of the carbonate under preflure being thereby eftablifhed. But from this very circumftance, a neceflity arofe of adding fome new devices to thofe already defcribed: for the carbonate, in fu- fion, {preading itfelf on the infide of the tube containing it, and the two uniting firmly together, fo'as to be quite infe- parable, it was impoflible, after the experiment, to afcertain the weight of the carbonate by any method previoufly ufed. I therefore determined in future to adopt the following ar- rangement. jud A sMALL tube of porcelain (i 4, fig. 23.) was weighed by means of a counterpoife of fand, or granulated tin ; then the car- bonate was firmly rammed into the tube, and the whole weighed again: thus the weight of the carbonate, previous to the ex- periment, was afcertained. After the experiment, the tube, with its contents, was again weighed; and the variation of weight obtained, independently of any mutual action that had taken place between the tube and the carbonate. ‘The balance which I ufed, turned, in a conftant and fteady manner, with one hundredth of a grain. Whenpounded chalk was rammed into this tube, I generally left part of it free, and in that fpace laid a fmall piece of lump-chalk (7), drefled to a cy- linder, with the ends cut flat and fmooth, and I ufually cut a letter on each end, the more effeCtually to obferve the effects produced by heat upon the chalk; the weight of this piece of chalk being always eftimated along with that of the powder contained in the tube. In fome experiments, I placed a cover of porcelain on the muzzle of the little tube, (this cover being weighed along with it), in order to provide againft the cate of ebullition : - MODIFIED ly COMPRESSION. 113 ~ ebullition: But as that did not often occur, I feldom took the trouble of this laft precaution. Ir was now of confequence to protect the tube, thus prepa- red, from being touched during the experiment, by any fub- - ftance, above all, by the carbonate of lime, which might adhere to it, and thus confound the appreciation by weight. This was provided for as follows : The fmall tube (Fig. 23. 7), with its pounded carbonate (é), and its cylinder of lump-chalk (7,), - was dropt into a large tube of porcelain (p , Fig. 24.). Upon this a fragment of porcelain (/), of fuch a fize as not to fall in between the tubes, was laid. Then a cylinder of chalk (7) was dreffed, fo as nearly to fit and fill up the infide of the large tube, one end of it being rudely cut into the form ofa cone. This mafs - being then introduced, with its cylindrical end downwards, was made to prefs upon the fragment of porcelain (/). I then dropped into the fpace (7), between the conical part of this mafs and the tube, a fet of fragments of chalk, of a fize beyond what could poffibly fall between the cylindrical part ‘and the tube, and prefied them down with a blunt tool, by which the chalk being at the fame time crufhed and rammed. into the angle, was forced into a mafs of fome folidity, which effectually prevented any thing from pafling between the large mafs of chalk and the tube. In practice, I have found this method always to anfwer, when done with care. I covered the chalk, thus rammed, with a ftratum of pounded flint (9), and that again with pounded chalk (f) firmly rammed. In this manner, I filled the whole of the large tube with alternate layers of filex and chalk; the muzzle being always occu- ' pied with chalk, which was eafily preffed into a mais of to- lerable firmnefs, and, fuffering no change in very low heats, excluded the fufible metal in the firft ftages of the experiment. Tue large tube, thus filled, was placed in the cradle, fome- times with the muzzle upwards, and fometimes the reverfe. I Vor. VI.—P. I. Ly have - \ EFFECTS of HEAT have frequently altered my views as to that part of the ar-"- rangement, each mode poffefling peculiar advantages and dif- advantages. With the muzzle upwards, (as fhewn in fig: 24. and > 25.) the beft fecurity is afforded againft the intrufion of the fufi- ble metal ; becaufe the air, quitting the air-tube in the working .. pofition, occupies the upper part of the barrel; and the fufible metal ftands as a liquid (at 7, fig. 25.) below the muzzle of the — tube, fo that all communication is cut off, between the liquid me- tal and the infide of the tube. On the other hand, by this arrange- ment, the {mall tube, which is the fiducial part of the appara- tus, is placed at a confiderable diftance from the breech of the barrel, fo as either to undergo lefs heat than the upper part, - or to render it neceflary that the barrel be thruft high into the | muffle. _. Wir the muzzle of the large tube downwards, the inner tube is placed (as fhewn in fig. 22.), fo as ftill to have its muzzle upwards, and in contact with the breech of the large tube. This: has the advantage of placing the {mall tube near . to the breech of the barrel: and though there is here lefs fe- curity againft the intrufion of liquid metal, I have found that: a point of little confequence; fince, when the experiment is a good one, and that the carbonic acid has been well con- fined, the intrufion feldom takes place in any pofition. In whichever of the two oppofite pofitions the large tube was placed, a pyrometer was always introduced, fo as to lie as near as poflible to the fmall tube. Thus, in the firft-mentioned — _pofition, the pyrometer was placed immediately below the large tube, and, inthe other pofition, above it ; fo that, in both eats it was feparated from the carbonate by the thicknefs only of the two tubes. Mucn room was unavoidably Bee Yeah by this method, which neceflarily obliged me to ufe fmall quantities of car- bonate, i a md ‘ MODIFIED ly COMPRESSION. - 115 bonate, the fubject of experiment feldom weighing more than Io or’12 grains, and in others far lefs * On the rith of April 1803, en a barrel of old fable _ iron having a bore of 0.75 of an inch, I made an experi- ment in which all thefe arrangements were put in prac- -ticé.’ .The large tube contained two fimall ones; one filled with fpar, and the other with chalk. I conceived that the heat had’ rifen to 33°, or fomewhat higher. On melting the metals, the cradle was thrown out with confiderable violence. The pyrometer, which, in this experiment, had been placed within the barrel, to my aftonifhment, indi- cated 64°. Yet all was found. The two little tubes came out quite clean and uncontaminated. The {par had loft 17.0 per cent.: The chalk 10.7 per cent.: The fpar was half funk down, and run againft the fide of the little tube: Its furface was fhining, its texture fpongy, and it was - compofed of a tranfparent and jelly-like fubfance : The chalk was entirely in a ftate of froth. This experiment extends our power of action, by fhewing, that compreffion, to a confider- able degree, can be carried on in fo great a heat as 64°. It feems likewife to prove, that, in fome of the late experiments with the fquare barrel, the heat had been much higher than was fuppofed at the time, from the indication of the pyrometer placed on the breech of the barrel; and that infome of them, particularly in the laft, it muft have rifen at leaft as high as in the prefent experiment. isis. P'2 On * I meafured the capacity of the air-tubes by means of granulated tin, ating as a fine and equal fand.. By comparing the weight of this tin with an equal bulk of water, I found that a cubic inch of it weighed 1330.6 grains, and that each grain of it corre{ponded to 0.00075 of a cubic inch. From thefe data I was - able, ee tolerable accuracy, to gage a tube by weighing the tin required to fill at! 116 EFFECTS of HEAT On the 21ft of April 1805, a fimilar experiment was madewith a new barrel, bored in a {quare bar of old fable, of about two and a half inch in diameter, having its angles merely rounded ; the inner tube being filled with chalk. The heat was main-. tained during feveral hours, and the furnace allowed to burn - out during the night. The barrel had the appearance of found- nefs, but the metals came off quietly, and the carbonate was entirely calcined, the pyrometer indicating 63°. On examina- tion, and after beating off the fmooth and even fcale of oxide peculiar to the old fable, the barrel was found to have yielded in its peculiar manner ; that is, by the opening of the longi- tudinal fibres. This experiment, notwithftanding the failure of the barrel, was one of the moft interefting I had made, fince it afforded proof of complete fufion. The carbonate had boil- ed over the lips of the little tube, ftanding, as juft defcribed, with its mouth upwards, and had run down to within half an inch of its lower end: moft of the fubftance was in a frothy ftate, with large round cavities, and a fhining furface; in other parts, it was interfperfed with angular mafles, which have evidently been furrounded by a liquid in which they floated. It was harder, I thought, than marble; giving no effervefcence, and not turning red like quicklime in nitric acid, which feemed to have no effe& upon it in the lump. It was probably a compound of quicklime with the fubftance of the tube. Wiru the fame barrel repaired, and with others like it, many fimilar experiments were made at this time with great fuccefs ; but to mention them in detail, would amount near- ly to a repetition of what has been faid. I fhall take notice of only four of them, which, when compared together, throw much light on the theory of thefe operations, and likewife _ feem to eftablifh a very important principle in geology. Thefe bs bgt four MODIFIED ly. COMPRESSION. 114 four experiments ‘differ from each other only in the heat em- _ ployed, and/in the quantity of air introduced. Tue firft of thefe experiments was made on the a, of A- pril 1803, in oneiof the large barrels. of old fable, with all the ’ above-mentioned arrangements. The heat had rifen, contrary to my imtention, to 78° and 79°. The tubes came out un- contaminated with fufible metal, and every thing bore the ap- -pearance of foundnefs. The contents of the little tube, con- - fifting of pounded chalk, and of a fmall piece of lump-chalk, came out clean, and quite loofe, not having adhered to the infide of the tube in the {malleft degree. There was a lofs of Al per cent., and the calcination feemed to be complete ; the fubftance, when thrown into nitric acid, turning red, without effervefcence at firft, though, after lying a few minutes, fome bubbles appeared. According to the method folfowed in all thefe experiments, and lately defcribed at length, (and fhewn in fig. 24. & 25.), the large tube was filled over the fmall one, with various mafles of chalk, fome in lump, and fome rammed into it in powder ; and in the cradle there lay fome pieces of chalk, filling up the fpace, fo that in the cradle there was a continued chain of carbonate of four or five inches in length. The fubftance was found to be lefs and lefs calcined, the more it was removed from the breech of the barrel, where the heat was greateft. ~A fmall piece of chalk, placed at the diftance of half an inch from the {mall tube, had fome faline fubftance in the heart, furrounded and intermixed with quicklime, dif. tinguifhed by its dull white. In nitric acid, this fubftance be- came red, but effervefced pretty brifkly; the effervefcence continuing till the whole was diffolved. The next portion of chalk, was in a firm ftate of limeftone ; and a lump of chalk in the cradle, was equal in perfection to any marble _ IT have obtained by compreifion : the two laft-mentioned pieces of chalk effervefcing with violence in the acid, and fhewing no 118 EFFECTS +of HEAT no rednefs when thrown into it. Thefe fads élearly. prove, 4 that the calcination of the contents of the fmall tube had been internal, owing to the violent’ heat which had feparated iG 3 acid from the moft heated part of the carbonate, according to the theory already ftated. The foundnefs of the barrel’ was’ © proved by the complete ftate of thofe carbonates which lay in lefs heated parts. The air-tube in this experiment had a ~ capacity of o. 29, nearly one-third of a cubic inch. - Tux fecond of thefe experiments was made on the 29th of © April, in the fame barrel with the laft, after it had afford- ed fome good refults. The air-tube was reduced to one- third of its former bulk, that is, to one-tenth of a cubic inch. The heat rofe to 60°. The barrel was covered externally with a black fpongy fubftance, the conftant indication of fai- lure, and a fmall drop of white metal made its appearance. The cradle was removed without any explofion or hifling. The carbonates were entirely calcmed. The barrel had yield- ed, but had refifted well at firft; for, the contents of the little tube were found in a complete ftate of froth, and running with the porcelain. | Tue third experiment was made on the 30th of insseil) in another fimilar barrel. Every circumftance was the fame as in the two laft experiments, only that the air-tube was now reduced to half its laft bulk, that is, to one-twentieth of a cu- bic inch. A pyrometer was placed at each end of the large tube. The uppermoft gave 41°, the other only 15°. The™ contents of the inner tube had loft 16 per cent., and were redu- ced to a moft beautiful ftate of froth, not very much injured by the internal calcination, and indicating a thinner ftate of fufion than had appeared. Tue fourth experiment was made’on the 2d of May, like the reft in all refpects, with a ftill fmaller air-tube, of 0.0318, being lefs than one-thirtieth of a cubic inch. The upper py- rometer -" MODIFIED ly COMPRESSION. 119 rometer gave 25°, and theunder one 169 :. The loweft mafles of carbonate were fcarcely affected by the heat: The contents of the little tube had loft 2.9 per cent. ; both the lump and the pounded chalk were ina fine faline flate, and, in feveral places had run and fpread upon the infide of the tube, which I had not expected to fee in fuch a low heat. On the upper furface of the chalk rammed into the little tube, which, after its in- ~trodudétion had been wiped fmooth, were a fet of white cryf- tals, with fhiming facettes, large enough to be diftinguifhed by the naked eye, and feeming to rife out of the mafs of car- ° bonate. I likewife obferved, that the folid mafs on which thefe cryftals ftood, was uncommonly traniparent. In thefe four experiments, the bulk of the included air was fuccellively diminifhed, and by that means its elafticity in- ‘creafed. -The confequence was, that in the firft experiment, ewhere that elafticity was the leaft, the carbonic acid was allowed to feparate from the lime, in an early ftage of the -rifing heat, lower than the fufing point of the carbonate, and complete internal calcination was effected. In the fecond experiment, the elaftic force being much greater, calcina- ‘tion was prevented, till the heat’ rofe fo high as to occa- fion the entire fufion of the carbonate, and its action on the tube, before the carbonic acid was fet at liberty by the failure of the barrel. In the third experiment, with ftill greater elaftic force, the carbonate was partly conftrained, ‘and its fufion accomplifhed, in a heat between 41° and 15°. In the laft experiment, where the force was ftrongeft of all, the carbonate was almoft completely protected from) decom- pofition by heat, in confequence of which it cryftallized and acted on the tube, in a temperature between 25° and 16°. On the other hand, the efficacy of the carbonic acid as a flux on the lime, and in enabling the carbonate to ad as a. ae on other Bia was clearly evinced.; fince the firft ex- . | periment a 120 EFFECTS of HEAT periment proved, that quicklime by itfelf, could neither be melted, nor,act upon porcelain, even in the violent heat of 79° ; ‘whereas, in the laft experiment where the carbonic acid was retained, both of thefe effects took place in a very low tempe- rature. . Wi Experiments in which Water was employed to increafe thé Elafticity of the included Air.—Cafes of complete Compreffion.—General Obferva- tions. —Some Experiments affording interefting refults ; in particular, foewing a mutual action between Silex and the Carbonate of Lime. Finpine that fuch benefit arofe from the increafe of elafti- city given to the included air in the laft-mentioned experi- ments, by the diminution of its quantity ; it mow occurred to - me, that a fuggeftion formerly made by Dr KenneEpy, of ufing water to aflift the comprefling force, might be followed with advantage: That while fufficient room was allowed for the expanfion of the liquid metal, a reacting force of any requi- red amount, might thus be applied to the carbonate. In this view, I adopted the following mode, which, though attended with confiderable difficulty in execution, I have often practi- — fed with fuccefs. The weight of water required to be intro- duced into the barrel was added to a {mall piece of chalk or baked clay, previoufly weighed. The piece was then dropped in- to atube of porcelain of about an inch in depth, and covered with pounded chalk, which was firmly rammed upon it. The tube was then placed in the cradle along with the fubject of expe- riment, and the whole was plunged into the fufible metal, previoufly poured ito the barrel, and heated fo as merely to render it liquid. The metal beimg thus fuddenly cooled, the oa —— ss MODIFIED by COMPRESSION. 121 the tube was encafed in-a folid mafs, before the heat had reached the mcluded moifture. The difficulty was to catch the fufible metal at the proper temperature; for when it was fo hot as not to fix in a few feconds, by the contaét of the cradle and its contents, the water was heard to bubble through the metal and efcape. I overcame this difficulty, however, by firft heating the breech of the barrel, (containing a fufh- cient quantity of fufible metal), almoft to rednefs, and then fetting it into a veffel full of water, till the temperature had funk to the proper pitch, which I knew to be the cafe when the hiffing noife produced in the water by the heated barrel cea- fed; the cradle, during the laft ftage of this operation, being held clofe to the muzzle of the barrel, and ready to be thruft into it. On the 2d of May, I made my firft experiment in this way, ufing the fame air-tube as in the laft experiment, -which was equal in capacity to one-thirtieth of a cubic inch. Half a grain of water was introduced in the manner juft defcribed. ‘The barrel, after an hour of red-heat, was let down by a ‘rope and pulley, which I took care to ufe in all experiments, _ in which there was any appearance of danger. All was found. The metals rufhed out {martly, and a flafh of flame accompa- nied the difcharge. The upper pyrometer gave 24°, and the lower one 14°. The contents of the inner tube had loft lefs ‘than 1 per cent., ftridly 0.84. The carbonate was in a ftate of good limeftone ; but the heat had been too feeble: The lower part of the chalk in the little tube was not agglutinated : The chalk round the fragment of pipe-ftalk (ufed to introduce the water), which had been more heated than the pyrometer, and the fmall rod, which had moulded itfelf in the boll of the ftalk, were in a ftate of marble. On the 4thof May, Imade an experiment like the laft, but with the addition of 1.05 grains water. After application of heat, the Vou. VI.—P. I. Q Re if 606 |. MOWER ERr ror ygAeAT fire was allowed to burn out till the barrel was black. The me- tal was difcharged irregularly. Towards the end, the inflam- mable air produced, burnt at the muzzle, with a lambent flame, during fome time, arifing doubtlefs from hydrogen gas, more © or lefs pure, produced by the decompofition of the water. The upper pyrometer indicated 36°, and the lower one 19°. The chalk which lay in the outer part of the large tube was in a ftate of marble. The inner tube was united to the outer one, by a ftar of fufed matter, black at the edges, and f{pread- ing all round, furrounding one of the fragments of porcelain which had fallen by accident in between the tubes. The in- ner tube, with the ftarry matter adhering to it, but without the coated fragment, feemed to have fuftained a lofs of 12-per cent., on the original carbonate introduced. But, the fub- ftance furrounding the fragment being inappreciable, it was impoffible to learn what lofs had been really fuftained. Exa- mining the little tube, I found its edges clean, no boiling over having taken place. The top of the fmall lump of chalk had funk much. When the little tube was broken, its contents gave proof of fufion in fome parts, and in others, of the neareft ap- proach to it. A ftrong action of ebullition had taken place all round, at the contact of the tube with the carbonate: in the heart, the fubftance had a tranfparent granular texture, with little or no cryftallization. The fmall piece of lump-chalk was united and blended with the rammed powder, fo that they could fcarce- ly be diftinguifhed. In the lower part of the carbonate, where the heat muft have been weaker, the rod had acted more feebly on the tube, and was detached from it: here the fubftance was ~ firm, and was highly marked in the fracture with cryftalline facettes. Wherever the carbonate touched the tube, the two fubftances exhibited, in their mixture, much greater proofs of fufion than could be found in the pure carbonate. At one - place, a ftream of this compound had penetrated a rent in | the MODIFIED by COMPRESSION. 123 the inner tube, which it had filled completely, conftituting a real vein, like thofe of ‘the mineral kingdom: which is {till diftin@tly to be feen in the fpecimen. It had then fpread it- felf upon the outfide of the inner tube, to the extent of half an inch in diameter, and had enveloped the fragment of por- _celain already mentioned. When pieces of the compound were thrown into nitric acid, fome effervefced, and fome not. I REPEATED this experiment on the fame day, with two grains of water. The furnace being previoufly hot; I conti- nued the fire during one half-hour with the muffle open, and another with a cover upon it. . I then let the barrel down by means of the pulley. The appearance of a large longitudinal rent, made me at firft conceive that the experiment was loft, and the barrel deftroyed: The barrel was vifibly fwelled, and in fwelling had burft the cruft of fmooth oxide with which it was furrounded; at the fame time, no exudation of metal had happened, and all was found. The metals were thrown out with more fuddennefs and violence than in any former experiment, but the rod remained in its place, _beimg fecured by a cord. The upper pyrometer gave 27°, the lower 23°.. The contents of the inner tube had loft 1.5 per cent. The upper end of the little lump of chalk, was _ rounded and glazed by fufions and the letter which I have been in the habit of cutting on thefe fmall pieces, in or- der to trace the degree of action upon them, was thus quite obliterated. On the lower end of the fame lump, the letter is {till vifible. Both the lump and the rammed chalk were ina good femitranfparent ftate, fhining a little in the fracture, but with no good facettes, and no where appearing to have acted on the tube. This laft circumftance is of confequence, fince it feems to fhew, that this very remarkable action of heat, under compreflion, was performed without the aflitt- . ance of the fubftance of the tube, by which, in many other Q2 experiments, 124 EFFECTS of HEAT experiments, a confiderable additional waa has been poems? municated to the carbonate. THESE experiments, and many chia made about the fame time, with the fame fuccefs, clearly prove the efficacy of wa- ter in aflifting the compreflion ; and refults approaching to thefe in quality, obtained, in fome cafes, by means of a very {mall air- tube, fhew that the influence of water on this. occafion has been merely mechanical. During the following fummer and autumn 1803, I was oc- . cupied with a different branch of this fubje@, which I fhall foon have occafion to mention. In the early part of laft year, 1804, I again refumed the fort of experiments lately defcribed, having in view principally to accomplith abfolute compreffion, in complete imitation of the natural procefs. In this purfuit, I did not confine myfelf to water, but made ufe of various other volatile fubftances, in order to aflift compreflion; namely, carbonate of ammo- nia, nitrate of ammonia, gunpowder, and paper impregna- ted with nitre. With thefe I obtained fome good refults, but none fuch as to induce me to prefer any of thefe compreffors to water. Indeed, I am convinced, that water is fuperior to them all. I found, in feveral experiments, made with a fimple air-tube, without any artificial compreflor, in which a very low red-heat had been applied, that the carbonate loft one or one and a half per cent. Now, as this muft have happened in a temperature fcarcely capable of inflaming gunpowder, it is | clear, that fuch lofs would not have been prevented by its pre- - fence: whereas water, beginning far below rednefs to aflume a gafeous form, will effectually refift any calcination, in low as well as in high heats. And as the quantity of water can very ealily be regulated by weight, its employment for this purpofe feems liable to no objection. On 0 a ee oe MODIFIED by COMPRESSION. 125 On the 2d of January 1804, I made an experiment with marble and chalk, with the addition of 1.1 grain of water. I aimed at a low heat, and the pyrometer, though a little bro- ken, feemed clearly to indicate 22°. Unluckily, the muzzle of the large tube, which was clofed as ufual with chalk, was placed uppermoft, and expofed to the ftrongeft heat. I found it rounded by fufion, and ina frothy ftate. The little tube came out very clean, and was fo nearly of the fame weight as when put in, that its contents had loft but 0.074 per cent. of the weight of the original carbonate. The marble was but feebly aggluti- nated, but the chalk was in a ftate of firm limeftone, though it muft have undergone a heat under 22°, or that of melting fil- ver. This experiment is certainly a moft remarkable one, fince a heat has been applied, in which the chalk has been chan- ged to hard limeftone, with a lofs lefs than the rooodth par of its weight, (exactly +,'--) 3 while, under the fame circum- ftances of preflure, though probably with more heat, fome of the fame fubftance had been brought to fufion. What lofs of. weight this fufed part fuftained, cannot be known. On the 4th of January, a fimilar experiment was made, like- — wife with 1.1 grain of water. The difcharge of the metal was accompanied with a flafh of flame. The pyrometer in- dicated 26°. The little tube came out quite clean. Its con- tents had been reduced from 14.53 to 14.46, difference 0.07 grains, being. 0.47 per ceat.. on the original carbonate, lefs. than one two-hundredth part of the original weight, (exactly s+): The chalk was in a ftate of firm faline marble, but with. no unufual qualities. THESE two laft experiments are rendered ftill more intereft-. ing, by another fet which I made foon after, which thewed, that one effential precaution in a point of fuch nicety had been neglected, in not previoufly drying the carbonate. In feyeral trials made in the latter end of the fame month, I 126 EFFECTS of HEAT I found, that chalk expofed to a heat above that of boiling water, but quite fhort of rednefs, loft 0.34 per cent.; and in another fimilar trial, 0.46 per cent. Now, this lofs of weight equals within o.or per cent. the lofs in the laft-mentioned ex- periment, that being 0.47; and far furpaffes that of the laft but one, which was but 0.074. There is good reafon, there- fore, to believe, that had the carbonate, in thefe two laft ex- periments, been previoufly dried, it would have been found during compreffion to have undergone no lofs. Tue refult of many of the experiments lately mentioned, feems fully to explain the perplexing difcordance between my experiments with porcelain tubes, and thofe made in barrels of iron. With the procelain tubes, I never could fucceed in a _ heat above 28°, or even quite up toit; yet the refults were often excellent. Whereas, the iron-barrels have currently ftood firm in heats of 41° or 51°, and have reached even to 470° or 80° without injury. At the fame time, the refults, even in thofe high heats, were often inferior, in point of fu- fion, to thofe obtained by low heats in porcelain. The rea- fon of this now plainly appears. In the iron-barrels it has always been.confidered as neceflary to ufe an air-tube, in con- fequence of which, fome of the carbonic acid has been fe- parated from. the earthy bafis by internal calcination: what carbonic acid remained, has been more forcibly attracted, ac- cording to M. BERTHOLLET’s principle, and, of courfe, more eafily comprefled, than when of quantity fufficient to faturate the lime: but, owing to the diminifhed quantity of the acid, the compound has become lefs fufible than in the natural ftate, and, of courfe, has undergone a higher heat with lefs effec. The introduction of water, by furnifhing a reacting force, has produced a ftate of things fimilar to that in the porcelain tubes ; the carbonate fuftaining little or no lofs of weight, eee El eee 4 742) Bice: ; Oe SS ee MODIFIED ly COMPRESSION. 127 weight, and the compound retaining its fufibility in low heats *. In the early part of 1804, fome experiments were made with barrels, which I wifhed to try, with a view to another feries of experiments. The refults were too interefting to be pafled over; for, though the carbonic acid in them was fay from being completely conftrained, they afforded fome of the fineft examples I had obtained, of the fufion of the carbonate, and of its union with filex. On the 13th of February, an experiment was made with pounded oyfter-fhell, in a heat of 33°, without any water be- ing introduced to aflift compreflion. The lofs was apparently of 12 per cent. The fubftance of the fhell had evidently been in vifcid fufion: it was porous, femitranfparent, fhining in fur- face and fracture; in moft parts with the glofs of fufion, in many others with facettes of cryftallization. The little tube had been fet with its muzzle upwards ; over it, as ufual, lay a fragment of porcelain, and on that a round mafs. of chalk. At the contact of the porcelain and the chalk, they had run together, and the chalk had been evidently in a very foft ftate ; for, refting with its weight on the porcelain, this laft had been prefled imto the fubftance of the chalk, deeper than its own breadth, a rim of chalk bemg vifible without the furface of _ the porcelain ; juft as when the round end of a knife is preffed upon * The retentive power here afcribed to the procelain tubes, feems not to accord with what was formerly mentioned, of the carbonic acid having been driven through the fubftance of the tube. But the lofs by this means has probably been fo fmall, that the native properties of the carbonate have not been fenfibly changed. ‘Or, perhaps, this penetrability may not be fo univerfal as‘ I have been induced to think, by having met with it in all the cafes which I tried. In this doubt, I ftrennoufly recommend a further examination of this fubje&t to gentlemen who have eafy accefs to fuch procelains as that of Drefden or of Seve. 128 | EFFECTS of HEAT upon a piece of foft butter. The carbonate had fpread very much on the infide of the tube, and had rifen round its lip, as fome falts rife from their folution in water. In this manner, a {mall quantity of the carbonate had reached the outer tube, and had adhered to it. The black colour frequently mention- ed as accompanying the union of the carbonates with the ~ porcelain, is here very remarkable. On the 26th of February, I made an experiment, in which the carbonate was not weighed, and no foreign fubftance was introduced to aflift the compreflion. The temperature was 46°. The pyrometer had been affected by the contact of a piece of chalk, with which it had united; and fome of the carbonate muft have penetrated the fubftance of the py- rometer, fince this laft had vifibly yielded to preflure, as ap- peared by a {welling near the contact. I obferved in thefe ex- periments, that the carbonate had a powerful action on the tubes of Cornifh clay, more than on the pounded filex. Per- haps it has a peculiar afhnity for argil, and this may lead to important confequences. The chalk had vifibly firft fhrunk upon itfelf, fo as to be detached from the fides, and had then begun to run -by fucceffive portions, fo as ftill to leave a pil- lar in the middle, very irregularly worn away ; indicating a fucceflive liquefaction, like that of ice, not the he of a mafs foftening all at once. On the 28th of February, I made an ped ier arte with oyfter-fhell unweighed, finely ground, and paffled through the clofeft fieves. The pyrometer gave 40°. The piece of chalk below it. had been fo foft, as to fink to the depth of half an inch into the mouth of the iron air-tube, taking its impreffion completely. A {mall part of this lump was contaminated with iron, but the reft was ina fme ftate. The tube had a rent in it, through which the carbonate, united with the mat- ter of the tube, had flowed in two or three places. The fhell MODIFIED ty COMPRESSION. 129 - fhell had fhrank upon itfelf, fo as to ftand detached from the fides, and bore very ftrong marks of fufion. The exter- nal furface was quite fmooth, and fhining like an enamel. The internal part confifted of a mixture of large bubbles and folid parts: the infide of the bubbles had a luftre much fuperior to that of the outfide, and equal to that of glafs. The general mafs was femitranfparent ; but fmall parts were vifible by the lens, which were completely tranfparent and colour- lefs. In feveral places this fmooth furface had cryftallized, fo as to prefent brilliant facettes, fteadily fhining in certain afpects. L obferved one of thefe facettes on the infide of an -air-bubble, in which it interrupted the fpherical form as if the little {phere had been preffled inwards at that fpot, by the contact of a plane furface. In fome chalk near the mouth of the large'tube, which lay upon a ftratum of filex, another very gnterefting circumftance occurred. Connected with its lower end, a fubftance was vifible, which had undoubtedly refulted. from the union of the carbonate with the filex. This fubftance was white and femitranfparent, and bore the appearance of chalcedony. The mafs of chalk having attached itfelf to that above it, had fhrunk upwards, leaving an. interval between it and the filex, and carrying fome of the compound up with it. From thence this’ laft had been in the a& of dropping in a vifcid ftate of fufion, as evidently appeared when the {peci- ‘men was entire; having a ftalactite and ftalagmite corre- fponding accurately to each.other. Unluckily I broke off the ftalactite, but the ftalagmite continues entire, in the form of a little cone. This new fubftance effervefced in acid, but not brifkly.. biwatched its entire folution ; a fet of light clouds remained: undiffolved, and’ probably Siené jelly was. formed ; for! 1 obferved, that a feries of air-bubbles. remained in shes form: of the fragment, and moved together without any vifible connection ; thus feeming to indicate a chemical. union be- LYVor. VI.—P.L ; me, tween; 130 EFFECTS of HEAT tween the filex and the carbonate. The fhell, fufed in the ex- periment, difiolved entirely in the acid, with violent effervef- cence. In the three laft experiments, abil in feveral others made at the fame time, the carbonate had not been weighed; but no water being introduced to affift the compreffion, it is. probable there was much lofs by internal calcination ; and owing doubt- lefs to this, the carbonates have crumbled almoft entirely co duft, while the rae eee which they had formed with filex remain entire. On the 13th of Mar om I made a fimilar experiment, in which, befides fome pounded oyfter-fhell, I introduced a mixture of chalk, with 10 per cent. of filex intermixed, and ground to- gether in a mortar with water, in a ftate of cream, and then well dried. The contents of the tube when opened, were difcharged with fuch violence, that the tube was broken to pieces; but I found a lump of chalk, then in a ftate of white marble, welded to the compound; which laft, in its fracture, fhewed that irregular black colour, interfperfed roughly through a cryftalline mafs, that belongs to the al- pine marbles, particularly to the kind called at Rome Cipoi- line. It was very hard and firm; I think unufually fo. It effer- veiced conftantly to the laft atom, in diluted nitric acid, but much more fluggifhly than the marble made of pure chal A cloudinefs appeared pervading all the liquid. When. the effervefcence was. over, a feries of bubbles continued during the whole day in the acid, withont any difpofition to burtt, or rife to the furface. After ftanding all next day and night, they maintained their ftation; and the folution being ftirred, was found to be entirely agglutinated into a tran{fparent jelly, breaking with fharp angles. This experiment affords a direct and pofitive proof of a chemical union having taken place be- tween the carbonate and filex. VI. MODIFIED ly COMPRESSION. 131 VI. Experiments made in Platina,—with Spar,—with Shells,—and with - Garbonate of Lime of undoubted purity. Since I had the honour of laying before this Society a fhort fketch of the foregoing experiments, on the 30th of Auguft laft (1804), many chemifts and mineralogifts of eminence have favoured me with fome obfervations on the fubjecét, and have fuggefted doubts which I am anxious toremove. It has been fuggefted, that the fufibility of the carbonates may have been the confequence of a mixture of other fubftances, either ori- ginally exifting in the natural carbonate, or added to it by the contact of the porcelain tube. WITH regard to the firft of thefe furmifes, I beg leave to obferve, that, granting this caufe of fufion to have been the real one, a material point, perhaps all that is ftriftly necefla- ry in order to maintain this part of the Huttonian Theory, was neverthelefs gained. For, granting that our carbonates were -_ impure, and that their impurity rendered them fufible, {till the fame is true of almoft every natural carbonate ; fo that our experimenats were, in that refpect, conformable to nature. And as to the other furmife, it has been fhewn, by com- paring together a varied feries of experiments, that the mu- tual action between the lime and the porcelain was oc- cafioned entirely by the prefence of the carbonic acid, fince, when it was abfent, no action of this kind took place. The fufion of our carbonates cannot, therefore, be afcribed to the porcelain. Be1nc convinced, however, by many obfervations, that the fafibility of the carbonate did not gi ca upon impurity, Ri2 I 132 EFFECTS of HEAT I have exerted myfelf to remove, by frefh experiments, every doubt that has arifen on the fubject. In order to guard againft natural impurities, I have applied to fuch of my friends as have - turned their attention to chemical analyfis, (a branch of the fcience-to which I have never attended,) to furnith me with carbonate of lime of undoubted purity. To obviate the con- tamination arifing from the contact of the porcelain tubes, I de- termined to confine the fubjecét of experiment in fome fubftance which had no difpofition to unite with the carbonate. I firft tried charcoal, but found it very troublefome, owing to its irregular abforption of water and air. ? ‘4 I THEN turned my thoughts to the conftrudion of tubes or cups of platina for that denen Being unable readily to, pro- cure proper folid veflels of this fubftance, I made ufe of thin laminated plates, formed into cups. My firft method was, to fold the plate exaétly as we do blotting-paper to form a filter’ (Fig. 26.); this produced a cup capable of holding the thin- neft liquid ; and being covered with a lid, formed of a fimilar thin plate, bent at the edges, fo as to overlap confiderably (Fig. 28.), the carbonate it contained was fecured on all fides from the contact of the porcelain tube within which it was placed. Another convenient device likewife occurred: I wrapt a piece of the plate of platina round a cylinder, fo as to. form a tube, each end of which was clofed by a cover like that juft defcribed (Fig. 27. and 29). (In figure 26. and 27. thefe cups are,reprefented upon a large fcale, and in 28. and 29. nearly of their actual fize). This laft conftruction — had the advantage of containing eight or nine grains of car- bonate, whereas the other would only hold about a grain and a half. On the other hand, it was not fit to retain a thin liquid; but, in moft cafes, that circumftance was of no confequence; and I forefaw that the carbonates, could not. thus MODIFIED by COMPRESSION. 133 thus efcape without proving the main sire under corifidera: tion, namely, their fufion. Tue reft of the apparatus was marrnes in’ all refpects as formerly defcribed, the fame precautions being taken to defend the platina veffel as had been ufed with the inner tubes of porcelain. In this manner I have made a number of experiments during this fpring and fummer, the refult of which is highly fatisfactory. They prove, in the firft place, the propriety of the obfervations which led to this trial, by fhewing, that the pute carbonate, thus defended from any contamination, is decidedly more refractory than chalk; fince, in many ex- periments, the chalk has been reduced to a ftate of marble, while the pure carbonate, confined in the platina veflel, has been but very feebly acted upon, having only acquired the in- duration of a fandftone. In other experiments, however, I have been more fuccefs- ful, having obtained fome refults, worthy, I think, of the at- tention of this Society, and which I fhall now fubmit to their infpection. The {pecimens are all inclofed, for fafety, in glafs tubes, and fupported on little ftands of wax, (fig. 31, 32, 33-). The fpecimens have, in general, been removed from the cup or tube of platina in which they were formed, thefe © devices having the advantage of fecuring both the veflel and its contents, by enabling us to unwrap the folds without vio- lence; whereas, in a folid cup or tube, it would have been dif- ficult, after the experiment, to avoid the deftruction either of the veffel or its contents, or both. Apri 16. 1805.—An experiment was made with pure calcareous {par from St Gothard, remarkably tranfparent, and having a ftrong double refraction. A temperature of 40° was applied ; but owing to fome accident, the weight was not known. ‘The conical cup came out clean and entire, filled not 134 EFFECTS of HEAT not quite to the brim with a yellowifh-grey fubftance, having _ a fhining furface, with longitudinal ftreaks, as we fometimes fee on glafs... This furface was here and there interrupted by lit- tle white tufts or protuberances, difpofed irregularly. On the ledge of the cup, formed by the ends of the folded plati- na, were feveral globular drops like minute pearls, vifible to the naked eye, the number of which amounted to fixteen. Thefe feem to have been formed by the entire fufion of what carbonate happened to lie on the ledge, or had been entangled amongft the extremities of the folds, drawing itfelf together, and uniting in drops ; as we fee when any fubftance melts un- der the blowpipe. This refult is preferved entire, without de- ranging the tube. I am forry to find that it has begun to fall to decay, in confequence, no doubt, of too great a lofs: of its carbonic acid. But the globules do not feem as yet to have _ fered any injury. . APRIL 25.—The fame fpar was ufed, with two grains of water, and a heat of 33°. I have reafon to fufpe@, how- ever, that, in this and feveral other experiments made at this time, the metal into which the cradle was plunged; on firft introduction into the barrel, had been too hot, fo as to drive off the water. There was a lofs of 6.4 per cent. The refult lay in the cup without any appearance of frothing or fwelling.. The furface was of a clean white;. but rough, having in one corner a fpace fhining like glafs. The cup being unwrapt, the fubftance was obtained found and: entire: where it had moulded itfelf on the platina, it had a fmall de- gree of luftre, with the irregular femitranfparency of faline marble: when broken, it preferved that character more com- pletely than in any refult hitherto obtamed; the fracture be- ing very irregular and angular, and fhining with facettes in various directions. I much regret that this beautiful fpecimen no MODIFIED by COMPRESSION. 135 no longer exifts, having crumbled entirely to pieces, notwith- ftanding all the care I took to inclofe it with glafs and wax. Apait 26. - An experiment was made with fome carbonate _of lime, purified by my friend Sir GeorGE Mackenzie. ‘Two grains of water were introduced, but were loft, I fufpect, as in the laft cafe. The heat applied was 32°. The lofs of weight was 10.6 per cent. Yet, though made but one day after the laft-mentioned fpecimen, it remains as frefh and entire as at firft, and promifes to continue unchanged. The external _furface, as feen on removing the lid of the conical cup, was found to fhine all over like glafs, except round the edges, which were fringed with a feries of white and rough {phericles, one fet of which advanced, at one fpot, near to the centre. The fhining furface was compofed of planes, which formed ob- tufe angles together, and had their furface ftriated’; the ftrie bearing every appearance of a cryftalline arrangement. When freed from the cup, as before, the fubftance moulded on the platina was found to have affumed a fine pearly furface. Some large air-bubbles appeared, which had adhered to the. cup, and were laid open by its removal, whofe internal furface had a beautiful luftre, and was full of ftrie like the outward fur- face. The mafs is remarkable for fémitranfparency, as feen particularly where the air-bubbles diminifh its thicknefs: a fmall part of the mafs being broken at one end, fhews an in- ternal faline ftrudcture. APRIL 29.——A cup of platina was filled with feveral large pieces of a periwinkle* fhell, the tharp point of the fpiral being made to ftand upright in the cup, (fig. 30.). A heat of 30° was applied, and no water was introduced. The carbonate loft no lefs than 16 per ceat. The fhell, particularly } the _* Turbo terebra, Lin.. 136 o EFFECY?S of HEAT the fharp end of the periwinkle, retained its original fhape in a great meafure, fo as to be quite difcernible ; but the whole was glazed over with a truly vitreous luftre. This glaze co- vered, at one place, a fragment of the fhell which had been ori-. ginally loofe, and had welded the twotogether. All the angles are rounded by this vitrifaction; the fpace between the en- tire fhell and the fragment being filled, and the angles of their meeting rounded, with this fhining fubftance. The colour is a pale blue, contrafted, in the fame little glafs, with a natural piece of periwinkle, which is of a reddifh-yel- low... One of the fragments had adhered to the lid, and had © been converted into a complete drop, of the fize of a muftard- feed. It is fixed on the wax (at 4), along with the other fpe- cimens of the experiment (fig. 32.). This refult fhews, as yet, no fign of decay, notwithftanding fo great a lofs of weight. Tue laft experiment was repeated on the fame day, and pre- pared in the fame manner, with large fragments of fhell, and the point of the periwinkle ftanding up in the cup. A heat of 34° was applied; alofs took place of 13° per cent. All the original form had difappeared, the carbonate lying in the cup as a com- plete liquid, with a concave furface, which did not fhine, but was ftudded all over with the white {phericles or tufts, like thofe feen in the former refults, without any {pace between them. When detached from the cup, the furface moulded on the platina, was white and pearly, with a flight glofs. -The mafs was quite folid; no veftige whatever appearing, of the original form of the fragments, (fig. 33.).. A fmall piece, bro- ken off near the apex of the cone, fhewed the internal ftruc- ture to be quite faline. Inthe act of arranging the fpecimen on its ftand, another piece came off ina new direction, which pre- fented to view the moft perfect cryftalline arrangement : the fhining plane extended acrofs the whole fpecimen, and was more than the tenth of an inch in all directions. This fracture, likewife, Oe teh page ot Ca . ae ae MODIFIED ty COMPRESSION. 134 Yikewife, fhewed the entire internal folidity of the mafs. Un- fortunately, this fpecimen has fuffered much by the fame de- cay to which all of them are fubje@ which have loft any con- fiderable weight. The part next the outward furface alone remains entite. I have never been able to explain, in a fatis- factory manner, this difference of durability ; the laft-men- tioned refult having loft more in proportion to its weight than this. | ‘. AsouT the beginning of June, I received from Mr Hatcuntt fome pure carbonate of lime, which he was fo good as to pre- pare, with a view to my experiments; and I have been conftant- ly employed with it till within thefe few days. My firft experiments with this fubftance were peculiarly un- fortunate, and it feemed to be lefs eafily acted upon than any fabftance of the kind I had tried. Its extreme purity, no doubt, contributed much to this, though another circumftance had likewife had fome effe&. The powder, owing to a cryftallization which had taken place on its precipitation, was very coarfe, and little fufceptible of clofe ramming ; the particles, therefore, had lefs advantage than when a fine powder is ufed, in acting up- on each other, and I did not choofe to run any rifk of contami- nation, by reducing the fubftance to a finer powder. Whatever be the caufe, it is certain, that in many experiments in which the chalk was changed to marble, this fubftance remained in a loofe and brittle ftate, though confifting generally of clear and | fhining particles. I at laft, however, fucceeded in ee fome very good refults with this carbonate. -Inan experiment made with it on the 18th of June, in a bedng heat, I ebtained a very firm’ mafs with a Taline fra@ure, moulded in féveral places on the platina, which was now ufed in the cylindrical form. On the 23d, ina fimilar experiment, the barrel failed, and the fubject of experiment was found in an entire ftate of froth, proving its former fluidity. Vor. VI,—P. I. S On 138 _ EFFECTS of HEAT On the 25th, in a fimilar experiment, a heat of 64° was ap- plied, without any water within the barrel... The platina. tube; (having been contaminated in a former experiment with fome fufible metal), melted, and the carbonate retaining its cylin- drical fhape, had fallen through it, fo as to touch the piece of porcelain which had been placed next to the platina tube. At the point of contaé, the two had run together, as a hot iron runs when touched by fulphur. The carbonate itfelf was very tran{parent, refembling a piece of fnow in the act of melting. . On the 26th of June, I made an, experiment with this car- bonate, which afforded a beautiful refult. One grain’ of wa- ter was introduced with great care; yet there was a lofs, of 6.5 per cent., and the refult, has fallen to decay. .The pyro- meter indicated 43°. On the outfide of the platina cylinder, and on one.of the lids, were feen a fet of globules, like pearls, as once before. obtained, denoting perfect fufion. . When the upper lid was removed, the fubftance was found to have funk almoft out of fight, and had aflumed a form not eafily defcri- — bed. (I have endeavoured to reprefent it in fig, 31. by anideal fection of the platina-tube and its contents, made through the axis of the cylinder). The powder, firft fhrinking upon itfelf in the a@ of agglutination, had formed a cylindrical rod,.a remnant of which (4c) ftood up in the middle of the tube. By the continued action of heat, the fummit of the rod (at a) had been ,rounded,.in fufion, and the mafs being now fof- tened, had funk by its weight, and {pread below, fo as to mould itfelf in the tube, and fill its lower part completely (dfge). At the fame time, the vifcid fluid adhering to the fides (at ¢ atid d), while the middle part was finking, had been in part left behind, and in part drawn out into, a thin but. tapering fhape, united by a curved furface (at 5 and c) to the middle rod. When the platina tube was unwrapt, the thin edges (at ¢ and d) were preferved all round, and in a ftate « —— - °° °° Pe Oe MODIFIED ty COMPRESSION. 139 ftate of beautiful femitranfparency. (I have attempted to re- prefent the entire {pecimen, as it ftood on its cone of wax, in fig. 34.). The carbonate, where moulded on the platina, had a clean pearly whitenefs, with a faline appearance externally, and in the fun, fhone with facettes. Its furface was interrupt- _ ted by a few fcattered air-bubbles, which had lain againft the tube. The intervening fubftance was unufually compact and hard under the knife. The whole furface (eb acd, fig. 31.), and the infide of the air-bubbles, had a vitreous luftre. Thus; every thing denoted a ftate of vifcid fluidity, like that: of hos ne th laft experimeits feem to obviate every fabs that re- mained with refped to the fufibility of the pureft carbonate, without the affiftance of any foreign fubftance. VIl. Meafurement of the Force required to conftrain the Carbonic Acid.—Ap- - paratus with the Muzzle of the Barrel upwards, and the weight acting «by a long Lever.—Apparatus with the. Muzzle downwards. Appa- ratus with Weight acting directly on the barrel.—Comparifon of various refults. \ In order to determine, within certain limits at leat, what force had been exerted in the foregoing experiments, and what was neceffary to enfure their fuccefs, I made a numbet of ex- periments, in a mode nearly allied to that followed by Count Rumrorp, in meafuring the explofive force of gunpowder. I BEGAn to ufe the following fimple apparatus in June 1803. I took one of the barrels, made as above defcribed, for the purpofe of compreffion, having a bore of 0.75 of an $2 inch, 140 ’ EFFECTS of HE“ZT inch *, and drefled its muzzle to a fharp edge. To this barrel: was firmly {crewed a collar of iron (aa, fig. 36.) placed at a diftance of about three inches from the muzzle, having two ftrong bars (44) projecting at right angles to the barrel, and drefled fquare. The barrel, thus prepared, was introduced, with its breech downwards, into the vertical muffle (fig. 35.) 5 its length being fo adjufted, that its breech fhould be placed in the ftrongeft heat; the two projecting bars above defcribed, refting on two other bars.(¢c, fig. 35.) laid upon the furnace to receive them ; one upoa each fide of the muffle. Into the _ barrel, fo placed, was introduced a cradle, containing. carbo- nate, with all the arrangements. formerly mentioned’; the rod connected: with it being of fuch length, as_juft to lie within the: muzzle of the barrel. The liquid metal was then poured in till it filled the barrel, and ftood at the muzzle with a convex fur- face; a cylinder of iron, of about an inch in diameter, and half. an inch thick, was laid on the muzzle (fig. 35. and 37.), and to: it a comprefling weight was inftantly applied. This. was firft. done by the preffure of a-bar of iron (de, fig. 35.), three-feet in. length, introduced. loofely into a hole (d), made: for the purpofe. in the wall againft which the furnace ftood ; the diftance between. this hole and the barrel being one foot. A weight was then fuf- pended at the extremity of the bar (¢), and thus a comprefling force was applied, equal to three times that weight. In the courfe of practice, a cylinder of lead. was fubftituted for that of iron, and a piece of leather was placed between it and the muzzle of the barrel, which laft being dreffed toa pretty, fharp. edge, made an impreffion in the lead : to affift, this: effe@, one, fmart blow of a hammer was ftruck upon the bar, dire@ly. over the barrel, as foon as the weight had been hung on. | Ir. * This was tlie fize of barrel ufed in all the following experiments, where thee faét is not otherwife expreffed. MODIFIED by COMPRESSION. I45E Ir was eflential, in this mode of operation, that the whole. of the metal fhould continue in a liquid ftate during the action: of heat ; but when I was fatisfied as to its intenfity and dura- tion, I congealed the metal, either by extinguifhing the furnace entirely, or by pouring water on the barrel. As foon as the heat began to act, drops of metal were feen to force themfelves- between the barrel and the leather, following each other with. more or lefs rapidity, according to. circumftances. In fome. - experiments, there was little exudation; but few of them were entirely free from it. To fave the metal thus extruded, I placed.a black-lead crucible, having its bottom perforated, round the barrel, and. luted clofe to. it, (fig. 37.); fome fand. being laid in this crucible, the metal was. collected on its fur-. face. On fome occafions, a found of ebullition was heard. during the action of heat; but this was.a certain fign of. fail-- ure. ‘hi wae Tue refults.of the moft important of thefe experiments, have been reduced to a common ftandard in the fecond table: placed in the Appendix ; to which reference is. made by the. following numbers. No. 1.—QOn the 16th of June 1803, I made an experiment; with thefe arrangements. I had tried to ufe.a weight of 30 lb.. producing a preflure of 90 lb., but I found this not fufficient. _I them hung on a.weight of 1 cwt., or 112.1b. ; by. which a com-. prefling force was applied of .3.cwt. or, 3361b. Very little metal was feen to efcape, and no found of ebullition was heard. ‘he chalk in the body of the large tube was reduced to quick- lime; but what lay in the inner tube was pretty firm, and ef-. fervefced to thelaft. One or two facettes, of good appearance, were likewife found. The contents of the {mall tube had loft. but 2.6 per cent.; but there was a fmall vifible intrufion of me- tal, and the refult, by its appearance, indicated a greater lofs. L.confidered. this, however, as one point gained ; that being the 142 | EFFECTS of HEAT firft tolerable compreflion accomplifhed by a determinate force. The pyrometer indicated 22°. THIS experiment wes repeated the itch day, when a ftill fmaller quantity of metal efcaped at the muzzle ; but the bar- rel had given way below, in the manner of thofe that have yielded for want of fufficient air. Even this refult was fatis- factory, by fhewing that a mechanical power, capable of for- cing fome of the barrels, could now be commanded. The car- bonate in the little tube had loft 20 per cent.; but part of it was in a hard and firm ftate, effervefcing to the laft. No. 2.—On the 2rft June, I made an experiment with ano: ther barrel, with the fame circumftances. I had left an empty {pace in the large tube, and had. intended to introduce its muzzle downwards, meaning that {pace to.anfwer as an air- tube; but it was inverted by miftake, and the tube entering with its muzzle upwards, the empty {pace had of courfe filled with metal, and thus the experiment was made without any included air. There was no pyrometer ufed ; but the heat was guefled to be about 25° where the fubje& of experiment lay. The barrel, when opened, was found full of metal, and the cradle being laid flat.on the table, a confiderable quantity of metal ran from it, which had undoubtedly been lodged in the vacuity of the large tube. When cold, I found that vacuity ftill empty, with a plating of metal. The tube was very clean to appearance, and, when fhaken, its contents were heard to rattle. Above the little tube, and the cylinder of chalk, I had put fome borax and fand, with a little pure borax in the middle, and chalk over it. The metal had not penetrated beyond the borax and fand, by a good fortune peculiar to this experiment; the intrufion of metal in this mode of execution, being extremely troublefome. The button of chalk, was found in a ftate of clean white car- bonate, and pretty hard, but without tranfparency. The little tube MODIFIED by COMPRESSION. 143 tube was perfectly clean. Its weight with its contents, feem- ed to have: fuffered no change from what it had been when firft introduced. Attending, however, to the balance with ferupulous nicety, a fmall preponderance did appear on the fide of the weight. This was done away by the additiom of the hundredth of a grain to the fcale in which the carbo- nate lay, and an addition of another hundredth produced in it a decided preponderance. Perhaps, had the tube, before its in- troduction, been examined with the fame care, as great a diffe- rence might have been detected ; and it feems as if there had. been no lofs, at leaft not more than one hundredth of a grain, which on 10.95 grains, amounts to 0.0912, fay 0.1 per cent. The carbonate was loofe in the little tube, and fell out by fha-. king. It had a yellow colour, and compact appearance, with: a ftony hardnefs under the knife, and a ftony fracture; but with very flight facettes, and little or no tranfparency.. In fome parts of the fpecimen, a whitith colour feemed to indicate partial . ‘ealcimation. On examining the fracture, I perceived, with the magnifier, a {mall globule of metal, not vifible to the naked. eye, quite infulated and fingle. Poflibly the fubftance may have contained others of the fame fort, which may have compenfated for a fmall lofs, but there could:not be many fuch, from the general clean appearance of the whole. In the fra@ure, I faw here and there {mall round holes, feeming, though imperfect-- ly, to. mdicate a beginning of ebullition. -- [ mabe a number of experiments in the fame manner, that is to fay, with the muzzle of the barrel: upwards, in fome of which I obtained: very fatisfa@tory refults; but it was. only by chance that the fubftance efcaped. the contamina-- tion of the fufible metal; which induced me to'think of ano-. ther mode of applying the comprefling weight with the muzzle of the barrel downwards, by which I expected to re-. pers, with a. determinate weight, all the experiments formerly. made +44 EFFECTS of HEAT ‘made in barrels clofed by congealed metal ; and that, by ma- king ufe of an air-tube, the air, rifing to the breech, would fe- cure the contents of the tube from any contamination. In this view, the barrel. was introduced from below into the - mufHe with its breech upwards, and retained in that pofition by means of a hook fixed to the furnace, till the collar was made to prefs up againft the grate, by an iron lever, loaded with a weight, and refting en a fupport placed im front. In fome experiments made in this way, the refult was obtained very clean, as had been expected ; but the force had been too feeble, and when it was increafed, the furnace yielded up- wards by the mechanical ftrain. : I rounp it therefore neceflary to ufe a frame of iron, fea in fig. 38.; the frame being reprefented feparately in fig. 39.), by which the brick-work was-relieved from the ievbeieiesl ftrain. This frame confifted of two bars (ad and fe, figs. 38. and 39.), fixed into the wall, (at aand f,) pafling horizontally under the furnace, one on each fide of the muffle, turning downwards at the front, (in band ¢e),and meeting at the ground, with a flat bar (¢d) uniting © the whole. In this manner, a kind of ftirrup (4c de) was formed in front of the furnace, upon the crofs bar (¢ d) of which a block of wood (44, fig. 38.), was placed, fupporting an edge of iron, upon which the lever refted ; the working end of the lever (g) acting upwards. A ftrain was exerted, by means of ‘the barrel and its collar, againft the horizontal bars, (a 4 and f ¢), which was effectually refifted by the wall (at a4 and f) at one end of thefe bars, and by the upright bars (¢ 4 and d ¢) at the other end. In this manner the whole ftrain was fuftained — by the frame, and the furnace ftood without injury. | Tue iron bar, at its working end, was formed into the fhape of a cup, (atg), and half filled with lead, the fmooth furfaceof which, was applied to the muzzle of the barrel. The lever, too; — swas lengthened, by joining to the bar of iron, a beam of wood, making — MODIFIED ly COMPRESSION. 145 making the whole ten feet in length. In this manner, a pref fure upwards was applied to the barrel, equal to the weight of To cwt. | | In the former method, in which the barrel ftood with its muzzle upwards, the weight was applied while the metal was liquid. In this cafe, it was neceflary to let it previoufly con- geal, otherwife the contents would have run out in placing the barrel in the muffle, and to allow the liquefaction eflential to thefe trials, to be produced by the propagation of heat from the muffle downwards. This method required, therefore, in every cafe, the ufe of an air-tube; for without it, the heat acting upon the breech, while the metal at the muzzle was ftill cold, would infallibly have deftroyed the barrel. A great number of thefe experiments failed, with very confiderable wafte of the fufible metal, which, on thefe occafions was nearly all loft. But a few of them fucceeded, and afforded very fatis- factory refults, which I fhall now mention. ! In November 1803, fome good experiments were made in this way, all with a bore of 0.75, and a preffure of 10 cwt. No. 3.—Own the 19th, a good limeftone was obtained in an experiment made ina ne epee of 21°, with a lofs of only I.I per cent. No. 4.—Own the 22d, in a fimilar experiment, there was little exudation by the muzzle. The pyrometer gave 31°. The carbonate was in a porous, and almoft frothy ftate. No. 5.—Iwn a fecond experiment, made the fame day, the heat rofe to 37° or 41°. The fubftance bore ftrong marks of fufion, the upper part having fpread on the little tube: the whole was very much fhrunk, and run againft one fide. The mafs fparkling and white, and.in a very good ftate. No.6.—On the 25th,an experiment was made with chalk,and fome fragments of {nail fhell, with about half a grain of water. ‘The heat had rifen to near 51° or 49°. The barrel had been Vor, VI.—P. I. £ held 146 EFFECTS of HEAT held tight by the beam, but was rent and a little fwelled at the breech. The rent was wide, and fuch as has always appeared in the ftrongeft barrels when they failed. The car- bonate was quite calcined, it had boiled over the little tube, and was entirely in a frothy ftate, with large and diftin@ly rounded air-holes. The fragments of fhell which had occupied the upper part of the little tube, had loft every trace of their original fhape in the act of ebullition and fufion. No. 7.—Own the 26th a fimilar experiment, was made, in which the barrel was thrown open, in fpite of this powerful comprefling force, with a report like that of a gun, (as I was told, not having been prefent), and the bar was found in a ftate of ftrong vibration. The carbonate was calcined, and fomewhat frothy, the heart of one piece of chalk ufed was in a ftate of faline marble. Ir now occurred to me to work with a corpteffing force; lid = no air-tube, trufting, as happened accidentally in one cafe, that the expanfion of the liquid would clear itfelf by gentle exu- dation, without injury to the carbonate. In this mode, it was neceflary, for reafons lately ftated, to place the muzzle upwards. Various trials made thus, at this time, afforded no remarkable refults. But I refumed the method, with the fol- lowing alteration in the application of the weight, on the 27th of April 1804. I concEIvED that fome inconvenience might arife from the mode of employing the weight in the former experiments. In them it had been applied at the end of the bar, and its effet propagated along it, fo as to prefs againft the barrel at its other extremity. It occurred to me, that the propagation of motion in this way, requiring fome fenfible time, a confi- derable quantity of carbonic acid might efcape by a fudden eruption, before that propagation had taken effect. I there- — fore thought, that more effectual work might be done, by placing MODIFIED ly COMPRESSION. 147 placing a heavy mafs, (fig. 40.), fo as to act directly and fimply upon the muzzle of the barrel ; this mafs being guided and commanded by means of a powerful lever, (a0). For this purpofe, I procured an iron roller, weighing 3.cwt. 7 [b., and fufpended it over the furnace, to the end of a beam of wood, refting on a.fupport near the furnace, with a long arm guided by a rope (¢ ¢) and pulley (4), by which the weight could be raifed or let down at pleafure. Wiru this apparatus I made fome tolerable experiments ; but I found the weight too light to afford certain and fteady refults of the beft quality. I therefore procured at the foundry a large mafs of iron (f), intended, I believe, for driving piles, and which, after allowing for the counterpoife of the beam, gave a direct preflure of 8.1 cwt.; and I could, at pleafure, diminifh the comprefling force, by placing a bucket (¢) at the extremity of the lever, into which I introduced weights, whofe effec on the ultimate great mafs, was known by trial. Many bar- rels failed in thefe trials: at laft, I obtained one of {mall bore, inch 0.54, which gaye two good refults on the 22d of June 1804. | No. 8.—Wsu1NnG to afcertain the leaft comprefling force by which the carbonate could be effectually conftrained in melt- ing heats, I firft obferved every thing ftanding firm in a heat of above 20°; I then gradually threw weights into the bucket, | till the. comprefling force was reduced to 2 cwt. Till then, things continued fteady ; but, on the preflure being {till further diminifhed, metal began to ooze out at the muzzle, with in- creafing rapidity. When-the preflure was reduced to 14 cwt. air rufhed out with a hifling noife. I then ftopped the experi- ment, by pouring water on the barrel. The piece of chalk ‘had loft 12 per cent. It was white and foft on the outfide, but firm and good in the heart. ) Li2 | No. g. 148 EFFECTS of HEAT No. 9.—AN experiment was made with chalk, in a little tube; to this, one grain ef water was added, I had intended to work with 4 cwt. only ; but the barrel was no fooner placed, than an exudation of metal began at the muzzle, owing, doubt- lefs, to the elafticity of the water. I immediately increafed the preflure to 8.1 cwt. by removing the weight from the bucket, when the exudation inftantly ceafed. I continued the fire for three quarters of an hour, during which time no exu- dation happened ; then all came out remarkably clean, with {carcely any contamination of metal. The lofs amounted to. 2.58 per cent. The fubftance was tolerably indurated, but had not acquired the character of a complete ftone. In thefe two laft experiments, the bore being fmall, a pyro- meter could not be admitted. On the 5th of July 1804, I made three very fatisfatory ex- periments of this kind, in a barrel with the large bore of 0.75 of.an inch. No. 1o.—was made with a comprefling force of only 3 cwt. A {mall eruption at the muzzle being obferved, water was thrown on the barrel: the pyrometer gave 21° : the chalk’ was. in a firm ftate of limeftone. No. 11.—witH 4 cwt. The barrel ftood without any erup- tion or exudation, till the heat rofe to 25°. There was a lofs of 3.6 per cent.: the refult was fuperiar, in hardnefs and tranf parency, to the laft, having fomewhat of a faline fradture. No. 12.—wira 5 cwt. The refult, with a lofs of 2.4 per cent., was of a quality fuperior to any of thofe lately obtained. THESE experiments appear to anfwer the end propofed, of af- certaining the leaft preflure, and loweft heat, in which lime- ftone can be formed. The refults, with various barrels of different fizes, agree tolerably, and tend to confirm’each other. The table fhews, when we compare numbers 1, 2, 8, 10, 11, 12, That a preflure of 52 atmofpheres, or 1700 en of fea, is ' capable | | MODIFIED ly COMPRESSION. 149 capable of forming a limeftone in a proper heat : That under 86 atmofpheres, anfwering nearly to 3000 feet, or about half a mile, a complete marble may be formed: and laftly, That with a preflure of 173 atmofpheres, or 5700 feet, that is, little more than one mile of fea, the carbonate of lime is made to undergo complete fufion, and to act powerfully on other earths. Vill. Formation of Coal.— Accidental occurrence which led me to undertake thefe Experiments.—Re/fults extracted from a former publication.—Explana- tion of fome difficulties that have been fuggefted.—The Fibres of Wood in. fome cafes obliterated, and in fome preferved under compreffion.—Re- femblance which thefe Refults bear to a feries of Natural Subftances de- Jeribed by Mr Harcuerr.—Thefe refults feem to throw light on the biftory of Surturbrand. As Ff intend, on: fome future occafion, to refume my ex- periments with inflammable fubftances, which I look upon as. far from complete, I fhall add but a few obfervations to what I have already laid before this Society, in the fketch I had the honour to read. inthis place on. the 30th of Auguft laft. Tue following incidental occurrence led me to enter upon this fubje& rather prematurely, fince I had determined firft to. fatisfy myfelf with regard. to the carbonate of lime. OBSERVING, in many of the laft-mentioned clafs.of experi- ments, that the elaftic matters made their efcape between the muzzle of the barrel and the cylinder of lead, I was in the ha- bit, as mentioned above, of placing a piece of leather between: the lead and the barrel; in which pofition, the heat to which the leather was expofed, was neceflarily below that of melting f lead. 150 EFFECTS of HEAT lead. In an experiment, made on the 28th November 1803, in _ order to afcertain the power of the machinery, andthe quantity of metal driven out by the expanfion of the liquid, there being nothing in the barrel but metal, I obferved, as foon as the com- prefling apparatus was removed, (which on this occafion was done while the lower part of the barrel was at its full heat, and the barrel ftanding brim full of liquid metal,) that all the leather which lay on the outfide of the circular muzzle of the barrel, remained, being only a little browned and crumpled by the heat to which it had been expofed. What leather lay within the circle, had difappeared; and, on the furface of the liquid metal, which ftood up to the lip of the barrel, I faw large drops, of a fhining black liquid, which, on cooling, fixed into-a crifp black fubftance, with a fhining fracture, exactly like pitch or pure coal. It burned, though not with flame. While hot, it fmelt decidedly of volatile alkali. The important circumftance here, is the different manner in which the heat has acted on the leather, without and within the rim of the barrel. The only difference confifted in compreffion, to which, therefore, the dif- ference of effect muft be afcribed: by its force, the volatile matter of the leather which efcaped from the outward parts, had within the rim, been conftrained to remain united to the reft of the compofition, upon which it had aéted as a flux, and the whole together had entered into a liquid ftate, in a very low heat. © Had the preflure been continued till all was cool, thefe fubftances muft have been retained, pro- ducing a real coal. | On the 24th April 1803, a piece of leather ufed in a fimilar manner, (the comprefling force being continued, however, till all was cold,) was changed to a fubftance like glue, owing doubtlefs to compreflion, in a heat under that of melting lead. Turse obfervations led me to make a feries of experi- ments with animal and vegetable fubftances, and with coal; the MODIFIED by COMPRESSION. BEE the refult of which I have already laid before the Society. I fhall now repeat that communication, as printed in Nicuot- son’s Fournal for October laft (1804). “ I wave likewife made fome experiments with coal, treated in the fame manner as the carbonate of lime: but I have found. it much lefs tractable; for the bitumen, when heat is applied to it, tends to efcape by its fimple elafticity, whereas the car- bonic acid in marble, is in part retained by the chemical force _ of quicklime. I fucceeded, however, in conftraining the bitu- minous matter of the coal, to a certain degree, in red heats, fo. as to bring the fubftance into a complete fufion, and to retain its faculty of burning with flame. But, I could not accomplifh this in heats capable of agglutinating the carbonate; for I have found, where I rammed them fucceflively into the fame tube, and where the vefiel has withftood the expanfive force, that the carbonate has been agglutinated into a good limeftone, but that the coal has loft about half its weight, together with its power of giving flame when burnt, remaining in a very compact ftate, with a fhining fracture. Although this experi- ment has not afforded the defired refult, it anfwers another purpofe admirably well. It is known, that where a bed of coal- is crofled by a dike of whinftone, the coal is found in a pecu- liar ftate in the immediate neighbourhood of the -whin: the fabftance in fuch places being incapable of giving flame, it is. diftinguifhed by the name of blind coal. Dr Hutton has ex- plained this fact, by fuppofing that the bituminous matter of the coal, has been driven by the local heat of whin, into places. of lefs intenfity, where it would probably be retained by diftil- lation. Yet the whole muft have been carried on under the action of a preflure capable of conftraining the carbonic acid. of the calcareous {par, which.occurs frequently in fuch rocks. ‘In the laft-mentioned experiment, we have a perfect reprefen- tation. 152 EFFECTS of HEAT tation of the natural fact; fince the coal has loft its petroleum, while the chalk in conta& with it has retained its carbonic acid. “ I HAVE made fomie experiments of the fame kind, with ve- getable and animal fubftances. I found their volatility much greater than that ef coal, and I was compelled, with them, to work in heats below rednefs; for, even in the loweft red-heat, they were apt to deftroy the apparatus. The animal fubftance I commonly ufed was horn, and the vegetable, faw-duft of fir. The horn was incomparably the moft fufible and volatile of thetwo. Ina very flight heat, it was converted into a yellow red fubftance, like oil, which penetrated the clay tubes through and through. In thefe experiments, I therefore made ufe of tubes of glafs. It was only after a confiderable portion of the fubftance had been feparated from the mafs, that the remainder affumed the clear black peculiar to coal. In this way I ob- tained coal, both from faw-duft and from horn, which yielded a bright flame in burning. . “ THE mixture of the two produced a fubftance having ex- aly the {mell of foot or coal-tar. Iam therefore ftrongly in- clined to believe, that animal fubftance, as well as vegetable, has contributed towards the formation of our bituminous ftrata. This feems to confirm an opinion, advanced by Mr Keir, which has been mentioned to me fince I made this exs periment. I conceive, that the coal which now remains in the world, is but a fmall portion of the organic matter originally depofited : the moft volatile parts have been driven off by the action of heat, before the temperature had rifen high enough to bring the furrounding fubftance into fufion, fo as to confine the elaftic fluids, and fubje@ them to compreffion. “ Tn feveral of thefe experiments, I found that, when the pref- fure was not great, when equal, for inftance, only to 80 at- mofpheres, that the horn employed was diflipated entirely, the glafs ~ . see » a MODIFIED by COMPRESSION. 153 glafs, tube which had contained it being left almoft clean: yet undoubtedly, if expofed to heat without compreflion, and pro- tected from the contact of the atmofphere, the horn would leave a cinder or coak behind it, of matter wholly devoid of volatility. Here, then, it would feem as if the moderate preflure, by keeping the elements of the fubftance together, had promoted the general volatility, without being ftrong enough to refift that expanfive force, and thus, that the whole had efcaped. This refult, which I fhould certainly not have forefeen in theory, may perhaps, account for the abfence of coal in fituations where its prefence might be expected on prin- ciples of general analogy.’’ SincE this publication, a very natural queftion has been put to me. When the inflammable fubftance has loft weight, or when the whole has been diflipated, in thefe experiments, what has become of the matter thus driven off ? I musT own, that to anfwer this queftion with perfed con- fidence, more experiments are required. But, in the courfe of practice, two circumftances have occurred as likely, in moft ca- - fes, to have occafioned the lofs alluded to. I found in thefe expe- riments, particularly with horn, that the chalk, both in powder and in lump, which was ufed to fill vacuities in the tubes, and to fix them in the cradle, was ftrongly impregnated with an oily or bituminous matter, giving to the fubftance the qua- lities of a ftinkftone. I conceive, that the moft volatile part of the horn has been conveyed to the chalk, partly in a ftate of vapour, and partly by boiling over the lips of the glafs tube; the whole having been evidently in a ftate of very thin fluidity. Having, in fome cafes, found the tube, which had been intro- duced full of horn, entirely empty after the experiment, I was induced, as above ftated, to conceive, that, under preflure, it had acquired a greater general volatility than it had in free- Vor, VI.—P. I. U | dom ; E54 - MO RRBECTS Of BRaT _ dom; and I find that, in the open fire; horn yields a charcoal equal to 20 per cent. of the original weight. But more expe- -— ximents muft be made on this fubjeé. } Anoruer caufe of the lofs of weight, lay undoubtedly in the excefs of heat employed in moft of them, to remove the cradle from the barrel. With inflammable fubftances, no air-tube was ufed, and the heats being low, the air lodged in inter- flices had been fufficient to fecure the barrels from deftruction, by the expanfion of the liquid metal. In this view, likewife, I often ufed lead, whofe expanfion in fuch low heats, I expected to be lefs than that of the fufible metal. And the lead requir- ing to melt it, a heat very near to that of rednefs, the fubje@? of experiment was thus, on removing the cradle, expofed in freedom to a temperature which was comparatively high. But, obferving that a great lofs was thus occafioned, I returned to the ufe of the fufible metal, together with my former method | of melting it, by plunging the barrel, when removed from the furnace, into a folution of muriate of lime, by which it could only receive a heat of 250° of FAHRENHEIT. | Tue effect was remarkable, in the few experiments via in this way. The horn did not, as in the other experiments, change to a hard black fubftance, but acquired a femifluid and vifcid confiftency, with a yellow-red colour, and a very offen-- five {mell. This fhews, that the fubftances which here: occa- fioned both the colour and fmell of the refults, had been: dri- ven off in the other experiments, by the too great heat applied. to the fubftance, when free from compreffion. I rounp that the organization of animal fubftance was aan tirely obliterated by a flight action of heat, but that a ftronger heat was required to perform the entire fufion of vegetable matter. This, however, was accomplifhed ; and in feveral | experiments, pieces of wood were changed to a jet-black and inflammable fubftance, generally very porous, in which no trace MODIFIED ty COMPRESSION. 155 trace could be difcovered of the original organization. In others, the vegetable fibres were ftill vifible, and are forced afun- der by large and fhining air-bubbles. Since the publication of the fketch of my experiments, I have had the pleafure to read Mr Hatcuertt’s very interefting account of various natural fubftances, nearly allied to coal ; and I could not help being ftruck with the refemblance which my refults bear to them, through all their varieties, as brought into view by that able chemift; that refemblance affording a prefumption, that the changes which, with true fcientific mo- defty; he afcribes to an unknown caufe, may have refulted from various heats acting under preflure of various force. The fubftance to which he has given the name of Retina/phaltum, feems to agree very nearly with what I have obtained from animal fubftance, when the barrel was opened by means of low heat. And the fpecimen of wood entering into fufion, but ftill retaining the form of its fibres, feems very fimilar to the in- termediate fubftance of Bovey-coal and Surturbrand, which Mr Hatcuett has aflimilated to each other. It is well known, that the furturbrand of Iceland, confifts of the ftems of large - trees, flattened to thin plates, by fome operation of nature hi- therto unexplained. But the laft-mentioned experiment feems to afford a plaufible folution of this puzzling phenomenon. In all parts of the globe, we find proofs of flips, and various relative motions, having taken place amongft great mafles of rock, whilft they were foft in a certain degree, and which _ have left unequivocal traces behind them, both in the derange- ment of the beds of ftrata, and in a fmooth and fhining fur- face, called lickenfide, produced by the direct friction of one mafs on another. During the adtion of fubterranean heat, were a fingle ftratum to occur, containing trees intermixed with animal fubftances, fhell-fith, &c. thefe trees would be reduced, to a foft and unctuous ftate, fimilar to that of the piece of wood 2. in oan EFFECTS of HEAT _ in the Jaft-mentioned experiment, whilft the fubftance of the - contiguous ftrata retained a confiderable degree of firmnefs. In _ - this ftate of things, the ftratum juft mentioned, would very na- turally become the fcene of a flip, occafioned by the unequal preflure of the furrounding mafles. By fuch a fliding motion, accompanied by great compreflion, a tree would be flattened, as any fubftance is ground in a mortar, by the combination of a lateral and direct force. At the fame time, the fhells along with the trees, would be flattened, like thofe defcribed by BErG- MAN; while thofe of the fame fpecies in the neighbouring limeftone-rock, being protected by its inferior fufibility, would retain their natural fhape. IX. Application of the foregoing refults to Geology.—The fire employed in the: _Huttonian Theory. is a modification of that of the Volcanoes.—This mo- dification muft take place in a lava previous to its eruption.—An Inter- nal Lava is capable of melting Limeftone.—The effects of Volcanic Fire on fubftances in a fubterranean and fubmarine fituation, are the fame as thofe afcribed to Fire in the Huttonian Theory.—Our Strata were once in a fimilar fituation, and then underwent the action of fire.—All the conditions of the Huttonian Theory being thus combined, the formation of all Rocks may be accounted for ina fatisfactory manner.—Concli- 10M. Havine inveftigated, by means of the foregoing experi- - ments, fome of the chemical fuppofitions. involved in the Hut- tonian Theory, and having endeavoured to affign a determi- nate limit to the power of the agents: employed ; I fhall now apply thefe refults to Geology, and inquire how far the events fuppofed MODIFIED by COMPRESSION. 157- fuppofed anciently to have taken place, accord with the exift- ing ftate of our globe. Tue moft powerful and eflential agent of the itont oe Theory, is Fire, which I have always looked upon as the fame | with that of volcanoes, modified by circumftances which muft, toa certain degree, take place in emince lava previous to its eruption. Tue original fource of internal fire is involved in great ob- {curity ; and no fufficient reafon occurs to me for deciding whether it proceeds by emanation from fome vaft central re- fervoir, or is generated by the local operation of fome chemi- cal procefs. Nor is there any neceflity for fuch a decifion = all we need to know is, that internal fire exifts, which no one can doubt, who believes in the eruptions of Mount Vefuvius. To require that a man fhould account for the generation of internal fire, before he is allowed to employ it in geology, is no lefs abfurd than it would be to prevent him from reafoning about the conftruction of a telefcope, till he could explain the nature of the fun, or account for the generation of light *. But while we remain in fufpenfe as to the prime caufe of this tremendous agent, many circumftances of importance with regard to it, may fairly become the fubjects of obfervation and difcuffion. Some authors (I conceive through ignorance of the facts) have alleged, that the fire of Atna and Vefuvius is merely fu- _perficial. But the depth of its action is fufficiently proved, by the great diftance to which the eruptive percuflions are felt, and ftill more, by the fubftances thrown out uninjured by fome eruptions _ * Tuts topic, however, has of late been much urged againft us, and an unfair advantage has been taken of what Mr PLayrair has faid upon it. What he gave as mere conjecture on a fubject of collateral importance, has been argued upon as: the bafis and fundamental doftrine of the fyftem. 158 - EFFECTS of HEAT eruptions of Mount Vefuvius. Some of thefe, as marble and gyp- fum, are incapable in freedom of refifting the action of fire. We shave likewife granite, fchiftus, gneifs, and ftones of every known -clafs, befides many which have never, on any other occafion, been found at the furface of our globe. The circumftance of thefe fubftances -having been thrown out, unaffected by ‘the fire, proves, that it has proceeded from a fource, not only as deep, but deeper, than their native beds; and as they exhibit f{pecimens of every clafs of minerals, the formation of which we pretend to explain, we need inquire no further into the ‘depth of the Vefuvian fire, which has thus been proved to reach below the range of our {peculationis. ) Votcanice fire is fubject to perpetual and irregular varia- tions of intenfity, and to fudden and violent renewal, after long periods of abfolute ceffation. Thefe variations and inter- miffions, are likewife effential attributes of fire as employed by Dr Hutton ; for fome geological fcenes prove, that the indura- ting caufe has acted repeatedly on the fame fubftance, and that, during the intervals of that action, it had ceafed entirely. This circumftance affords a complete anfwer to an argument lately urged againft the Huttonian Theory, founded on the wafte of heat which muft have taken place, as it is alleged, © through the furface. For if, after abfolute ceflation, a power of renewal exifts in nature, the idea of wafte by continuance is quite inapplicable. jak ks THE external phenomena of volcanoes are fufficiently well known ; but our fubject leads us to inquire into their internal actions. This we are enabled to do by means of the foregoing experiments, in fo far as the carbonate of lime is concerned. SoME experiments which I formerly * laid before this So- ciety and the public, combined with thofe mentioned in this paper, * Edinburgh TranfaGions, Vol. V. Part I. p.60—66. MODIFIED ty COMPRESSION: - ‘159. paper, prove, that the feebleft exertions of volcanic fire, are of fufiicient intenfity to perform the agglutination, and even the. entire fufion, of the carbonate of lime, when its. carbonic acid is effectually confined by preflure; for though lava, after its. fufion, may be made, in our experiments, to congeal into a - glafs, in a temperature of 16° or 18° of WEDGWwooD, in which _ temperature the carbonate would {carcely be affected ; it muft. be obferved, that a fimilar. congelation is not to be looked for in nature ; for the mafs, even of the {malleft ftream of lava, is. too great to admit of fuch rapid cooling. And, in fact, the external part of a lava is not vitreous, but confifts of a fub-- ftance which, as my experiments have. proved, muft have been. congealed in a heat of melting filver, that is, in 22° of WEDG- woop ; while its internal parts bear a character indicating that they congealed in 27° or 28° of the fame fcale. It follows, that no part of the lava, while it remained liquid, can have been lefs hot than 22° of WepGwoop. Now, this happens to be a heat, in which I have accomplifhed the entire fufion of the carbonate of lime, under preffure. We muft therefore conclude, that the heat of a running lava is always of fuffi-. cient intenfity to perform the fufion of limeftone. In every active volcano, a communication muft exift between. the fummit of the mountain and the unexplored region, far below its bafe, where the lava has been melted, and whence it has been propelled upwards; the liquid lava rifing through this internal channel, fo-as to fill the crater to the brim, and flow. over it. On this occafion, the fides of the mountain muft un- dergo a violent: hydroftatical preflure outwards, to which they often yield by the formation of a vaft rent, through which the. Jaya is difcharged in a lateral eruption, and flows in a continued. fiream fometimes during months. On Aitna moft of the erup- tions are fo performed; few lavas flowing from the fummit, | but generally Rene out. laterally, at. very. elevated. ftations.. : At. map. EFFECTS of HEAT At the place. of delivery, a quantity of gafeous matter is pro- pelled violently upwards, and, along with it, fome liquid lava ; which laft, falling back again in a {pongy ftate, produces one of thofe conical hills which we fee in great number on the vaft fides of Mount Etna, each indicating the difcharge of a parti- cular eruption. At the fame time, a jet of flame and {moke iffues from the main crater, proving the internal communication be- tween it and the lava; this difcharge from the fummit gene- — . rally continuing, in a greater or a lefs degree, during the in- tervals between eruptions. (Fig. 41. reprefents an ideal fection of Mount Etna; a2 is the direct channel, and dc is a lateral branch)en: » Ler us now attend to the ftate of the lava within the moun- tain, during the courfe of the eruption ; and let us fuppofe, that a fragment of limeftone, torn from fome ftratum below, has been included in the fluid lava, and carried up with it. By the laws of hydroftatics, as each portion of this fluid fuftains pref- fure in proportion to its perpendicular diftance below the point of difcharge, that preflure muft increafe with the depth. The fpecific gravity of folid and compadt lava is nearly 2.8 ; and its weight, when in a liquid ftate, is probably little different. The table fhews, that the carbonic acid of limeftone cannot be conftrained in heat by a preflure lefs than that of 1708 feet of - fea, which correfponds nearly to 600 feet of liquid lava. As foon, then, as our calcareous mafs rofe to within 600 feet of the furface, its carbonic acid would quit the lime, and, affuming a vafeous form, would add to the eruptive effervefcence. And this change would commonly begin in much greater depths, in confequence of the bubbles of carbonic acid, and other fubftan- ces in a gafeous form, which, rifing with the lava, and through it, would greatly diminifh the weight of the column, and would render its preflure on any particular fpot extremely variable. | With all thefe irregularities, however, and interruptions, the preflure MODIFIED ty COMPRESSION. 161 preflure would in all cafes, efpecially where the depth was confi- derable, far furpafs what it would have been under an equal depth of water. Where the depth of the ftream, below its point of delivery, amounted, then, to 1708 feet, the preflure, if the heat was not of exceffive intenfity, would be more than fuf- ficient to conftrain the carbonic acid, and our limeftone would fuffer no calcination, but would enter into fufion; and if the eruption ceafed at that moment, would cryftallize in cooling along with the lava, and become a nodule of calcareous fpar. The mafs of lava, containing this nodule, would then conftitute a real whinftone, and would belong to the kind ‘called amygda- loid. In greater depths ftill, the preflure would be propor- tionally increafed, till fulphur, and even water, might be con- {trained ; and the carbonate of lime would continue undecom- pofed in the higheft heats. Ir, while the lava was in a liquid ftate, during the eruption or previous to it, a new rent (dé, fig. 41.), formed in the folid country below the volcano, was met by our ftream (at d@), it is obvious that the lava would flow into the aperture with great rapidity, and fill it to the minuteft extremity, there being no air to impede the progrefs of the liquid. In this manner, a ftream of lava might be led from below to approach the bot- tom of the fea (f/), and to come in contac with a bed of loofe fhells (¢¢), lying on that bottom, but covered with — beds of clay, interftratified, as ufually occurs, with beds of fand, and other beds of fhells. The firft effe@ of heat would be to drive off the moifture of the loweft fhell-bed, in a ftate of va- pour, which, rifing till it got beyond the reach of the heat, would be condenfed into water, producing a flight motion of ebullition, like that of a veflel of water, when it begins to boil, and when it is faid to fimmer. The beds of clay and fand might thus undergo fome heaving and partial derangement, but would ftill poffefs the power of yee or of very much im- Vou. VI.—P. I. peding, 162 Vo “EFFECTS of HEAT peding, the defcent of water fromthe fea above; fo that the water which had been driven from the fhells at the bottoni, would not return to them, or would return but flowly ; and they would be expofed' dry»to the:action of heat *. In this cafe, one of two,things would inevitably ar sea Ei- ther the carbonic acid of the fhells would »beidriven off by the heat, producing an incondenfable elaftic fluid,-which, heaving up or penetrating the fuperincumbent beds, would. force its way to the furface of the fea, and produce a fubmarine erup- tion, as has happened at Santorini and elfewhere ;, or the vo- latility of the carbonic acid would be reprefled by the weight of the fuperincumbent water (4), and the fhell-bed, being fof- tened or fufed by the action of heat, would be converted. inte a ftratum of limeftone. Tue foregoing experiments enable us to dsdideri in any parti- cular:cafe, which of thefe two events muft, take place, when the heat of the lava and:the depth of the fea are known. - Tue table fhews, that under,a fea no, deeper than 1708 feet, near one-third of a mile,.a limeftone would be formed by proper heat; and that, in,a depth, of little more than one mile, it would enter into,entire fufion.. Now, the .com- mon foundings of mariners extend to 200 fathoms, or 1200 feet. Lord. Muierave + found bottom at 4680 feet, or nearly nine-tenths of a mile; and Captain Exxis let down a fea-gage to the depth of 5346 feet,{...It thus, appears; | | Sha * Tus fituation of things, is fimilar to what ‘haypens when {mall- Pal is moi- -ftened, in order to make it cake. The dutt, drenched with water, is laid. upon the fire, and remains long See while the heat below fuffers little or no abatement, | t Voyage jeg the North Pole, p- 142. 4 Pbilofophical Tranfaciial, 1751, pe 2t2.- 7 ———. Fra MODIFIED by COMPRESSION. 163 that at the bottom of a fea, which would be founded by a line much lefs than double of the ufual length, and lefs than half the depth of that founded by Lord Mutcrave, lime- ftone might be: formed by‘ heat ; and that, at the depth ‘reach- ed by Captain Exuis, the entire fufion would be accomplith- ed; if the bed of thells were touched by a lava:at the extre- mity of its courfe, when its heat was loweft. Were the heat of the lava greater, a greater depth of fea would, of courfe, be requifite to conftrain' the carbonic acid effectually; and future experiments may determine what depth is required to co-ope- rate with any given temperature. It is enough for our prefent purpofe to have fhewn, that the refult is poffible in any cafe,’ and to have circumfcribed the neceflary force of thefe agents within moderate limits. - Atthe fame time it muft be obferved, that we have been far from ftretching the known facts; for when: we compare the fmall extent of fea in which any foundings can be found, with that of the vaft unfathomed ocean, it is ob- vious, that in afluming a depth of one mile or two, we fall very {hort of the medium. -M. pz ta Puracez, reafoning from the phenomena of the tides, ftates it as highly probable that this medium is not lefs than eleven Englifh miles *. Ira great part or the whole of the fuperincumbent mafs confifted, not of water, but of fand or clay, then the depth re- quifite to produce thefe effects would be leffened, in the inverfe ratio of the {pecific gravity. If the above-mentioned oc- currence took place under a ma{s compofed of ftone firmly bound together by fome previous operation of nature, the power of the fuperincumbent mafs, in oppofing the efcape of K'2 carbonic * “ On peut donc regarder au moins comme trés probable, que la profondeur * moyenne de la mer n’eft pas au-deffous de quatre lieues.”. De 1a Prac, Hiift. de l’_Acad. Roy. des Sciences, année 1776, 464 | EFFECTS of HEAT carbonic acid} would be very much increafed by that union and by the ftiffmefs or tenacity of the fubftance. We have feen numberlefs examples of this power in the courfe of thefe experiments, in which barrels, both of iron and: porcelain, whofe thicknefs did not exceed one-fourth of an inch, have exerted a force fuperior to the mere weight of a mile of fea. Without fuppofing that the fubftance of a rock could in any cafe ad with the fame advantage as that of a uniform and con- nected barrel ; it feems obvious that a fimilar power muft, in many cafes, have been exerted to a certain degree. We know of many calcareous maffes which, at this mo- ment, are expofed to a preflure more than fufficient to accom- plith their entire fufion. The mountain of Saleve, near Geneva, is 500 French fathoms, or nearly 3250 Englifh feet, in height, from its bafe to its fummit. Its mafs confifts of beds, lying nearly horizontal, of limeftone filled with fhells. Independent- ly, then, of the tenacity of the mafs, and taking into account its mere weight, the loweft bed of this mountain, muft, at this moment, fuftain a preflure of 3250 feet of limeftone, the fpecific gravity of which is about 2.65. This preflure, therefore, is equal to that of 8612 feet of water, being nearly a mile and a half of fea, which is much more than adequate, as we have fhewn, to accomplifh the entire fufion of the carbonate, on. the appli- cation of proper heat. Now, were an emanation from a volcano, to rife up under Saleve, and to penetrate upwards to its bafe, and ftop there; the limeftone to which the lava approached, would inevitably be foftened, without being cal- cined, and, as the heat retired, would cryftallize into a faline marble. Some other circumftances, relating to this fubject, are very deferving of notice, and enable us ftill further to compare the ancient and modern operations of fire. It MODIFIED ly COMPRESSION. 165 Ir appears, at firft fight, that a lava having once pene- trated the fide of a mountain, all fubfequent lavas fhould conti- nue, as water would infallibly do, to flow through, the fame aperture. But there is a material difference in the two cafes. As foon as the lava has ceafed to flow, and the heat has begun to abate, the crevice through which the lava had been pafling, remains filled with a fubftance, which foon agglutinates in- ‘to a mafs, far harder and firmer than the mountain itfelf. This mafs, lying in a crooked bed, and being firmly welded to the fides of the crevice, muft oppofe a moft powerful refiftance to any ftream tending to purfue the fame courfe. The injury done to the mountain by the formation of the rent, will thus be much more than repaired; and in a fubfequent eruption, the lava muft force its way through another part of the moun- tain or through fome part of the adjoining country. The action of heat from below, feems in moft cafes to have kept a channel open through the axis of the mountain, as appears by the fmoke and flame which is habitually difcharged at the faummit during intervals of calm. On many occafions, how- ever, this {piracle feems to have been entirely clofed by the confolidation of the lava, fo as to fupprefs all emiflion. This happened to Vefuvius during the middle ages. All appearance of fire had ceafed for five hundred years, and the crater was covered with a foreft of ancient oaks, when the volcano open- ed with frefh vigour in the fixteenth century. THE eruptive force, capable of overcoming fuch an ob- ftacle, muft be tremendous indeed, and feems in fome cafes to have blown the volcano itfelf almoft to pieces. It is im- poffible to fee the Mountain of Somma, which, in the form of a erefcent, embraces Mount Vefuvius, without being convinced that it is a fragment of a large volcano, nearly concentric with 166 EFFECTS of HEAT with the ‘prefent inner cone, which, in fome:great eruption, . had been deftroyed all but this fragment. In our own times, an event of no {mall magnitude has taken: place on the fame {pot ; the inner cone of Vefuvius having undergone fo great a change during the eruption in 1794, that it now bears no re- femblance to what it was when I faw it in 1785. ‘Tue general or partial ftagnation of the internal lavas at the clofe of each eruption feems, then, to render it neceflary, that in every new difcharge, the lava fhould begin by ma- king a violent laceration. And this is probably the caufe of thofe tremendous earthquakes. which precede all great erup- tions, and which ceafe as foon as the lava has found a vent. It feems but reafonable to afcribe like effects to like caufeés, and _to believe that the earthquakes which frequently defolate coun- tries not externally volcanic, likewife indicate the protrufion from below of matter in liquid fufion, penetrating the mafs of rock. - Tue injection of a whinftone-dike into a frail mafs of thale and fandftone, muft have produced the fame effects upon it that the lava has juft been ftated to produce on the loofe beds of volcanic {coria. One ftream of liquid whin, having flowed into fuch an aflemblage, muft have given it great additional weight and ftrength : fo thata fecond ftream coming like the firft, would be oppofed by a mafs, the laceration of which would produce an earthquake, if it:were overcome ; or by which, if it refifted, the liquid matter would be compelled to penetrate fome weaker mafs, perhaps at a great diftance from the firft. , The internal fire being thus compelled perpetually to change the {cene, of its action, its influence might be carried to an indefinite, extent : So that the intermittance in point of time, as well as the verfa- tility in point of place, already remarked as common to the Huttonian and Volcanic fires, are accounted for on our princi- ples. a ee a MODIFIED by COMPRESSION. 167 ples “And:it‘thus appears, that whinftone poflefles all the pro- perties which we are led by theory. to afcribe to an internal lavaeiii ls: ti apne Tuts conection ‘1s argh iMoiiasad By an izitenmddisite cafe between the refults of external and internal-fire, difplayed in-anadual fection of the ancient part of Vefuvius, which oc- curs in the Mountain of Somma mentioned above. I formerly defcribed this fcene in my paper on Whioftone and Lava; and I muft beg leave once more to prefs it upon the notice of the public, as affording to future travellers:a) moft interefting field of geological inquiry. ? ‘Tue fection is feen in the bode vertical cliff, feveral hundred feet im height, which Somma prefents to the view from the littlé-valley, in form of a crefcent, which lies. between Somma and the interior cone of Vefuvius, called the Atrio del Cavallo. (Figs42. reprefents /this ‘fcene, done from the recollection of whatrl: faw in-1785- abc is the interior cone of Vefuvius; af g the mountain of Somma; and ¢de the Atrio del Cavallo). By means of this cliff. (fd in figure 42. and which is repre- fented ‘feparately in fig. 44:),; we fee the internal ftructure of the mountain, ‘compofed of thick beds (&) of loofe fcoria, which have fallen in fhowers ; hetween which thin but firm ftreams:(m m) of lava are interpofed, which have flowed down the outward conical fides of the mountain. (Fig: 43. is an ideal féGion-of Vefuvius and Somma, through the axis of the cones, fhewing: the manner in which the beds of fcoria and of lava lie upon each other; the extremities of which beds are feen edgewifein the, cliff at mm and kk, fig. 42, 43, and’ 44.). ‘Tuts aflemblage of {coria and lava is traverfed abruptly and wertically J by !ftreams of folid lava (am, fig. 44.), reaching from top to bottom of the cliff. Thefe laft I conceive to have flowed in rents of the ancient mountain, which rents had acted as 168 EFFECTS of HEAT as pipes through which the lavas of the lateral eruptions were conveyed to the open air. This fcene prefents to the view ~ of an attentive obferver, a real fpecimen of thofe internal ftreams which we have juft been confidering in f{peculation, and they may exhibit circumftances decifive of the opinions here ad- vanced. For, if one of thefe ftreams had formerly been con- nected with a lateral eruption, difcharged at more than 600 feet above the Atrio del Cavallo, it might poflibly contain the carbonate of lime. But could we fuppofe that depth to extend to 1708 feet, the interference of air-bubbles, and the action of a ftronger heat than was merely required for the fufion of the carbonate, might have been overcome. Peruaps the height of Vefuvius has never been great enough for this purpofe. But could we fuppofe Aitna to be cleft in two, and its ftructure difplayed, as that of Vefuvius has juft been defcribed, there can be no doubt that internal ftreams of lava would be laid open, in which the preflure muft have far exceeded the force required to conftrain the carbonic acid of limeftone ; fince that mountain occafionally delivers lavas from its fummit, placed 10.954 feet above the level of the Mediterra- nean *, which wathes its bafe. Irecolle& having feen, in fome parts of Aitna, vaft chafms and crags, formed by volcanic re- volutions, in which vertical ftreams of lava, fimilar to thofe of Somma, were apparent. But my attention not having been turned to that objec till many years afterwards, I have only now to recommend the inveftigation of this interefting point to future travellers. Wuar lias been faid of the heat conveyed by internal volca- nic ftreams, applies equally to that deeper and more general heat by which the lavas themfelves are melted and propelled upwards. * Phil. Tranf. 1977, p. $95. MODIFIED by COMPRESSION. _ X69 upwards. That they have been really fo propelled, from a : ; . great internal mafs of matter, in liquid fufion, feems to admit of no doubt, to whatever caufe we afcribe the heat of volcanoes. It is no lefs obvious, that the temperature of that liquid muft be of far greater intenfity than the lavas, flowing from it, can re- tain when they reach the furface. Independently of any actual eruption, the body of heat contained in this vaft mafs of liquid, muft diffufe itfelf through the furrounding fubftances, the in- tenfity of the heat being diminifhed by flow gradations, in pro- portion to the diftance to which it penetrates. When, by means of this progreffive diffufion, the heat has reached an aflemblage of loofe marine depofites, fubject to the preflure of a great fu- perincumbent weight, the whole muft be agglutinated into a mafs, the folidity of which will vary with the chemical com- pofition of the fubftance, and with the degree of heat to which each particular fpot has thus been expofed. At the fame time, analogy leads us to fuppofe, that this deep and extenfive heat muft be fubject to viciflitudes and intermiflions, like the exter- nal phenomena of volcanoes. We have endeavoured to explain fome of thefe irregularities, and a fimilar reafoning may be ex- tended to the prefent cafe. Having fhewn, that fmall in- ternal ftreams of lava tend fucceflively to pervade every weak part of a volcanic mountain, we are led to conceive, that the great mafles of heated matter juft mentioned, will be fucceffive- ly directed to different parts of the earth; fo that every loofe afflemblage of matter, lying in a fubmarine and fubterranean fituation, will, in its turn, be affected by the indurating caufe ; and the influence of internal volcanic heat will thus be cir- cumferibed within no limits but thofe of the globe itfelf. A sERIES of undoubted facts prove, that all our ftrata once day ina fituation fimilar in all refpects to that in which the _ marine depofites juft mentioned have been fuppofed to lie. Tue inhabitant of an unbroken plain, or of a country form- ed of horizontal ftrata, whofe obfervations have been confi- Vert. Vi.—P. I. y ned 170 EFFECTS of HEAT ned to his native fpot, can form no idea of thofe truths, which at every ftep in an alpine diftri@ force themfelves on the mind of a geological obferver. Unfortunately for the progrefs of geology, both London and Paris, are pla- ced in countries of little intereft; and thofe fcenes by which the principles of this fcience are brought into view in the moft ftriking manner, are unknown to many perfons beft capable of appreciating their value. The moft important, and at the fame time, the moft aftonifhing truth which we — learn by any geological obfervations, is, that rocks and moun- tains now placed at an elevation of more than two miles above the level of the fea, muft at one period have lain at its bottom. This is undoubtedly true of thofe ftrata of lime- ftone which contain fhells ; and the fame conclufion muft be ex- tended to the circumjacent ftrata. The imagination ftruggles againft the admiflion of fo violent a pofition; but muft yield to the force of unqueftionable evidence ; and it is proved by the example of the moft eminent and cautious s obfervers, that the conclufion is inevitable *. ANOTHER queftion here occurs, which has been well treat- ed by Mr Pravratr. Has the fea retreated from the moun- tains? or have they rifen out of the fea? He has fhewn, that the balance of probability is incomparably in favour of the latter fuppofition ; fince, in order to maintain the former, we mutt difpofe of an enormous mafs of fea, whofe depth is feveral miles, and whofe bafe is greater than the furface of the whole fea. Whereas the elevation of a continent out of a fea like ours, would not change its level above a few feet; and even were a great derangement thus occa-- fioned, * Saussure, Voyages dans les Alpes,tom. ii. p. 99.—104. . MODIFIED by COMPRESSION. ryt fioned, the water would eafily find its level without the af- fiftance of any extraordinary fuppofition. The elevation of the land, too, is evinced by what has occafionally happened in volcanic regions, and affords a complete folution of the con- tortion and erection of ftrata, which are almoft univerfally ad- mitted to have once lain in a plane and horizontal pofition. WuaATEVER opinion be adopted as to the mode in which the land and the water have been feparated, no one doubts of the ancient fubmarine fituation of the ftrata. AN important feries of facts proves, that they were likewife fub- terranean. Every thing indicates that a great quantity of matter has been removed from what now conftitutes the furface of our globe, and enormous depofites of loofe fragments, evidently de- tached from mafles fimilar to our common rock, evince the action of fome very powerful agent of deftrution. Analogy too, leads us to believe, that all the primary rocks have once been covered with fecondary ; yet, in vaft diftrids, no fecondary rock ap- pears. In fhort, geologifts feem to agree in admitting the ge- neral pofition, that very great changes of this kind have taken place in the folid furface of the globe, however much they may differ as to their amount, and as to their caufes. Dr Hotton afcribed thefe changes to the action, during very long time, of thofe agents, which at this day continue flowly to corrode the furface of the earth; frofts, rains, the or- dinary floods of rivers, &c. which he conceives to have acted always with the fame force, and no more. But to this opinion I could never fubfcribe, having early adopted that of Saus- SURE, in which he is joined by many of the continental geologifts. My conviction was founded upon the infpection of thofe facts in the neighbourhood of Geneva, which he has adduced in fupport of his opinion. I was then convinced, we and 172 EFFECTS of HEAT and I ftill believe, that vaft torrents, of depth fufficient to over- top our mountains, have {wept along the furface of the earth, excavating vallies, undermining mountains, and carrying away whatever was unable to refift fuch powerful corrofion. If fuch agents have been at work in the Alps, it is difficult to conceive that our countries fhould have been fpared. I made it therefore my bufinefs to fearch for traces of fimilar operations here. I was. not long in difcovering fuch in great abundance ; and, with the help of feveral of my friends, I have traced the indications of yaft torrents in this neighbourhood, as obvious as thofe I formerly faw on Saleve and Jura. Since I announced my opi- nion on this fubject, in a note fubjoined to my paper on Whin- ftone and Lava, publifhed in the fifth volume of the 7ran/- actions of this Society, I have met with many confirmations. of thefe views. The moft important of thefe are derived from the teftimony of my friend Lord SeLxirx, who has lately met with a feries of fimilar facts in North America. Ir would be difficult to compute the effects of fuch an agent ; but if, by means of it, or of any other caufe, the whole mafs. of fecondary ftrata, in great tracts of country, has been remo- ved from above the primary, the weight of that mafs alone muft have been fufficient to fulfil all the conditions of the Huttonian Theory, without having recourfe to the preflure of the fea. But when the two preflures were combined, how great muft have been their united ftrength! We are authorifed to fuppofe, that the materials of our ftrgta,. in this fituation, underwent the action of fire. For volcanoes have burnt long before the earlieft times recorded in hiftory, as. appears by the magnitude of fome volcanic mountains; and it can fcarcely be doubted, that their fire has acted without any material ceflation ever fince the furface of our globe acquired its. prefent. MODIFIED ly COMPRESSION. 173 prefent form. In extending that fame influence to periods of ftill higher antiquity, when our ftrata lay at the bottom of the fea, we do no more than afcribe permanence to the exifting laws of nature. | Tue combination of heat and compreffion refulting from. thefe circumftances, carries us to the full extent of the Hutto- nian Theory, and enables us, upon its principles, to account for the igneous formation of all rocks from. loofe marine depo- fites. Tue fand would thus be changed to fandftone ; the fhells to: fimeftone ; and the animal and vegetable fubftances to coal. Oruer beds, confifting of a mixture of various fubftances, would be ftill more affected by the fame heat. Such as con- tained iron, carbonate of lime, and alkali, together with a mix- ture of various earths, would enter into thin fufion, and, pe- netrating through every crevice that occurred, would, in fome cafes, reach what was then the furface of the earth, and con- ftitute lava: in other cafes, it would congeal in the internal rents, and conftitute porphyry, bafalt, greenftone, or any other of that numerous clafs of fubftances, which we comprehend under the name of whinffone. At the fame time, beds of fimi-- lar quality, but of compofition fomewhat lefs fufible, would. enter into a ftate of vifcidity, fuch as many bodies pafs. through in their progrefs towards fufion. In this ftate, the particles, though far from pofleffing the fame freedom as in a. liquid, are fufceptible of cryftalline arrangement * ; and the fubftance * Tus ftate of vifcidity, with its numberlefs modifications, is deferving of. great attention, fince it affords a folution of fome of the moft important geologi- cal gueftions. The mechanical power exerted by fome fubftances, in the aét of. affuming a cryfialline form, is well known. I have feen a fet of large and broad cryftals 174 EFFECTS of HEAT fubftance, which, in this fluggifh ftate, would be little difpofed to move, being confined in its original fituation by contiguous beds of more refractory matter, would cryftallize, without undergoing any change of place, and conftitute one of thofe beds of whinftone, which frequently occur interftratified with fandftone and limeftone. In other cafes where the heat was more intenfe, the beds of fand, approaching more nearly to a ftate of fufion, would ac- quire fuch tenacity and toughnefs, as to allow themfelves to be bent and contorted, without laceration or fra@ture, by the in- fluence of local motions, and might affume the fhape and’ character of primary {chiftus: the limeftone would be highly cryftallized, and would become marble, or, entering into thin fufion, would penetrate the minuteft rents in the form of cal- careous fpar. Laftly, when the heat was higher ftill, the fand itfelf would be entirely melted, and might be converted, by the fubfequent effects of flow cooling, into granite, fienite, &c. ; in fome cafes, retaining traces of its original ftratification, and conftituting gneifs and ftratified granite ; in others, flowing into the crevices, and forming veins of perfect granite. In confequence of the action of heat, upon fo great a quan- tity of matter, thus brought into a fluid or femifluid ftate, and in which, notwithftanding the great preflure, fome fubftances would be volatilized, a powerful heaving of the fuperincumbent mafs muft have taken place; which, by repeated efforts, fuc- ceeding cryftals of ice, like the blade of a knife, formed ina mafs of clay, of fuch ftiffnefs, that it had juft been ufed to make cups for chemical purpofes. In many of my former experiments, I found that a fragment of glafs made from whinftone or lava, when placed in a muffle heated to the melting point of filver, affumed a cryftalline arrangement, and underwent a complete change of character. During this change, it became foft, fo as to yield to the touch of an iron rod; yet retained {uch ftiffnefs, that, lying untouched in the muffle, it preferved its fhape entirely ; the fharp angles of its fracture not being in the leaft blunted. EEE MODIFIED ly COMPRESSION. 175 ceeding each other from below, would at laft elevate the ftrata into their prefent fituation. Tue Huttonian Theory embraces fo wide a field, and com- prehends the laws of fo many powerful agents, exerting their influence in circumftances and in combinations hitherto un- tried, that many of its branches muft ftill remain in an unfi- nifhed ftate, and may long be expofed to partial and plaufible objections, after we are fatisfied with regard to its fundamen- tal doctrines. In the mean time I truft, that the objet of our purfuit has been accomplithed, in a fatisfa€tory manner, by the fufion of limeftone under preffure. This fingle refult af- fords, I conceive, a {trong prefumption in favour of the folu- tion which Dr Hutton has advanced of all the geological phe- nomena; for, the truth of the moft doubtful principle which he has aflumed, has thus been eftablifhed by dire experi- ment. APPEN- ey a ie Sotlnintarho Seon Py A bts pe) Cae a cr yy tise nigh? : ees Ee hie pox Es oe Si i noe soght RES AE soe ae “ os shoe: Sa ras : on fi % “a Me ar" % A Be oe oy $ Ar. aL a 4 eds : er? ete 42) af, ( : re nigh Ss ss a ‘ vbr “I t i wily oa Ten co ees by a , _. ¥ 4 roi > pF Oa ee VAMP eet Ls ewe “¢ . M b A ,. or + 7 4 ‘ . =e ian att tS ie Ur I pig eit ' i. a Py aie) i - si a bi Bian - ° a 1 MF er Ee, ee} a ae 7 kA \ Bry 9 : y ‘4 4 é “ La 2 Eg s ~~ a ’ Mi — D * ‘ . 2 s * me : - 3 o 4 doh ATO uM Ds veh 4 | vy 4 PAG MAS Ray Teen aD eS 7 et Neg . : «28 ite Alay tat Ge £ 5 Pre ra & : f AAA | - ¢ ' v1 " i ; é 4 - o & ; ti MS + 7 ¢ is re ; La — . 3 x J * . r 4 * SS x + s Trans. B.S. Edin? Vol 6 P2185, | | Carbonate.) Mi 4 = SASS Siler \ Sip ger es ee’ BOTA 00200 Carhonatle.| Borax. Carbonate iy. I. a BOTOX. sseeevch Silex and bottle glass LITT LE ese Stlex and. bottle glass SCL eo Silex and bottle glass PLT TEN recede WS CLELEE sinw cee Carbonate....1/ Silex: and Y bottle glass\ Carbonate WA: gt PLATE JI. AM A =) SSS sam RS Bdin7 Vol b*P 185. ms. BS. Edin” Vol. 6° R185. ; PLATE IV. ae a + a i PLATE V. RS. Edin? Vol. 6" P 185. Ss SF | iN 735 ws BE <} aed I Te oi 5 esta ‘’ 4 gh y ene 7 | ' 4 q i 1 ' t 1 “os + a e bd Ye > a ae oe : =) ' miteg i * : id h . i os f F ‘ m rt W Figs by “a g ‘ ~/ ‘ ; ‘ ‘ ‘ - i mi : i ‘ , . / IV.. Of the Sotivs of Greatest ATTRACTION, or those which, among all the Soutps that have certain Properties, Attract with the greatest Force in a given Direction. By Joun Prayratir, F.R.S. Lond. and Edin. and Professor of Natural Philosophy in the University of Edinburgh. [Read 5th January 1807.] ° YHE inveftigations which I have at prefent the honour of fubmitting to the Royal Society, were fuggefted by the experiments which have been made of late years concerning the gravitation of terreftrial bodies, firft, by Dr Masketyne, on the Attraction of Mountains, and afterwards by Mr Caven- DISH, on the Attraction ef Leaden Balls. In refle@ing on thefe experiments, a queftion naturally enough occurred, what figure ought a given mafs of matter to have, in order that it may attract a particle in a given direc- tion, with the greateft force poflible? This feemed an inquiry not of mere curiofity, but one that might be of ufe in the fur- ther profecution of fuch experiments as are now referred to. On confidering the queftion more nearly, I foon found, though - it belongs to a clafs of problems of confiderable difficulty, which the CaLtcuLus VaRIATIONUM is ufually employed to re- folve, that it neverthelefs admits of an eafy folution, and one leading to refults of remarkable fimplicity, fuch as may intereft Vou. VI.—P. II. Aa Mathematicians 188 : Of the SOLIDS Mathematicians by that circumftance, as well as by their con- nection with experimental inquiries. In the problem.thus propofed, no condition was joined to that of the greateft attraction, but that of the quantity of ho- mogeneous matter being given. This is the moft general ftate of the problem. It is evident, however, that other conditions may be combined with the two preceding ; it may be required that the body fhall have a certain figure, conical, for example, cylindric, &c. and the problem, under fuch reftrictions, may be ftill more readily applicable to experiments than in its moft general form. Tuoucu the queftion, thus limited, belongs to the common method of Maxima and Minima, it leads to inveftigations that are in reality confiderably more difficult than when it is propofed in its utmoft generality. AmonG the following inveftigations, there are alfo fome that have a particular reference to the experiments on SCHEH ALLIEN. A few years ago, an attempt was made by Lord Wexs Sry- mour and myfelf, toward fuch a furvey of the rocks which compofe that mountain, as might afford a tolerable eftimate of their {pecific gravity, and thereby ferve to correé the con- clufions, deduced from Dr Mask ELYNE’s obfervations, concern- ing the mean denfity of the earth. The account of this furvey, and of the conclufions arifing from it, belongs naturally to the Society under whofe direction the original experiment was made ; what is offered here, is an inveftigation of fome of the theorems employed in obtaining. thofe conclufions. When a new element, the heterogenity of the mafs, or the unequal di- ftribution of denfity in the mountain, was to be introduced into the calculations, the ingenious methods employed by Dr Hur- Ton could not always be purfued. The propofitions that re- late to the attraction of a half, or quarter cylinder, on a par- ticle placed in its axis, are intended to remedy this inconveni- ence, Of GREATEST) ATTRACTION. 189 ence, and will probably be found of ufe in all inguiries con- cerning the difturbance of the direction of the plumb-line by in- equalities, whether in the figure or denfity of the exterior cruft of the globe. Tue firft of the problems here refolved, hs beeA urciafesk of by Boscovicn; and his folution is mentioned m the catalogue of his works, as publifhed m the memoirs of a philofophical fo- ciety at Pifa. I have never, however, been able to procure a fight of thefe memoirs, nor to obtain any account of the folu- tion juft mentioned, and therefore am fenfible of hazarding a good deal, when I treat of a fubject that has paffed through the hands of fo able a mathematician, without knowing the conclu- fions which he has come to, or the principles which he has em- ployed in his inveftigation. In fuch circumftances, if my re- fult is juft, I cannot reafonably expect it to be new; and I fhould, indeed, be much alarmed to be told, that it has not been anticipated. The other problems contained in this paper, as far as I know, have never been confidered. .. I. To find the folid into which a mafs of homogeneous matter muft be formed, in order to attract a particle given)in pofition, with the greateft force poflible, in a given direction, Let A (Fig. 1. Pl. 6.) be the particle given in pofition,; AB the direction in which it is to be attra@ed; and ACBH a fec- tion of the folid required, by a plane pafling through AB. SINCE the attraction of the folid is a maximum, by hypothe- fis, any {mall variation in the figure of the folid, provided the quantity of matter remain the fame, will not change the attrac- tion in the direction AB. If, therefore, a fmall portion of mat- ter be taken from any point C, in the fuperficies of the folid, and Paced at D, another point in the fame fuperficies, there a2 will 190 Of the SOLIDS will be no variation produced in the force which the folid exerts on the particle A, in the direction AB. Tue curve ACB, therefore, is the locus of all the points in which a body being placed, will attract the particle A in the direction AB, with the fame force. Tuis condition is fufficient to determine the nature of the curve ABC. From.C, any point in that curve, draw CE per- pendicular to AB ; then if a-mafs of matter placed at C be call- 3 ed m’, ee will be the attraction of that mafs on A, in the di- rection AC, and ene will be its attraction in the direction AB. As this is conftant, it will be equal to aS and therefore. AB?-< AE = AG?. Aut the fections of the required folid, therefore, by planes. pafling through AB, have this. property, that AC? =AB’*XAE ; and as this equation is fufficient to determine the nature of the eurve to which it belongs, therefore all the fections.of the fo- lid, by planes that pafs through AB, are fimilar and equal curves ; and the folid of confequence may be conceived to be generated by the revolution of ACB, any one of thefe curves, about AB as an axis. Tue folid fo generated may be: called the Solid of greatef Attraction ; and the line ACB, the Curve of equal Attraction. Il. To find the equation between the co-ordinates of ACB, the curve of equal attraction. From Of GREATEST ATIRACTION. 19 From GC (Fig. 1.) draw CE perpendicular to AB; let AB=a, -AE=x, EC=y. We have found AB?XAE=AC;, that is, a x= (a +- yy, or a* x*= (x°+ ¥), which is an equation to a line of the 6th order. | 4 2 3 Bee To have y interms of 7, a +y’=a?*a?,y=a and y = ww pie a HENCE y=0, both when x=0, and whenx=a. Alfo if « be fuppofed greater than a, y is impoffible. No part of the. curve, therefore, lies beyond B. Tue parts of the curve on oppofite fides of the line AB, are fimilar and equal, becaufe the pofitive and negative values of y are equal. There is alfo another part of the curve on the fide of A, oppofite to B, fimilar and equal'to ACB; for the values of y are the fame whether x be pofitive or negative. LEE: THE curve may eafily be conftructed without having recourfe to the value of y juft obtained. Let AB =a, (Fig. 1.) AG=2; and the angle BAC=¢@. Then AE=AC X cofg=2zcofg, and fo a zcofg=z3, or a cof@ =z’; hence z =aN cofo. From this formula_a value of AC or z may be found, if @ or. the angle BAC be given; and if it be required to find z in numbers, it may be conveniently calculated from this expref- fion. A geometrical conftruction may alfo be’ eafily derived from it. For if with the radius AB, a circle BFH be defcribed. from the centre A; if AC be produced to meet the circumfe- rence 192 “Of the SOLIDS ence in F,:and if FG be drawn at right angles to AB, then AGS Wai 18° UG pres oe. t= AB = cofg, and fo z =aXN AG —/ AB x AG = AC. THEREFORE, if from the centre A, with the diftance AB, a circle BFH be defcribed, and if a circle be alfo defcribed on the diameter AB, as AKB, then drawing any line AF from A, meeting the circle BFH in F, and from F letting fall FG per- pendicular on AB, interfecting the femicircle AKB in K; if AK be joined, and AC made equal to AK, the point C is in the curve. For AK=VABx AG, from the nature of the femicircle, -and therefore AC = V AB x AG, which has been fhewn to be a property of the curve. In this way, any number of points of the curve may be determined ; and the Solid of greatef attrac- tion will be defcribed, as already explained, by the revolution of this curve about the axis AB. 7 IV. To find the area of the curve ACB. 1. Ler ACE, AFG (Fig. 2.) be two radii, indefinitely near to one another, meeting the curve ACB in C and F, and the circle, defcribed with the radius AB, in E and G. Let AG=z as before, the angle BAC = 9, and AB=a. Then GE= aq, and the area AGE = +.a’@, and fince AE’: AC*: : Seét. AEG : Sect. ACF, the feGor ACF=1 279. But 2*=a* cof g, (§ 111.), whence the fector ACF, or the fluxion of the area ABC=+a"@ cof % and confequently the area ABC = 4a fing, to which no con- ftant quantity need be added, becaufe it vanifhes when g= 0, or when: the area ABC vanithes. THE Of GREATEST ATTRACTION. 193 Tue whole area of the curve, therefore, is 4a’, or + AB’; for when ¢ is a right angle fing=1. Hence the area of the curve on both fides of AB is equal to the fquare of AB. 2. THE value of x, when y is a maximum, is eafily found. For |= } 4 when. y, and therefore y* is a maximum, 74? es Be, OF 4 ES : a a au —¢"*, thatis#= —= é | 37 AY 27 Hwce, calling 4 the value of y when a maximum, | ' 3 on + 2 , weg x ia ae ® =a C2) — 2% and ne ; Bae A 27 Bie nae 27 and therefore a:0:: N 247 :/ 2, or as 11: 7 nearly. 3. Ir is material to obferve, that the radius of curvature at A’ 4 4 mache ss, 4 acy pee ye a3 isinfinite. For fince y’= 4? «?—x?,"-— = —-— x.. But when es % x is very fmall, or y indefinitely near to A,t becomes the dia-: meter of the circle having the fame curvature with ACB at A, | : 4: and when x vanifhes, this value of ©, or zt — x, becomes infi- - x 3 nite, becaufe of the divifor x? being in that cafe =o. The dia- meter, therefore, and the radius of curvature at A are infinite. In other words, no circle, having its centre in AB produced, and pafling through A, can be defcribed with fo great a radius, but that, at the point A, it will be within thé curve of equal - attraction. THE ¥94 Of the SOLIDS “TueE folid of greateft attraction, then, at the extremity of its ‘axis, where the attracted particle is placed, is exceedingly flat, approaching more nearly to a plane than the fuperficies of any {phere can do, however great its radius. 4. To find the radius of curvature at B, the other extremity of 3 24 the axis, fince y* = a* x* — x’, if we divide by a— «x, we have 2 ANE 2 J =—4 **—*, But at B, when a—x, or the abfcifla a a—x . . i * reckoned from B vanithes, aoa is the diameter of the circle having the fame curvature with ACB in B. But when a—x=0, or 4=x, both the numerator and denominator of 2 : a® x3 —x’ the fraction . vanifh, fo that its ultimate value does not appear. To remove this difficulty, let a—x=2z, or x —=a—z, then we have y= at (a—z)? —(a—z). But when x is extremely fmall, its powers, higher than the firft, may be rejected; and therefore (a—z)* = 4* G —*) = 2 . a? (1— 2* &c.) Therefore the equation to the curve becomes | =: Pye iee 4. 2 2 2 in this cafe, y*= a? Xa? G- = )-e+ 2a2 erga a+ 2an =F az, HENCE Of GREATEST ATTRACTION. 1gs in f 3 2 HENCE Z, or the radius of curvature at B= 4% The % curve, therefore, at B falls wholly without the circle BKA, de- {cribed on the diameter AB, as its radius of curvature is greater. This is alfo evident from the conftruction. Vv. To find the force with which the folid above defined attracts the particle A in the direction AB. Let 4 (Fig. 2.) be a point indefinitely near to B, and ies the curve Acb be defcribed fimilar to ACB. Through C draw Cc D per- pendicular te AB, and fuppofe the figure thus conftructed to re- volve about AB; then each of the curves ACB, Acé will gene- rate a folid of greateft attraction; and the excefs of the one of thefe folids above the other, will be an indefinitely thin fhell, the attraction of which is the variation of the attraction of the folid ACB, when it changes into Ac J, AGAIN, by the line DC, when it revolves along with the reft of the figure about AB, a circle will be defcribed ; and by the part Cc, a circular ring, on which, if we fuppofe a folid of in- definitely {mall altitude to be conftituted, it will make the ele- ment of the folid fhell ACc. Now the attraction exerted by this circular ring upon A, will be the fame as if all the matter of it were united in the point C, and the fame, therefore, as if it were all united in B. But the circular ring generated by Cc, is =a (DG — Dc’) = 27DGx Cec. Now 2DC xXCe'is the variation of y’, or DC’, while DC pafles into De, and the curve BCA into the curve bc A; that is 2 DCX Cc is the fluxion of y’*, or of at xx", Vout. VI.—P. II. Bb | taken t96 Of th SOLIDS taken on the fuppofition that x is conftant and a4 variable, VIZ. 4 ata x oa Therefore the {pace generated by Cc = 3 | | 2° cy, pe ee 3 x Ir this expreffion be multiplied by x, we have the element of the fhell = = a? x? a In order to have the folidity of the fhell ACB4c, the above expreffion muft be integrated relatively to x, that is, fuppofing only « eae and it is then : ses ae xe a + C.: But C=o, becaufe the fluent vanifhes when x vanithes, therefore the por- tion of the fhell ACc = : xi a3 a, and when x=a, the whole thell = 47 a? a. 5 Now, if the whole quantity of matter in the fhell were unit- ed at B, its attrative force exerted on A, would. be the fame with that of the fhell; therefore the whole force of the fhell — 474. The fame is true for every other indefinitely thin fhell into which the folid may be fuppofed to be divided ; and: therefore the whole attraction of the folid is equal. to rf 7s a, fuppofing a variable, that is = e a. HENCE Of GREATEST ATTRACTION. 19% HENCE we may compare the attraction of this folid with that of a {phere of which the axis is AB, for the attraction of = 6 ADBH, (Fig. 1.) is, therefore, to that of the {phere on the fame that fphere = - a3 xis = a. The attraction of the folid -axis as 47 a to ra or as 6 to 5. 5 VI. To find the content of the folid ADBH, we need only inte- grate the fluxionary expreflion for the content of the fhell, viz. = aa. We have then a a} = the content of the folid ig 7 6 the content of the folid ADBH is to that of the fphere on the fame 4) a; ADBH. Since the folidity of the fphere on the axis 4 is = : 5 . I AXIS as a a} to 2 a? ; that is, as + to 4, or as 8 to 5: 1g VII. Last.Ly, To compare the attraction of this folid with the at- traction of a {phere of equal bulk, let m = any given mafs of matter formed into the folid ADBH ; then for determining AB, ee ee [sa 3 : we have this equation, 2 a? =m, anda=mn = ; and there- yi : ; w Bb 2 fore 198 Of the SOLIDS fore alfo the attraction of the folid, (which is Bg a\= = m v ie z AGAIN, if m be formed into a fphere, the radius of that mat) : : : : {phere = 7 Ni a and the attraction of it on a particle at its m 3 + ee 6 )3 furface = : (3 \i = m a e m? \-— a 47 9* HENCE the attraction of the folid ADBH, is to that of a {phere equal to it, as m os" aie to m ym (202) 5 3. that is, as (27)? to (25)3, or as 3 to the cube-root of 25. THE ratio of 3 to N25, is nearly that of 3 to 3 ——, or 27 of 81 to 79; and this is therefore alfo nearly equal to the ratio of the attraction of the folid ADBH to that of a fphere of equal magnitude. VITl. Ir has been fuppofed in the preceding inveftigation, that the particle on which the folid of greateft attraétion exerts its force is in contact with that folid. Let it now be fuppofed, that the. diftance between the folid and the particle is given; the folid being Of GREATEST ATTRACTION. 199 being on one fide of a plane, and the particle at a given di- ftance from the fame plane on the oppofite fide. The mafs of - matter which is to compofe the folid being given, it is required to conftruct the folid. Let the particle to be attracted be at A (Fig. 3.), from A draw AA’ perpendicular to the given plane, and let EF be any ftraight line in that plane, drawn through the point A’; it is evident that the axis of the folid required muft be in AA’ produced. Let B be the vertex of the folid, then it will be\demonftrated as has been done above, that this folid is generated by the re- volution of the curve of equal attraction, (that of which yo dniel2 : the equation is y” = a? x ? — x’°), about the axis of which one extremity is at A, and of which the length muft be found from the quantity of matter in the folid. THE folid required, then, is a fegment of the folid of great- eft attraction, having B for its vertex, and a circle, of which A’E or A’ F is the radius, for its bafe. To find the folid content of fuch a fegment, CD being =y, 4 2 and AC =x, we have y* = 4? x7 —x’,and ry’ % = oO Pe: ae wx % = the cylinder which is the element of the folid feg- ment. | THEREFORE f ay x, or the folid fegment intercepted be- es 3 tween B and D muft be é ee xe — are +C. This muft vanifh when x = 4, or when C comes to B, and therefore C = 4r 3 = ee IL 15. 200 . Of th SOLIDS oie 7 a3. The fegment, therefore, intercepted between B and I 7 ae ees wT C, the line AC ee x, 1S 7 as Te at xi +7 x3, 3 Tuts alfo gives re a+, for the content of the whole folid, when « = 0, the fame value that was found by another me- thod at § v1. ; Now, if we fuppofe x to be = AA’, and to be gi- ven = 34, the folid content of the fegment becomes yaaa ic et oe fe 7 63, which muft be made equal to the 15 5 3 given folidity which we fhall fuppofe = m?, and from this equation a, which is yet unknown, is to be determined. gis then, for a? we put uw, we have ¢ oe u oe: BF uy’ a : "| Set m3 ai 35 pe aa Me Te and w — 29% Z 3 Tue fimpleft way of refolving this equation, would. be by the rule of falfe pofition. In fome particular cafes, it may be refolved more eafily ; thus, if oe — 36 =o, a as =o, anda? = 2 33, that is a? = 9 Dd? ora= 4 4 4 D\F 5/229. ea) Ne IX, Of GREATEST ATTRACTION. 201 IX. 1. Ir it be required to find the equation to the fuperficies of the folid of greateft attraction, and alfo to the fections of it pa- rallel to any plane pafling through the axis ; this can readily be done by help of what has been demonftrated above. Let AHB (Fig. 4.) be a fection of the folid, by a plane through AB its axis. Let G be any point in the fuperficies of the folid, GF a perpendicular from G on the plane AHB, and FE a per- pendicular from F on the axis. Let AE =x, EF =z, FG =, then x, z, and v are the three co-ordinates by which the fuper- ficies is to be defined. Let AB =a, EH =y, then, from the na- ture of the curve AHB, y* = a3 x3 — x. But becaufe the plane GEH is at right angles to AB, G and H are in the circumfer- ence of a circle of which E is the centre; fo that GE = EH =y. Therefore EF*-+ FG* = EN’, that is, z°-+- v* =’, and by 2 eta x; 4 fubftitution for y* in the former equation, z°-+ v* = a? x 3 ee : or (x? +27 + 0’) = a+ x, which is the equation to the fuperfi- cies of the folid of greateft attraction. 2. Ir we fuppofe EF, that is z, to be given = 4, and the {o- lid to be cut by a plane through FG and CD, (CD being paral- lel to AB), making on the furface of the folid the fe@ion DGC ; and if AK be drawn at right angles to AB, meeting DC in K, then we have, by writing 4 for z in either the preceding equa- er 3 tions, 0+ vt =a? equation of the curve DGC, the co-ordinates being GF and FK, becaufe FK is equal to AE or x. 4 2 =x, and v= a? «3 — x*—ZJ* for the Tuis 202 Of the SOLIDS Tuis equation alfo belongs to a curve of equal attraction ; the plane in which that curve is being parallel to AB, the line in which the attraction is eftimated, and diftant from it by the | {pace 0. InsTEAD of reckoning the abfcifla from K, it may be made to begin atC. If AL or CK=4/, then the value of 4 is deter- mined from the equation 4* = ai bs — b’, and if w=h+u4, u being put for CF, 07 =a? (b +4)? —(b-+u)'—a? be +2, or ot (b+tay +h Hat (64-0), or (+ (btn +5)3= a‘ (b+u)’. WHEN 3 is equal to the maximum value of the ordinate EH, (zv. 2.) the curve CGD goes away into a point ; and if b be fup- pofed greater than this, the equation to the curve is impoflible. Xx. Tue folid of greateft attraction may be found, and its pro- perties inveftigated, in the way that has now been exemplified, whatever be the law of the attracting force. It will be fufhi- cient, in any cafe, to find the equation of the generating curve, or the curve of equal attraction. Tuus, if the attraction which the particle C (Fig. 1.) exerts on the given particle at A, be inverfely as the m power of the di- I AG then the attraction in the direction AE ftance, or as AE will be ——— AG, m? , and if we make this = ——, we haye ee AB AC / Of GREATEST ATTRACTION. 203 tor making AE=x, EC=y, and AB=g, as before, m: Sh! m at d an SB aa aie BRR art eee % » an re ye es zm 2 2m 2 qutt XML or y = qutt emEE yy? Ir m=1, or m+ 1= 2, this equation becomes y* = a x — x’, being that of a circle of which the diameter is AB. If, therefore, the attracting force were inverfely as the diftance, the folid of greateft attraction would be a fphere. Ir the force be inverfely as the cube of the diftance, or m = 3, and m-+- 1= 4, the equation is y° = renin at x, which belongs to a line of the 4th order.” & 912 5 Ir m= 4, and m-+ 1 = 5, the equation is y’ = a* x alg al 5 which belongs to a line of the roth order. In general, if m be an even number, the order of the curve is m+ 1X23 but if m be an odd number, it is m+ 1 fimply. pats In the fame manner that the folid of greateft attraction has been found, may a great class of fimilar problems be refolved. Whenever the property that is to.exift in its greateft or leaft degree, belongs to. all the points of a plane figure, or to all the points of a folid, given in magnitude, the queftion is reduced to. _ the determination of the locus of a certain equation, juft as in the preceding example. | Vou. VI.—P. I. , Ge Let 204 Of the SOLIDS Let it, for inftance, be required to find a folid given in mag= nitude, fuch, that from all the points in it, ftraight lines being drawn to any afligned number of given points, the fum of the {quares of all the lines fo drawn fhall be a minimwn. It will be found, by reafoning as in the cafe of the folid of greateft at. traction, that the fuperficies bounding the required folid muft be fuch that the fum of the fquares of the lines drawn from any point in it, to all the given points, muft be always of the fame magnitude. Now, the fum of the fquares of the lines drawn from any point to all the given points, may be fhewn by plane geometry to be equal to the fquare of the line drawn to the centre of gravity of thefe given points, multiplied by the number of points, together with a given {pace. The line, there- fore, drawn from any point in the required fuperficies to the centre of gravity of the given points, is given in magnitude, _and therefore the fuperficies is that of a fphere, having for its centre the centre of gravity of the given points. Tue magnitude of the {phere i is next detetmined from the condition, that its folidity is given. In general, if x, y, and 2, are three rectangular co-ordinates - that determine the pofition of any point ofa folid given in magnitude, and if the value of a certain function Q, of x, y and z, be computed for each point of the folid, and if the fum of all thefe values of Q added together, be a maximum or a mini- mum, the folid is bounded by a fuperficies in which the func- tion Q is every where of the fame magnitude. That is, if the triple integral J a Ry I) [ Qz be the greateft or leaft poffible, the fuperficies bounding the —- is fuch that oa A, a con- ftant quantity. Tur fame holds of plane agiiohes ; the spoons is then fimpler, as there are only two co-ordinates, fo that me x ba Q Fi 1s the Of GREATEST 2 ee ee 405 the quantity that is to be a maximum or a minimum, and the line bounding the figure is defined by the equation Q= A Aut the queftions, therefore, which come under this fietetint tion, though they belong to an order of problems, which re- quires in general the application of one of the moft refined in- ventions of the New Geometry, the Calculus Variationum, form a particular divifion admitting of refolution by much fimpler means, and direétly reducible to the conftruGion of loci. In thefe problems alfo, the fynthetical demonftration will be found extremely fimple. In the inftance of the folid of great- eft attraction this holds remarkably. Thus, it is obvious, that (Fig. 1.) any particle of matter placed without the curve ACBH, will attra& the particle at A in the direction AB, lefs than any of the particles in that curve, and that any particle of matter within the curve, will attract the particle at A more than any particle in the curve, and more, @ fortior?, than any particle without the curve. The fame is true of the whole fuperficies of the folid. Now, if the figure of the folid be any how changed, while its quantity of matter remains the fame, as much matter muft be expelled from within the fur- face, at fome one place C, as is accumulated without the fur- face at fome other point H. But the action of any quantity of matter within the fuperficies ACBH on A, is greater than the action of the fame without the fuperficies ACBH. The {folid ACBH, therefore, by any change of its figure, muft lofe more attraction than it gains ; that is, its attraction is diminithed by every fuch change, and therefore it is itfelf the folid of great- eft attraction. .Q. E. D. Ce 2 XII. 206 Of the SOLIDS ah XII. Tue preceding theorems relate to the folids, which, of all folids whatfoever of a given content, have the greateft attrac- tion in a given direction. It may be interefting alfo to know, among bodies of a given kind, and a given folid content, for example, among cones, cylinders, or parallelepipeds, given in magnitude, which has the greateft attractive power, in the di- rection of a certain ftraight line. We fhall begin with the cone. Let ABC (Fig. 5.) be a cone of which the axis is AD, re- quired to find the angle BAC, when the force which the cone exerts, in the direction AD, on the particle A at its vertex, is greater than that which any other cone of the fame folid con- tent, can exert in the direction of its axis, on a particle at its vertex. Ir is known, if x be the femicircumference of the circle of which the radius is 1, that is, if = 3.14159, &c. that the at- traction of the cone ABC, on the particle A, in the direction ADo i= 29 X (AD —_4> . (Srmpson’s Fluxions, vol. ii. ATED 397.) | Ler AD=x, AB =z, the folid content of the cone = m3,. and its attraction — A. THEen A272 (x —=), and x x (z°—x"°) = 3m, . x ° . THE quantity x——, is to be a maximum, and therefore, ° CAH re : XZK— HS z fe eS . = 0, OF see 2 tea. z x AGAIN, Of GREATEST ATTRACTION. 204 Acain, from the equation nx (z"—x) = 3m3, we have eae pbusi Z 302 od by fubhti 24228 x%— 3X x =O, and = ==» and by fubfti- x tuting this value ef * in the former equation, we have ° . ° ° x 2 As this equation is homogeneous, if we make = ms we will obtain an equation involving w only, and therefore deter- mining the ratio of z to x, or of AB to AD. © Subftitu- ting, accordingly, wz for x in the laft equation, we have ge — Dutt 3 uw3z%* = o, and 1 — Sy43 u3 =O. 2 2 2 2 ‘THis equation is obvioufly divifible by s— 1, and when fo divided, gives 3 “+ 3 u—I=0, or wt+u= ; whence ee eee 2 12 Tuis is the value of = , and as = muft be lefs than unity, becaufe AB is greater than AD, the negative value of uw, or aE BN GP is €xéluded ; fo that «= —- + a = 45761 nearly. Now z= erate the cofine of the angle BAD, or half the angle of the cone; therefore that angle = 62°, 46’ nearly. As 208 "Of the SOLIDS As the tangent of 62°.46' is not far from being double of the radius, therefore the cone of greateft attraction has the ra- dius of its bafe nearly double of its altitude. To compare the attraction of this cone with that of a {phere containing the fame quantity of matter, we muft exprefs the attraction in terms of z, the ratio of x to. z, which has now been found. : Zz 7 BECAUSE 7% (%:— x’) = 35, and z= -, 7x (= —x’) = / “ u = , I — wz u* a (irate = 3m, and xm. peewee Te re NeG=*) x2 x Yc aoe Now, we have A=2z- («—=), and fmce'\- = 4, = z Pe z 3 Bu" aid : ale ase Vaayrttan ellos eel =27m.(1—n) ahs ; wherefore, A} = 8 7° m? (1—u)3 <——t")} 2. 2 rely ats Ope ™(I—4d2’) I+u But if A’ be the attraction of a {phere of which the mafs is ° id e 3 16 / m3, on a particle at its furface, A’ = m Vv “, and A? = 9 Ph SL Therefore A? : AY? :: 24 u' (1 — 4) 16 : 9 L-+4 9 a7u (I—U) Sues *(r—n)". Cea ae 1; and confequently A: A’:: 3 (iS (1 pay Ir, in this expreflion, we fubftitute .45761 for u,-we thall have A: A’ :: .82941 : 1, fo that the attraction of the cone, | when Of GREATEST ATTRACTION. 209 when a maximum is about 2 of the attraction of a {phere of equal folidity. XIII. Or all the cylinders given in mafs, or quantity of matter, to find. that which fhall attraé a particle, at the extremity of its axis, with the greateft force. Let DF (Fig. 6.) be a cylinder of which the axis is AB, if AC be drawn, the attraction of the cylinder on the particle A is 2”xX (AB+ BC — AC) *, and we have therefore to find when AB + BC — AC is a maximum, fuppafing AB. BC’ to be say to a given folid. Let AB=x, BC =y, then AC = Vx* +)’, and the one ty that is to be a maximum is x y— Ney y’» We have therefore PA EIU lr A = 0, and (x+y) (x +y)2= N x? ey? wee, or Ge) (xb yr)? =—x+y.2. x x But fince xy? = m3, 2xyy-y*x = 0, or 2uym—ye, pnd 2, eS v 22 THEREFORE * Princip. Lib. I. Prop. 91. Alfo Simpson’s Fluxions, vol. II. § 379. In the former, the conftant multiplier 2 is omitted, as it is in fome other of the theorems relating to the attraction of bodies. This requires to be particularly attended to, when thefe propofitions are to be employed for comparing the at- traction of folids of different fpecies. 210 Of the SOLIDS ) ie a RBCS AT ings Se ae THEREFORE (2 - -) Goepy)? soe aay ay! Of (2a—y) (+S)? =20°—y. As this equation is homogeneous, if we make 4 =z, or x y =u, both w and y may be exterminated.. For we have by” fubftituting wx for y, (27—uer) (’+2° x)? = 20° — a’ a’, or (2 4°—wa") (1-4)? = 20" in" w*, and dividing by 2’, (2—a)+ (1+ w)? = 2—u’; whence fquaring both fides, (4—4u-+u’) (+2) = 4—4u' 44 From this, by multiplying and reducing, we get 44° — 9 u —— 4, org Fut; and ow 2. THE two values of w in this formula create an ambiguity which cannot be removed without fome farther inveftigation. If A be the attraction of the cylinder, then A= 27 («+y — Vx +"), into which expreffion, if we introduce u, and exter- minate both w and y, by help of the equations 7 wy? = m3, and: 2 14+4—Vi+u a u, we get A= 2:73 m Ercan ph tes v z “ NoTWITHSTANDING the radical fign in this formula, there is but one value of A, correfponding to each value of w, as the. pofitive root of Wz —u* is not applicable to the phyfical pro- 3 blem.. Of GREATEST ATTRACTION. — 243 blem. This is evident, becaufe the attraction muft vanifh both when y = 0, and when x = 0; that is, both when z is nothing, and when it is infinite: This can only happen when V7 -++ a* is negative. FARTHER, the value of A is always pofitive (as it ought to be), 1+ being greater than V 1I-+ ua’, becaufe it is the {quare- root of r++ 2u-pu. 3 ssasibite the relation between A and w will be beft con- ceived, by fuppofing A to be the ordinate of a curve in which the ab{fciff are reprefented by the fucceflive values of wu» Thus, if OP (Fig. 7.) =z, and PM= A, the locus of M is a curve of the figure OMM’, which interfects the axis at O, and has the or- dinate PM a maximum, when OP = 2—¥*7 ot 7, 3 beyond PM’ the ‘curve has a point M of contrary flexure, Evie it becomes convex toward the axis OR, and afterwards approaches OR continually. It has alfo another branch mm'n, correfponding to the af- firmative values of Y1-+-*, which has the perpendicular OQ_ for an aflymptote; and has the ordinate P’m’ a minimum, when z= ares, After paffing the point where P’m’ is a minimum, this branch of the curve recedes continually from the axis OR. Befides thefe, there are other two branches of the fame curve, on the oppofite fide of OQ, anfwering to the negative values of wu. It is, however, only the firft-mentioned of thefe four branches that is connected with the mechanical queftion confidered here. VOL. VI.—P. th D d THE 212 Of the SOLIDS ‘THE attradlion is a maximum, therefore, when a ed that is, when y is to x, or the radius of the bafe of the cylinder, to its altitude, as g—\/17 to 8, or as 5 to 8 nearly. Therefore alfo the diameter of the bafe is to the altitude, when the at- traction of the cylinder is greateft, as 9 —/ 17 to 4, or as 5 ta 4 nearly. 5. THE attraction of the cylinder, when a maximum is now to be compared with that of a {phere of equal folid content, Ur are nc us FIRsT, to compute the quantity , when y= I= N17 = 6096, fince w= .37161, 1-2’ = 1.37161, and VI = 1.17116; fo that 1+ u—Ni-+u=.43844. 2 Atso becaufe w*=.37161, uw? =.718945; and therefore rptu—Nité aa Ba Therefore A= ea m Joke oN Ew 2 usr 2 10M ses a areca ae Afj=aar4 nx (—)>, and BA DA ee Brag OVER atte Of GREATEST ATTRACTION, 333 8768 7189 the cylinder, even when its form is moft advantageous, does not exceed that of a fphere, of the fame folid content, by more than a hundred and eighty-third part. : 1.2114, or as 1218 to 1211.43 fo that the attraction of 6. In a note on one of the letters of G. L. Le Sace, pub- lifhed by M. Prevost of Geneva *, the following theo- rem is given concerning the attraction of a cylinder and a fphere: If a cylinder be circum{fcribed about a fphere, the particle placed in the extremity of the axis of the cylin- der, or at the point of contact of the fphere, and the bafe of the cylinder, is attracted equally by the fphere, and by that portion of the cylinder which has for its altitude two-thirds of the diameter of the fphere, and of which the folidity is there- fore juft equal to that of the fphere. WE may inveftigate this theorem, by feeking the altitude of fuch a part of the circumfcribing cylinder as fhall have the fame attraction with the {phere at the point of conta@. If 7 be the radius of the {phere, the attraction at any point. of its furface, is ye 3 and if w be the altitude of the cylinder, and the radius of its bafe 7, then its attraction on a particle at the extremity of its axis is 27(@+7—Na"-+/7’). Since thefe attractions are fuppofed equal, 27 («+7 —Na' +r) =4*7, 3 and a+ 7—Na +r Pat Lind witiet saath: adel and = +", WOR 3. 9 3 Dd2 Lp THE _ * Novice de la vie de G. L. Lz Sace de Genéve, par P, PREVOST, p. 391. 214 ; Of \the SOLIDS Tue altitude of the cylinder is therefore : of the radius, or a of the diameter of the fphere, which is Lr Sace’s Theo- rem. Tuts cylinder is alfo known to be equal in folidity to the fphere ; but its attraction is not greater than that of the latter, becaufe the proportion of its altitude to the diameter of its bafe is not that which gives the greateft attraction. Its alti- tude is to the diameter of its bafe, as rite far jay 4 to 63 in order to have the greateft effet, it muft be as 4 to 5 nearly, we ae oe DING, therefore, chat the form of the one of thefe cylinders is confiderably different from that of the other, their attractions are very nearly equal; the one of them being the fame with that of the fphere, and the other greater than it by about the 183d part. On each fide of the form which gives the maximum of attraction, there may be great variations of figure, without much change in the attracting force. A fimi- lar property belongs to all quantities near their greateft or leaft ftate, but feems to hold efpecially in what regards the attrac- tion of bodies. | XIV. In confidering the attraction of the Mountain Shehallien, in fuch a manner as to make a due allowance for the heteroge-- neity of the mafs, it became neceflary to determine the attrac- tion of a half cylinder, or of any fector of a cylinder, on a point fituated in its axis, ina given direction, at right angles to Of GREATEST ATTRACTION. 215 to that axis. The folution of this problem is much connected with the experimental inquiries concerning the attraction of mountains, and affords examples of maxima of the kind that form the principal obje&t of this paper. The following lemma is neceflary to the folution. Let the quadrilateral DG (Fig. 8.) be the vadbanicaly. {mal] bafe of a column DH, which has everywhere the fame fection, and is perpendicular to its bafe DG. Ler A be a point at a given diftance from D, in the plane DG ; it is required to find the force with which the column DH attracts a particle at A, in the direction AD. Let the diftance AD =r, the angle DAE = 9, DE (fuppo- fed variable) =y, and let EF be a fection of the folid parallel, and equal to the bafe DG; and let the area of DG = m’. Tue element of the folid DF is m’ 75 and fince DE, or g fo that the element of the: cole yor tang, j=rtang=r. folid = mr. —2_.. col@ Tuis quantity divided by AE’, that is, fince AE: AD:: 1 cofg, by ae gives the element of the attraction in the direc- aan AS Dts tion AE equal tO ote x “ a ~*. To: reduce this to the direction AD, it muft be multiplied-into the cofine of the angle DAE or ¢, fo that the element of the attra@ion of the column. in the direction. AD. is = 9 cof g, and the attraction itfelf — % X fecofp=" fing. WHEN. 216 Of th SOLIDS 1% WHEN @ becomes equal to the whole angle fubtended by the column, the total attraction is equal to the area of the bafe di- vided by the diftance, and multiplied by the fine of the angle of elevation of the column. Ir the angle of elevation be 30°, the attraction of the co- -lumn is juft half the attraction it would have, fuppofing it ex- tended to an infinite height. In this inveftigation, m* is fuppofed an infinitefimal ; but if it be of a finite magnitude, provided it be fmall, this theorem will afford a fufficient approximation to the attraction of the column, fuppofing the diftance AD to be meafured from the centre of gravity of the bafe, and the angle ¢ to be that which is fubtended by the axis of the column, or by its Hh se cpg 5c height above the bafe. XV. Let the femicircle CBG (Fig. 9.), having the centre A, be the bafe of a half cylinder ftanding perpendicular to the horizon, AB a line in the plane of the bafe, bifecting the femicircle, and reprefenting the direction of the meridian; it is required to find the force with which the cylinder attracts a particle at A, in the direction AB, fuppofing the radius of the bafe, and the alti- tude of the cylinder to be given. Let DF be an indefinitely {mall quadrilateral, contained be- tween two arches of circles defcribed from the centre A, and two radii drawn to A; and let a column ftand on it of the fame height with the half cylinder, of which the bafe is the femi- circle CBG. Let z = the angle BAD, the azimuth of D; v = the vertical angle fubtended by the column on DF; a= the Of GREATEST ATTRACTION. 217 the height of that column, or of the cylinder, AD =x, AB, the radius of the bafe, =7. By the laft propofition, the column ftanding on DF, exerts on A an attraction in the direction AD, which is = ———= xX fin v. Now Dd=3%, Df=xz,andDdxDfaxzx. Therefore the attraction in the direction AD is wee chit piesa V, a and reduced to the direction AB, it is x z finv x cof. Tuis is the element of the attraction of the cylindric fhell or ring, of which the radius is AD or x, and the thicknefs «; and ~ area Syn, on the epposition that z only is variable, at x a v cantare it gives « fin v vf 2 zcofz=%#finv X fing a8 the attra@ion of the fhell:’ When z= go, and finz = 1, we have the attraction of a quadrant of the fhell = « fin v, and therefore that of the whole femicircle = 2 % fin v. Next, if x be made variable, and confequently v, we have 2 y x fin v for the etiaction of the femi-cylinder. Nowtheanglev would havea for its fineifthe radius were/a’-+x", and fo finv= Wp ge = ==; wherefore the above expreffion is 2a% f a+ x = 3; and as this muft va- mith when x = 0, 24La+Cx0, and C=2—2a4La, fo that the 218 Of the SOLIDS : xtNaax : } the fluent is 2a L stares M8 which, when x =7, gives the attraction of the femi-cylinder = 2¢L con Ba: Tuis expreflion is very fimple, and very convenient in cal- culation. It is probably needlefs to remark, that the loga- rithms meant are the hyperbolic. XVI. Let it be required to find the figure of a femi-cylinder gi- ven in magnitude, which fhall attract a particle fituated in the centre of its bafe with the greateft : syne poffible, in the di- rection of a line bifecting the bafe. i . Tue attraction of the cylinder, as jutft eee: is 2aL aE Bee ; and becaufe the folid is fuppofed to be given ; __ m3 a in magnitude, we may put 47°=m?, ora= ae fo that the formula m rN 9b i : r+ 2m? re 4t+Nro4 me above becomes Mie aera ea Now we may fuppofe m=1, and then the attraction of the pari cylinder = “ L (73 + 7° 1). THIs Of GREATEST ATTRACTION. 219 Turis formula vanifhes whether 7 be fuppofed infinitely great or infinitely fmall, and, therefore, there muft be fome magnitude of r in which its value will be the greateft pof- fible. eee DBM Ir r is very {mall in refpet of 1,Vi1+7r° =1 ae and : -6 fo rsa+V¥itre = rtre+5, or fimply = 1-+73. But L(1+7°), if 7 is very {mall in refped of 1, is 73 5 and there- fore the ultimate value of the formula, when 7 is infinitely fmall, is 2, X 73 = 27, which is alfo infinitely fmall. 7, AGAIN, let 7 be infinitely great; then V7* +1 = 73, and fo the formula is = L.2r3, or = 3 L.2r. But the logarithm of an infinitely great quantity 7, is an infinite ofan order in- comparably lefs than r, as is known from the nature of logarithms, (Grec. Fontan# Difquifitiones Phyf. Math. de Infinito Logarithmico, Theor. 4.); fo that . Lar is lefs 6 i 6 6 ° ° » : » . ° than “7 of than =%, But as infinitely {mall, 7 being infinite- , . ba ly great, and therefore, when the radius of the cylinder be- comes infinitely great, its folid content remaining the fame, its attraction is lefs even than an infinitefimal of the firft or- def. THE determination of the maximum, by the ordinary me- thod, leads to an exponential equation of confiderable difficulty, if an accurate folution is required. It is, however, eafily found Vou. VI.—P. II. Bc by 220° Of the SOLIDS by trial, that, when the function ah (r3>-+-NT+7*) is a I 2 6, maximum, 7 is nearly = : Therefore, becaufe a = =,25 30° = 6 7 is nearly to a as : to ma or as 216 to 125 and this of con- fequence, is, nearly, the ratio of the radius of the bafe, to the altitude of the half-cylinder, when its attraction, eftimated ac-. cording to the ol apes of the problem, is the greateft pof- fible. axVil. To determine the oblate {pheroid of a given: folidity which: fhall attract.a particle at its pole with the greateft force.. Let there be an oblate fpheroid generated by the revolution. of the ellipfis ADBE (PI. 7. Fig. 10.), about the conjugate axis AB, and let F be the focus; then if AF be drawn, and: the arch CG defcribed from the centre A, the force with. which the fpheroid draws a particle at A, in the direction AC, 5 At AC .CD* Crs ‘Ler this force =F, AC=a, CD.=4, the angle CAF=9; (CF—CG *).. iMacs atone? s Fluxions, § 650).. anal then CF = a4 tang, and F = aa (tang—o) a= 4zb tang—o@ 4» tan. @? 3 Now if m? be the folidity of the fpheroid, fince that folidity is two-thirds of the cylinder, having CD for the radius of its bafe,. * Tue multiplier 27, omitted. by Macxauriy, is reftored as above, $ XIII. Of GREATEST ATTRACTION. 224 ‘bafe, and AB for its altitude; therefore m? = : Kab X24 —4¢ab’; fo that = ti 3 atid 2.223" 3 +aa 47a a 47a But becaufe AF: AC::1:cofg, or b:a::1:cofg, & = a Dig cof g”’ Tae = colo” a , cof 9” , and alfo 2’ = ret we have—“— Now fince 2’ = ¥ cof @ t 3 3 a and 43 = re . cof ¢’, or if ~ =3,a5 = 23 col ¢’, ya and a =n cof oF. 2% 2 b a HENCE, as —- = a ig b° __ acolo ee bole 4 9 tore 15, 4° ey cof By fubftituting this value of = in the value of F, we have F eee tan ¢ fin 9? a and. becanie, a Se nae a pe ecaufe tan @ cof o>” - 2an.(tang@—).cofo? _ 2xn(tang— @¢) cof g* fin 93 x cof ge? =e 5 2an(tane—).cofe?.fing. Now when the product of any number of factors is a maxi- mum, if the fluxion of each factor be divided by the fadtor it- Ee 2 felf, 222 . Of Whe SOU TINS felf, the fum of the quotients is equal to nothing. Therefore ? wo 2 roth Bt cof ¢ : 5 cole . colo Say 3 fin ¢. fin ae ie 9 (1 —cof¢") sofng 39cofo _ cofg’(tang—¢) 3col@ fin. Qe ..-- Tyerid ou fin-¢° SM. .-3-cef pasa sa cofg’(tang—¢) 3cofo fing ras fin @ i org cof @ Aad aE p= cofg (tang—)° 3 3 fin ¢” tang—o > + 300 OD. Lp ee ca = ae Be ea = tan 9 g, and @ = tan o 3 tan @ bee 5tang-+9tan @. cot g’— 3 tan aa 5+9.cotg 5+gcotg’ oat 2 tan @-+ gcot s5t+togcotg — 9 meee Ser ge at Abii: i oe oe Let tang=?/, theng= 9 - 5 i? >] maximum. ala ae & which, therefore, is the value of @ when F is a THE Of GREATEST ATTRACTION. 223 Tue value of g, now found, is remarkable for being a near approximation to any arch of which ¢ is the tangent, provided that arch do not exceed 45°. The lefs-the arch is, the more near is the approximation ; but the expreflion can only be con- fidered as accurate when ¢ = o. Tuts will be made evident by comparing the fraction Bian 20} with the feries, that gives the arch in terms of the Ga 5f . : t3 Lie hi? £ tangent ¢, viz. 9 = t— a a 7 — a +, &c. The fraction i(o- at) eae -- Sib oui fghi Te af &c. The two firft ama 3 a 3-9 terms of thefe feries agree; and in the third terms, the differ- ence is inconfiderable, while ¢ is lefs than unity ; but the agree- ment is never entire, unlefs ¢ = o, when both feries vanifh. Tue attraction, therefore, or the gravitation at the pole of an oblate fpheroid, is not a maximum, until the eccentricity of the generating ellipfis vanifh, and the {pheroid pals into a {phere. ~ From the circumftance of the value of ¢ above found, agree- ing nearly with an indefinite number of arches, we muft con- clude, that when a {phere paffes into an oblate f{pheroid, its at- traction varies at firft exceeding flowly, and continues to-do fo till its oblatenefs, or the eccentricity of the generating ellipfis, become very great. This may be fhewn, by taking the value of F, and fubftituting in it that of g, in terms of tan ¢. We have poate? and ‘fince @ = tan @ — cof o? ——- 3 tan @ Jie 3 224 Of the SOLIDS tan 93 ‘tan 95 tan 93 tan car a pare , &c. tan @—e = ie aad = +,&c. and _4an ane! _ ener 4 taney I cof 93 3 5 7 cancel z Arn mets —me ca ae ). When 9 =0, F = ped 8 = 64%3.3 ee 3 Meer 27-47 and fince pin a halve Svar e Az m 4f Se, which is the attraction at the furface of a {phere of the folidity m3, as was already fhewn. This laft is the con- clufion we had to expect, the fpheroid, when it ceafes to have any oblatenefs, becoming of neceflity a {phere. Ir is evident alfo, that the variations of 9 will but little af- fect the magnitude of F, while ¢ and tan @ are fmall, as the leaft power of tan ¢ that enters into the value of F is the {quare. For, inftead of cof. 33, we may, when 9 is very {mall, write can tang’; fothat Fo4rn. (ahs tng) (F— -L tang! —, &c. ). 7 Ir the oblatenefs of a {pheroid diminifh, while its quantity of matter remains the fame, its attraction will increafe till the oblatenefs vanifh, and the fpheroid become a fphere, when the attraction at its poles, as we have feen, becomes a maxi- mum. If the polar axis continue to increafe, the fphe- roid tan ues Of GREATEST ATTRACTION. 225 roid becomes oblong, and the attraction at the poles again di- minifhes. This we may fafely conclude from the law of con- tinuity, though the oblong f{pheroid has not been immediately confidered. VILL: To find the force with which a particle of matter is attracted _ by a parallelepiped, in a direction perpendicular to any of its fides. First, let EM (Fig. 11.), be a parallelepiped, having the thicknefs CE indefinitely fmall, A, a particle fituated anywhere without it, and AB a perpendicular to the plane CDMN. The attraction in the direction AB is to be determined. Let the folid EM be divided into columns perpendicular to the plane NE, having indefinitely fmall re@angular bafes, and let CG be one of thofe columns. Ir the angle CAB, the azimuth of this column relatively to AB, be called z, CAD, its angle of elevation from A, e, and m’, the area of the little rectangle CF; then, as has been already me the attraction of the column CG, in the direction AC, is. x at - fine; and that fame attraction, reduced to the direction AB, is aay -fine.cofz. This is the element. of the attraction. of the folid, and if we call that attraction f, f= re .fine.cofz. Now, if AB=a, becaufe 1: cofz:: AC: AB, AC= fo that f = — « fine ..cof2’.. Bur 226 Of th SOLIDS _ Bur BC =a.tanz; and therefore KC, the fluxion of BC, is ; a6 then, CE Senpul = GE x CK =e eee cof z pan a re core. and fubftituting this for m’, we get f—nz. fine. Next, to exprefs fin ¢, in terms of z, if we make E = BAL, the angle fubtended by the vertical columns, when it is great- eft, or the inclination of the plane ADM, to the plane ACN, then we may confider the angle CAD, as meafured by the fide of a right angled {pherical triangle, of which the other fide is go—z, and E the angle, adjacent to that fide, and therefore tane = fin(go — fe tanE = cofz.tanE. But tanE = BL tan 2 bad = Bee > fappofing BL, or CD = 8. THEREFORE Ati rege fe 4 Par aoe e tie ; cole” a’ ; pus = 2 . cof 2 or fine’ —_ b of 2” d HENCE cole = ? aie eae e an 2 b . fine = Li .cof 2? — fine’.—, cofz?; and therefore fine = 1 a’ a . b . col z a 5 i = cof 2’ Vit oe Ir this value of fine be fubftituted for. it, we have Ate Tete bnzcolz fans fre= aNit+ F cof z* LET Of GREATEST ATTRACTION. 284 Ler @ bm fin z, then “= 2 cof z, and cof z" = =I—u “3 where- aa fore, again, by fabititution, f= Kis a BBs uD est bnu ———, Let a +2, or AL) 2; . then ah JOE EF ui ; i | bnu bnu 7 2 ,— 2 S fe bu LS Ape c bu Ir, therefore, 9 be fuch an arch, that os =fing, = C bn u giors1: ‘1 ve Thigpen midst of f then ret a F9L8iN a8 Laie) Mineo a Di Vinke — = if ¢ ible ag. a hee a and fs no +8, B being’ a conftant SIT ne fs _ Now, fince fin ¢ =— ts pres ° fin %,@ is nothing when z is nothing s ; and. as f may be fuppofed to begin when z begins, we have likewife B=0; and, forxzg=n multiplied i into an arch, the fine of which is. to the fine of 2, in.the given ratio of ¢ to = em : BG BL. BC b. Or fis fuch that = = fin’ = = =X 55.=A1 x xe “Vor. VIP. IL hap Peo Mutrtipry 228 Of the SOLIDS Hence this rule, multiply the fine of the greateft elevation, into the fine of the greateft azimuth of the folid; the arch of which this is the fine, multiplied into the thicknefs of the fo- lid, is equal to its attraction in the direction of the perpen- dicular from the point attracted. Tue heighth and the length of the parallelepiped, are, there- fore, fimilarly involved in the expreffion of the force, as cey ought evidently to be from the nature of the thing. XIX. Tus theorem leads directly to the determination of the at- traction of a pyramid, having a rectangular bafe, on a particle at its vertex. \ For if we confider EM (Fig. 11.) as a flice of a pyramid parallel to its bafe, A being the vertex, then the flice behind EM fubtending the fame angles that it does, will have its. force of attraction = 2‘ 9, a’ being its thicknefs, and fo of all the reft ; and, therefore, the fum of all thefe attractions, if p denote the whole height of the folid, or the perpendicular’ from A ‘on its bafe, will be pg. But as n@ is only the attraction of the part HB, it muft be doubled to give the attraction of the whole folid EM, which is, therefore, 279; and this mutt again be doubled, to give the attraction of the part which is on the fi fide of AB, oppofite to EM; thus the element of the attraction of the pyramid is 47 ¢, and the whole attraction i a to the depth pis 471% OU Ir the folid is the frudum ofa T Wecnid Lae depth i is “y ; ‘ae vertex A, the angle 9 being determined as before, the attraction on A is i Q. Ir Of GREATEST ATTRACTION. 229 » Ir we fuppofe BC and BL to be equal, and therefore the angle BAL = the angle BAG, calling either of them ,, then fin 9 = fin 7, by what has been already fhewn; and from this equation, as 7 is fuppofed to be given, ¢ is determined. Tuis expreflion for the attraction of an ifofceles pyramid, having a reCtangular bafe, may be of ufe in many computations concerning the attraction of bodies. Ir the folidity of the pyramid be given, from the equations f=4/¢?, and fin g = fin 7’, we may determine 1, and f, that is, the form of the pyramid when fis a maximum. Let the folidity of the pyramid = m3, then #, being the al- titude of the pyramid, and 7 half the angle at the vertex ptany= half the fide of the bafe, (which is a fquare), and _therefore the area of the bafe = 4 p’ tan 7’, and the folidity of the pyramid 3 f3 tany 3; fo that p3tany = m3, Now tan y* = ane, and fin ?=finy, alfo 1—fng = fin 1 — fin’ = cofy, therefore tan 7* = ———*— ; fo that #3 = 5 I—fing ‘ 44;, _fing ey 1— fing Ss a? sig IT Gn and p? = ‘ m3, oo ae 65 py m / see, we have, therefore, f, that is 4p9 = amo if 3-(t— fing) This lat is, therefore, a maximum 4 fin 9 Ff 2 by 230 Of the SOLIDS by hypothefis; and, confequently, its cube, or 64 m3 93 X 3G) , or omitting the conftant multipliers, 93 ae muft be a maximum. Ir we take the fluxion of each of thefe multipliers, and di- vide it by the multiplier itfelf, and put the fum equal to no- thing, we fhall have, RS sel AMONG 8 8 o, or 2= Q 1 —fin@ fin @ Q cof @ cofg __cofg. fing + cofg — cofg.fing i—fng fing” fin 9 (1 — fin 9) cof @ , and inverting thefe fra@ions ? fing (4 — fin 9) 3 ioe ae tang (1—fing), or p= 3tang(I —fing). TueE folution of this tranfcendental equation may eafily be obtained, by approximation, from the trigonometric tables, if we confider that 1 — fing is the coyerfed fine of g¢ Thus taking the logarithms, we have Le= L.3+L.tane+L.coverf. ¢. From which, by trial, it will foon be difcovered, that ¢ is nearly equal to an arch of 48°. To obtain a more exact va- Ine of 9, let = arc (48° + 6), @ being a number of mi- nutes to be determined. Becaufe arc.48° = .8377580, and arc (48° + 8) = .8377580 + .0002909 @, therefore log. arc (48° + 8) = 9.9231186 + .0001506 B. In Of GREATEST ATTRACTION. 2a% In the fame manner, Ltan (48° + 8) = 0.0455626 + .0002540 £6, and L.coverf. (48° + 6) = 9.4096883 — .0003292 6 . L3 = 0.4771213 Sum = 9.9323722 — .0000752 6 Subtract Log arc (48° +8) = 9.9231186 + .0001506 6 Remainder = .0092536 — .0002258 B8=0. Whence, 6 = 283° = 41’ nearly. A sECOND approximation will give a correction =— 20%, 2 fo that @= arc. 48°. 40° Me and fince fing = fin’, fing — N fin 9, fo that 7 => 76°. 30’, and 27, or the whole angle of the pyramid = 153°. An ifofecles pyramid, therefore, with a f{quare bafe, will at- tract a particle at its vertex with greateft force, when the in- clination of the oppofite planes to one another is an angle of 153° XX. To return to the attraction of the parallelepiped, it may be remarked, that the theorem concerning this attraction already inveftigated, 4 xv111. though it applies only to the cafe when the parallelepiped is indefinitely thin, leads, neverthelefs, to fome very general conclufions. It was fhewn, that the attraction. which the folid EL (Fig. 11.) exerts on the particle A, in the di- rection AB, is 7.9, @ being an arch, fuch that fing = fin BAC x fin BAL = fin z. fin E; and, therefore, if B be the centre of a toe Of the SOLIDS a rectangle, of which the breadth is 2 BC, and the height 2 BL, the attraction of that plane, or of the thin folid, having that plane for its bafe, and 2, for its thicknefs, is 42.9. Now, 9, which is thus proportional to the attraction of the plane, is al- fo proportional to the fpherical furface, or the angular fpace, {ubtended by the plane at the centre A. For fuppofe PSQ_ (Fig. 12.) and OQ _to be two quadrants of great circles of a {fphere, cutting one another at right angles in Q; let QO9=E, and QR=z. Through S, and Othe pole of PSQ, draw the great circle OST, and through P and R, the great circle PTR, interfeGting OS in T. The fpherical quadri- lateral SQRT, is that which the rectangle CL (Fig. 11.) would fubtend, if the fphere had its centre at A, if the point Q was in the line AB, and the circle PQ: m the vertical plane ABL. Now, in the fpherical triangle PST, right angled at S, cof T = cof PS x fin SPT = fin QS x fin QR = fn E.X fin 2. But this is alfo the value of fin g, and therefore ¢ is the complement of the angle T, or p= 90—T. But the area of the triangle PQR, in which both Q and R are right angles, is equal to the rectangle under the arch QR, which meafures the angle QPR, and the radius of the fphere. Alfo the area SPT = arc.(S+T-+P—180°)73 that is, be- caufe S is a right angle, = arc.(T + P — 90) Xr = arc .(T+QR—g0) Xr; and taking this away from the triangle POR, there remains the area QSTR = arc.(QR —T—QR +90°)Xr=(90—T)r=@xX~r. Thearch 9g, therefore, mul- tiplied into the radius, is equal to the {pherical quadrilateral QSTR, fubtended by the rectangle BD. Tuis propofition is evidently applicable to all rectangles whatfoever. For when the point B, where the perpendicular from A meets the plane of the rectangle, falls anywhere, as in Fig. 15. then it may be fhewn of each of the four rectangles BD, Of GREATEST ATTRACTION. 233 BD, BM, BM’, BD’, which make up the whole rectangle DM’, that its attraction in the direGtion AB is expounded by the area of the f{pherical quadrilateral fubtended by it, and, therefore, that the attraction of the whole rectangle MD’, is expounded by the fum of thefe {pherical quadrilaterals, that is, by the whole quadrilateral fubtended by MD’. In the fame manner, if the perpendicular from the attracted particle, were to meet the plane without the rectangle MD’, the difference between the {pherical quadrilaterals fubtended by MC and M’C, would give the quadrilateral, fubtended by the rectangle MD’, for the va- lue of the attraction of that rectangle. THEREFORE, in general, if a particle A, eravitate to a rec- tangular. plane, or to a solid indefinitely thin, contained between two parallel rectangular planes, its gravitation, in the line per- pendicular to those planes, will be equal to the thickness of the solid, multiplied into the area of the spherical quadrilateral sub- tended by either of those planes at the centre A. _ Tue fame'may,be ‘extended _to all planes, by; whatever figure they be bounded, as they may all be refolved into reCtangles of indefinitely {mall breadth, and having et imine ee to’ a ftraight line? given: in pofition. © | oTHe! gravitation’ ‘ofa point toward any’ riddle 2 in'a line per- pendicular to it} is, therefore, equal to n, a quantity that éx- prefies the intenfity of ‘the attraction, multiplied into the ‘area of the fpherical figure, or, as “it may be called, the’ angular {pace -fubtendedyby the given-plane, «+ Tuus, in the cafe of a triangular plane, where’ the angles fubtended at A, by the fides of the triangle, are a, 4 and c; fince EvtEer has demonftrated * that the area of the fpherical triangle contained by thefe arches, is equal to the reétangle un- ; der a * Nov. Acta Petrop, 1792, p. 47. 234 Of the SOLIDS der the radius, and an arch A, fuch ‘that‘ cof : a ee if A be computed, the attraction cet ee 2 i Mise Ln Iw the cafe of a circular plane, our general propofition agrees with what SirIs aac Newron has demonftrated. IfCFD (Fig. 13.) be a circle, BA a line perpendicular to the plane of it from its centre B; A, a particle. anywhere in that line; the force with Sphichi Ali atumatied. in he tore eee ae a G i aD # in which the multiplier 2 7 is fupplied, being left out in the inveftigation teferred to, where a quantity only proportional to the attraction is required. ‘Now aD is the cofinie of, the. angle BAD, and, therefore, I— ae is is its verted fine 5 Jand, therefore, ¢ the arch -GEK ye deferibed: frou, the ceritre. ‘Asi with the radius 1, and if the fine, GH, arid.thée. chord,-EG, be. drawn, HE is the -verfed. fine; of BAD, and. the attra@ion —27EH.. But 2.EH = EG’, becaufe 2 is the diameter of the circle GEK; ; therefore the irra Ghent =z. EG’ = the area of the circle of which EG is the radius, or the fpherical furface, included by the’ cone, which has.A ifr ‘its — on the circle CFD. for its bafe. . is Yo Sisd odd Hi euHT bee + » * Princip, Lib. i. Prop. go. Of GREATEST ATTRACTION. 238 XXI. From the general propofition, that the attraction of any plane figure, whatever its boundary may be, in a line perpen- dicular to the plane, is at any diftance proportional to the angular fpace, or to the area of the fpherical figure which the plane figure fubtends at that diftance, we can eafily deduce a demonftration of this other propofition, that whatever be the figure of any body, its attraction will decreafe in a ratio that approaches continually nearer to the inverfe ratio of the {quares of the diftances, as the diftances themfelves are greater. In other words, the inverfe ratio of the {quares of the diftances, is the limit to which the law by which the attraGtion decreafes, continually approaches as the diftances increafe, and with which it may be faid to coincide when the diftances are infi- nitely great. TuIs propofition, which we ufually take for granted, with- out any other proof, I believe, then, fome indiftin@ perception of what is required by the law of continuity, may be ri- goroufly demonftrated from the principle juft eftablithed. Let B (Fig. 14.) be a body of any figure whatfoever, A a particle fituated at a diftance from B vaftly greater than any of the dimenfions of B, fo that B may fubtend a very {mall angle at A; from C, a point in the interior of the bedy, fup- pofe its centre of gravity, let a ftraight line be drawn to A, and let A’ be another point, more remote from B than A is, where a particle of matter is alfo -placed. THE directions in which A and A’ gravitate to B, as they muft tend to fome point within B, muft either coincide with AG, or make a very {mall angle with it, which will be always the lefs, the greater the diftance. Vor. VI.—P. IT. Gg LEY 236 : Of the SOLIDS Let the body B be cut by two planes, at right angles to AC, and indefinitely near to one another, fo as to contain between them a flice or thin fection of the body, to which A and A’ may be confidered as gravitating, nearly in the direction of the line AG perpendicular to that fection. THE gravitation of A, therefore, to the aforefaid fection, will be to that of A’ to the fame, as the angular fpace fubtended by that fection at A, to the angular fpace fubtended by it at A’. But thefe angular fpaces, when the diftances are great, are in- verfely as the {quares of thofe diftances, and therefore, alfo, the eravitation of A toward the fection, will be to that of A’, in- verfely as the fquares of the diftances of A and A’ from the fection. Now thefe diftances may be accounted equal to CA and CA’, from which they can differ very little, wherever the fection is made. THE gravitations of A and A’ toward the faid fection, are, act Ar the gravitation to all the other fections, or laminz, into which the body can be divided by planes perpendicular to AC; there- fore the fums of all thefe gravitations, that is, the whole gravi- tations of A to B, and of A’ to B, will be in that fame ratio, therefore, as ——; And the fame may be proved of that is, as ——., or inverfely as the fquares of the di- E AG ' ° XG ftances from C. Q.E. D. Ir is evident, that the greater the diftances AC, A’C are, the nearer is this propofition to the truth, as the quantities rejected in the demonftration, become lefs in refpect of the reft, in the fame proportion that AC and AC increafe. It is here afflumed, that the angular fpace fubtended by the fame plane figure, is inverfely as the fquare of the diftance. This Of GREATEST ATTRACTION. 39 This propofition may be proved to be rigoroufly true, if we confider the inverfe ratio of the fquares of the diftances, as a limit to which the other ratio conftantly converges. _ Ir is a propofition alfo ufually laid down in optics, where the visible space fubtended by a furface, is the fame with what we have here called the angular space fubtended. by it, or the portion of a fpherical fuperficies that would be cut off by a line paffing through the centre of the fphere, and revolving round the boundary of the figure. The centre of the {phere is fuppofed. to coincide with the eye:of the obferver, or with the lace of the particle attracted, and its radius is fuppofed to be unity. THE propofitions that have been jut now demonftrated con- cerning the attraction of a thin plate contained between paral- lel planes, have an immediate application to fuch inquiries concerning the attraction of ee as were lately made by Mr CAVENDISH. In fome of the experiments inftituted by that ingenious and profound philofopher, it became neceflary to determine: the at- traction of the fides of a wooden cafe, of the form of a parallel, epiped, on a body placed any where within it. (Philofophical Tranfactions, 1798, p. 523.). The attraction in the direction perpendicular to the fide, was what occafioned the greateft dif- ficulty, and Mr Cavenprisu had recourfe to two infinite feries, in order to determine the: quantity of that attraction. The de- termination of it, from ae eae eee: | is eafier and more accurate.’ : Let MD’ (Fig. 15.) replant a thin rectangular plate, A, a particle attracted by it, AB a perpendicular on the plane MD’, NBC, LBL’, two lines “teen through B parallel to the fides of the rectangle MD’. Let AC, AL, AN, AL’, be drawn. Gg 2 “i THEN, 238 Of the SOLIDS THEN, if we find g fuch that fin g = an ee the attrac- tion of the rectangle CL is 7. 9, n denoting the thicknefs of the plate. _ BLY BN So alfo, if fine'= =i X AN’ the attraction of LN is I U 2.Q. Ir fing’ = x aa the attraction of NL’ is = 2. Q”. 2 Bia BE ; erat, If in = AL x AC the attraction of L'C tl 2.0", Tuus the whole effect of the plane MD, or f = n(o+te+o"+e"). WE may either fuppofe ¢, 9 &c. defined as above, or by the following equations, where 7, 7', 7", &c. denote the angles fub- tended by the fides of the rectangles that meet in B, beginning with BC, and going round by L, N and L’ to C. fine = fing .fin, fing’ = fin,’ .finy’ fing’ = fin,’ .fin 7" fine” = fin,” .finyg. Ir the computation is to be made by the natural fines, it will be better to ufe the following formule : fin p= ~cof (x 1 J—< cof (n =P ah, fing = Ecol (7 — 9") —F cof i’ + 2°) fin ¢ Of GREATEST ATTRACTION. 239 fin 9” = = cof (1° — a") — = cof Cn + 2”) fin 9” = : cof (7""— ) ee cof (1 +7). By either of thefe methods, the determination of the attrac-. tion is reduced to a very fimple trigonometrical calculation. XXIT. THE preceding theorems will alfo ferve to determine the at- traction of a parallelepiped, of any given dimenfions, in the di- rection perpendicular to its fides. Let BF (Fig. 16.) be a parallelepiped, and A, a point in BK, the interfection of two of its fides, where a particle of matter is. fuppofed to be placed; it is required to find the attraction in the direction AB. Tuoucu the placing of A in one of the interfections of the. planes, feems to limit the inquiry, it has in reality no fuch ef- fect ; for wherever A be with refpect to the parallelepiped, by. drawing from it a perpendicular to the oppofite plane of the fo- lid, and making planes to pafs through this perpendicular, the whole may be divided into four parallelepipeds, each having AB for an interfection of two of its planes; and being, there- fore, related to. the given particle, in the fame way that the: parallelepiped BF is to A. Let GH be any fection of the folid parallel to EC, and let it. reprefent a plate of indefinitely fmall thicknefs. Let AB =x, Bd, the thicknefs of the plate =%. Then o being fo determined, that fin g = fin B’AH x fin BAG, the at- traction of the plate GH is @%, which, therefore, is the ele- ment 240 Of the SOLIDS ment of the attraction of the folid. If that attra@ion =F, thea F= fos. But [ersox— fre; and the determi- nation of F depends, therefore, on the integration of x Q. 2 Ne rapa ; . kfing Now ¢cofg= fin Qg, and, therefore, x ¢ = Sie? 7 ’ ' ' b b Ir BG =4, and BH =8, then fin BAG = —~ = ——__. AG NB? +e x? agate | ue ygeied le ‘ Petre and fin BAH = 35 = ————; fo that fing = —_____ x A N Q* +. x? ND? x? : Lanjh evi, rae pe Jepe ™? = FEC PH arr bit s b* B? ie Bo oa SA DERI logos ah ak ac eUtO+s) sco tNPEP ES CFC THY NG EH Lay AGAIN, becaufe iy oe ee @ fine= —hxs j* 1. wt 2 a? 3. Nb + x7 NB +x (0? + x)? seirute 2 al Bx x x b 7 @+xe)? C+e)* (tx? HENCE ate or ¢ = ( — OB xa tee bRxs ) Grey x@+xyF C49? x (+0)? , x Of GREATEST ATTRACTION. 241 oes GRA Cee LO ae ROE it ick Oleg 213 JJG c? being put for *+ @”. (e? + x) (c ot Ce ia ‘ LBxex THEREFORE « 9 SS (b+ x*) (c* + x*)? bBx x | (@° + x*) (c? ++ x")? Now, (Se = bLog oath +C; G+ 2) +x)? S ecieen (Harmonia Menfurarum, Form. 1x.) ; and f paeceu =. (2°-+x*) (c°+- x") * b+Not x L P p Log a +O THEREFORE fue = oe b4NG pa Lo Bee Te L qt eee oe @} and et EEE ARAL oO Ce B4NGpx bE OE ay et Dy AOG, se err Spe sa eG, ox °8 NB x g & NB + x? Ir, then, we determine C, fo that the fluent may begin at K,. and end at B; if, alfo, we make 7 the value of 9, that corre- fponds to AB or 4; and 7’, the value of it that correfponds to AK 242 Of the SOLIDS AK or a’, we have the whole attraction of the folid, or F = | B4+NC+a + NP+ a NP Fa B+Ne +a" b4+NCO4a4 - NG + a" ( Ve +a b+VYe +a” 44— na’ —b Log — p Log Ir, in this value of F, we invert the ratios, in order to make the logarithms affirmative, and write like quantities, one under the other, we have F = 7a — 7a’ B+Ve+a* _ Vb +a a OF 64+V%c +a Vb + a" 2 b+Noifa®, NB ta +@L AES Bod es ¥ °8 hasN PEE Ve +a" Tue firft two terms of this expreflion deferve particular at- tention, as y is an arch, fuch that fin, = fin BAE x fin BAC; therefore, by what has been before demonftrated, 4 is the mea- fure of the angular fpace fubtended at A by the retangle BD. The firft term in the value of F, therefore, is the product of the diftance AB, into the angular fpace fubtended by the rec- tangle BD. In like manner, the fecond term, or 7a’, is the product of the diftance AK, into the angular {pace fubtended by the rectangle KF. Tue relation of the quantities exprefling the ratios, in the two logarithmic terms, will be beft conceived by fubftitu- ting for the algebraic quantities the lines that correfpond to them in the diagram. Becaufe c\ = J°+ 6 = EB’ + BC: Of GREATEST ATTRACTION. 243 BC? = EC’, therefore c= EC or BD. So alfo, c’+ a aa BD‘ + BA’ =AD*%, becaufe ABD is a right angle, &c. Thus, (AF+FN)AE (AD + DE) (AN (AF + FM) AC (AD + DC) AM’ Fo7a—74+BE. Log BC. Log Tuis expreflion for the attraction of a parallelepiped, though confiderably complex, is fymmetrical in fo remarkable a de- gree, that it will probably be found much more manageable, in inveftigation, than might at firft be fuppofed. That it fhould be fomewhat complex, was to be expected, as the want of con- tinuity in the furface by which a folid is bounded, cannot but introduce a great variety of relations into the expreflion of its attractive force. The farther fimplification, however, of this theorem, and the application of it to other problems, are fubjects on which the limits of the prefent paper will not permit us to enter. The determinations of certain marima de- pend on it, fimilar to thofe already inveftigated. It points at the method of finding the figure, which a fluid, whether elaftic or unelaftic, would aflume, if it furrounded a cubical or prifma- tic body by which it was attracted. It gives fome hopes of be- ing able to determine generally the attraction of folids bound- ed by any planes whatever ; fo that it may, fome time or other, be of ufe in the Theory of Chryftallization, if, indeed, that theory fhall ever be placed on its true bafis, and founded, not on an hypothefis purely Geometrical, or in fome meafure arbi- trary, but on the known Principles of Dynamicks. Vou. VIL—P. I. le 6 v. i] ’ * ? _, seb rtoldsdsseio1 ot. nt Leaking jai Ag: Fins ‘the. > Pbigisa ed Z te tee | 2 He Sagran. Pe: a ae Prt acy 7 choi bata ae wep i serge map: alqatoa ek wee ie Bcage oe aia -bluodk aitedT . int sedd coi. Le +o “2 9b Hileyedti a e. rae e j eee omit ‘ai is Trans RS LAM, Vol Vi page 220. SI - 4 H 4 4 4 i , gaia | Pipher im iperameth sein y's ae mee Don . a [ee t tite iene . V. An Account of a very extraordinary Effect of Rerrac- TION, observed at Ramsgate, by the Reverend S. Vince, -. A.M. F.R.S. Plumian Professor of Astronomy and Ex- perimental Philosophy at Cambridge. Communicated by Parrick Witson, Esq; F. R.S. Evin. [Read 5th January 1807.| “HE: phenomenon about to be defcribed, was feen on Au- guft 6. 1806, about feven in the evening; the air being very ftill, and a little hazy. The tops of the four turrets:_of Dover Caftle ufually appear above the hill, lying. between Ramfgate and Dover; but, at the above-ftated time, not only the tops were vifible, but the whole of the Caftle, appearing as if it were fituated on the fide of the hill next to Peat and rifing as much above the hill as ufual. Let AB (Plate VIII. Fig. 1.) reprefent the termination oe the hill; v, x, Ww, y, the tops of the four turrets of the Caftle, as they ufually appear. - But, at the time above mentioned, befides thus feeing the turrets, the whole Caftle murs was vifible, and ap- peared as if it had been brought over and placed on the Ramfgate fide of the hill, as|reprefented in the figure.’ Thisi phenomenon was fo very fingular and unexpected, that, at firft fight, I ehovgnt | it to be fome illufion ; but, uponcontinuing my obfer- Hihi8 co: vation, 246 EXTRAORDINARY EFFECT vation, I was fatisfied that it was a real image of the Caftle. Upon this I gave the telefcope to a perfon. prefent, who,.upan attentive examination, faw alfo a very clear image of the Caftle, exactly as I had defcribed it. He continued to obferve it for about twenty minutes, during which time the appearance re- mained precifely the fame; but rain coming on, we were pre- vented’ from making any further obfervations. Between us and the land; from which the hill rifes,-there was about fix miles; of} feay:and from thence'to the top of the hill about the fame diftance, and we were about| feventy , feet. above, the fur- face of the water. THE hill itfelf did not appear through the image, which, it might have been expected to do. The image of the Caftle ap- peared very ftrong, and well defined; and although the rays from the hill behind it, muft undoubtedly have come to the eye, yet fo it was, that the ftrength of the image of the Caftle fo far obf{cured ‘the: back-ground, that it made no fenfible i im- preflion upon us. Our) attention was of courfe principally di- rected to the image of the Caftle ;. but if the shill’ behind had been at all vifible,|it could not haveefcaped our !obfervation; 4s. we: continued to look at it for a confider able time Wish a. eed telefcope. lliv stow atjo3 adit ‘A .PHENOMENON ost this kind I do not remember to have feen defcribed; and it muft have been ‘a? very extraordinary’ ftate of the air to have produced. it.’ ‘Tt is manifeft, that’a ray of light coming from the top of the°hill,' muft have comie to the eye ina curve lying between the 'two curves deferibed by the rays coming from the a and bottom of the Caftle; in or- der to produce the-effe@tian 970 Shyu Hires | . Ler AB (Plate VIII. Bigg 2) repretent ‘lie Caftle, EG the! Cliff (at Ramfgate), BTD the Hill, DC the Sea, E the place of the {pectator, Tothe top of the hill) AywE’ a rayof light coming from the top of the Caftle to the {pectator, Bewk DI1TLIG of REFRACTION. 247 BxwkE a ray coming from the bottom, and TxzE a ray coming from the top of the hill, falling upon the eye at E, in a direction between thofe of the other two rays; then it is manifeft, that fuch a difpofition of the rays will produce the obferved appearance. To effec this, there muft have been a very quick variation of the denfity of the air which lay be- tween the two curves yv E, x w E, fo as to increafe the curva- ture of the ray TxzE, after it cuts BwE in x, by which means, the ray T x z E, might fall between the other two rays. The phenomenon cannot be otherwife accounted for. As there are not, that I know of, any records of a phenomenon of this nature, the conftitution of the air muft have been fuch as but very rarely happens, or fuch an appearance would before have been taken notice of. THE phenomena which I faw at the fame place, and which I defcribed in the Philofophical Tranfactions of the Royal So- ciety for the year 1798, I explained upon the fame principle, that of a quick variation of denfity; and this was afterwards confirmed by fome very ingenious experiments made by Dr Wo.taston. Perhaps this phenomenon may afterwards be fubjected to an experimental illuftration. Vr. 7 ay iq Le a Pes ETA, MR is fon penoasedg ome Settkcinett a a “total aoe cerlefit aK ay * A 2 magigalbrandhtad ibe sds ' Sieais seco feon dele eitend- aud, it rer Taghed Hees ah Vey : Spy sitesi Me ashe ‘hae Oe ly ‘sibel rr ar Oe il ig oa Ai yeh Ne a Sidaesseues aia ie sng tet give iene ices Peta 2 ee Gea . ria OBR. Lendaitnor fern tae" aN abd: & atbate of aie Hiaise,, aeagt 9 en antic: , ees ati ae et cay a ex: Ae oon Vie ig) ence. ee Cer ae gia Fath ¥ es ee, Se the Sent, al F. a GPG ipevhinndt Thue eg ar eet Boe: Ath ae 4 ri ae eae GR | De it ERY: OF, ‘oe Gutile Bi ‘Ke (pentoceia ss B, ae f Perna 7 i = + _ ey ; it : b “ ie i i H a j Hi f Ha i i ii mi : hi Pirate VI | 7 a et WT Lrans RS dim?’ Vol II pp. 248 = = “yl I = VI. Some Account of the Large Snake ALEA-azaGuR, (Boa Constrictor of Linnavus), found in the Province of Tipperah. Communicated by Mr James Russewt. Ez- tracted from the Memorandum Book. of Joun Cof#Se Scort, Esq. [Read 28th April 1807]. iy tee 1. 1787. Larce) aca of this {pecies. was brought to Comillah. ‘It meafured 15 feet 3 inches in length, and 18 inches in a a about the middle. This: meafurement, however, waried confiderably by the, wreathings:and contortions it ae, in order to free itfelf from confinement. Tue cefophagus, from the mouth to the pylorus, or bottom of the ftomach, meafured altogether 9 feet:3 inches, and was fo wide as to take in a man’s head with eafe: The ftomach was eafily, diftinguithed: by the thicknefs of its coats, or the number of rugz on its internal furface.; But. there was »no contraction. atthe. cardia or entrance of. the ftomach.., The outlet or pylo- rus, however, was fo narrow as hardly to admit two fingers. r THe, head of the fnake was fmall.in proportion to its body. And. J was curious, to obferve the mechanifm of the jaw, by which it can, fo eafiky take into.its mouth ally fubftance as large as the thickeft part of its pody, poonti £ : THE ar #0 «ACCOUNT of a4 LARGE SNAKE. Tue lower jaw contiits of two bones, connected anteriorly by ie the anterior ends can be feparated an inch from each other. The pofterior extremity, or condyle of each lower jaw- bone, is likewife connected, ,to the head. in fuch a manner, as to ‘allow’ of cénfidérable fepar: ation. The two bones which com- ‘\pole the upper jaw) are capable only of a very fall here’ of -feparation at the fymphifis or anterior pait. —” . #8Duis-fingulan degree of laxity in the fttu@ture of ‘the arti- culations, permits of a degree of diftenfion which is) incompa- cible with the firmnefs requifite to perform the function of ma- ftication. July 7. 1790. A snakE of the allea {pecies was brought in, of a very un- common thicknefs in proportion to its length, which induced me\to open ite A very large guana was extracted from the gullet.and ftomach;' for’the °animal was gorged to the throat. The. guana, from the nofe to the tip of the tail, meaftired 4 feet 3, inches, and in circumference round the belly 1 foot 6 inches; and yet the :fnake; after the guana° was taken’ out; ‘meafured only 8 feet 6 inches in length. ©) °° } FOE, _ Tue circumference of this fnake is not given’; but if it’ bore the fame proportion-to its: length that it did in’ the former inake, it would be nearly 10 inches. In this inftance, there- fore, the fnake had {wallowed ‘an animal of si magnitude than itfelf almoft im the proportion of 9 to 5.’ dies ‘On the 16th of the fame month another fnake was brought in, having nearly the ‘fame appearance as the laft, but ftill more diftended. It was opened while yet alive, and an entire fawn of one year old extracted. The fawn meafured 1 foot 8 inches round the belly ;' and the extreme “Tength ‘ats the Hake was only g feet 3 inches. April ACCOUNT of a LARGE SNAKE. 250 April 5. 1791. A sNAKE of the fame fpecies was brought to Comillah and opened, from which a fawn was taken ftill larger than the one juft mentioned ; but the fnake was 10 feet 6 inches in length. Ir is the general opinion, that fnakes break the bones of their prey before they {wallow it, if the animal be of any confi- derable fize. This, however, I am difpofed to doubt, as in none of the above inftances had the animal fuffered fuch offifraction, if I may be allowed the expreflion. The mechanifm of the jaws, and the width of the gullet above defcribed, render fuch violence unneceffary. THE animal is fwallowed very gradually, being firft, I fu- fpect, well lubricated with flime, with which this kind of large. {nake appears abundantly provided. THESE circumftances may undoubtedly be deemed rather fa- bulous by thofe who have never feen nor examined large fnakes. But they are facts not to be denied, and are well authenticated by every one who has had opportunities of feeing and opening fuch fnakes. Durine Mr LeEck1E’s refidence at Comillah, I have learned from undoubted authority, that a fnake of the above mentioned {pecies was found dead, with the horns of a large deer fticking in his throat, fuppofed to be the caufe of his death. The fnake and the horns were both brought to Comillah in this fituation ; but in a putrid ftate. The fnake meafured above 17 feet in. tength; and the bones of it were afterwards fent to Mr CHARLES CoLLinson of Banleak. Vou. VI.—P. II. Ti VIN. + i , Bs f wget. att | , ‘tail 3 If © peatecia Sitks a tille - - ee Ge meer es nee taal bay Larale so bait wicks anes sie “eat a ms thea tye «| a ee on > vant 4 eo shit gna se Seis Patel aH ely hai stlvait AUT DRS ai SERB Se oy 65184, Nagi askin at Rit o9'sHhoe Mad Sra? 2 iene “Dieren niarcanntade Sree wake trie i ieee we ail ott ie werk week we a * Seblact Vee pike adware aga rie Aas Bipaae ieee am pes land iN we ‘ifte aie hes ne al The aie MEN Oi FEN bg TERN § ‘ cs 2 ta . ? I ee VII... Chemical Analysis of a Buacx Sanp, from the River Dee in Aberdeenshire; and of a Coprer.Orz, from Ar- _threy in Stirlingshire. By Tuomas Tuomson, M.D. Lecturer on Chemistry, Edinburgh. [Read 18th May 1807.| HE fpecimen which formed the fubject of the firft of the following analyses, was brought from the banks of the river Dee, about feven years ago, by my friend Mr James Miu, who at that time refided in Aberdeenfhire. By him I was informed, that confiderable quantities of it are found in different parts of the bed-of that river,—that it is called by the inhabitants iron- ‘sand,—and that they ufe it for fanding newly written paper. I tried fome experiments in the year 1800, in order to afcertain ‘its nature ; but’ was ‘too little fkilled at that time, both in mi- neralogy and practical chemiftry, to manage an analyfis of any confiderable difficulty. Tue black powder is mixed with a good many fmall whitith, reddifh, and brownifh grains, which, when examined by means of a glafs, prove to be pieces of quartz, felfpar, and mica. From this it would appear, that the fand of the river Dee confifts chiefly of the detritus of granite or gneifs. “WHEN a magnet is pafled over the fand, fome of the black grains adhere to it, and are by this means eafily obtained fepa- liz rate. 254 ANALYSIS of a BLACK SAND rate. But after all that can be attracted by the magnet is re- moved, the greater part of the black powder ftill remains. This refidue is indeed attracted by a powerful magnet, but fo very feebly, that it is not poflible by means of it to feparate it from the grains of fand with which it is mixed. Thus we learn, that the black matter confifts of two diftin@ fubftances; one of which is powerfully attracted by the magnet, the other not. As this fecond fubftance was obvioufly {pecifically heavier-than the grains of fand with which it was mixed, I placed a quanti- ty of the powder onan inclined plane, and by expofing it cau- tioufly, and repeatedly, to a jet of water, I fucceeded in wathing away moft of the grains of fand, and thus obtained it in a ftate of tolerable purity. Tue firft of thefe minerals we may call leony and the fecond iserine, as they belong to mineral fpecies which oryc- tognofts have diftinguifhed by thefe names. jesTRON-SA ND .-enicow ccowsk ideal TuE iron-fand is much fmaller in quantity than. the iferine, and does not exceed one-fourth of the mixture at moft. Its co- lour is iron-black. It is in very {mall angular grains, common- ly pretty fharp-edged, and fometimes having the fhape of, im- perfect oftahedrons. The furface is rough ; the luftre is feebly glimmering and metallic; the fracture, from the fmallnefs of the grains, could not be accurately afcertained, but it feemed to be conchoidal. Opake, femihard, brittle, eafily reduced to pow- der. Powder has a greyifh-black colour; powerfully attrac- ted by the magnet ; {pecific gravity 4.765. 1. As acids were not found to ac upon this mineral, 100 grains of it were reduced to a fine powder, mixed with twice its from the RIVER DEE. 255 its weight of carbonate of potafh, and expofed for two hours to a red heat, ina’ porcelain crucible. The mals, being foftened in water, was. digefted, in muriatic acid. By repeating this procefs twice, the whole was diffolved in muriatic acid, ex- cept a brownifh-white matter, which being dried in the open air, bara I9t grains. ” st by 2. THE muriatic acid folution, which had a deep yellowith- brown colour, was concentrated almoft to drynefs, and then di- luted with water. It affumed a milky appearance ; but nothing was precipitated. Being boiled for fome time, and then fet afide, a curdy-like matter fell. Jt was of a milk-white colour, weighed, when dry, 7 grains, and f aescan the properties of oxide of titanium. 9G 4 Tue seGidea) Liquid being fuperfaturated with ammonia, a dark reddifh-brown matter precipitated, which being feparated bythe filter, dried, drenched in oil, and heated to rednefs, af- fumed the appearance’ of a black matter, ftrongly attracted - by the magnet. It weighed 93.7 grains, and was oxide of iron. -4. THE 19.5 grains of refidual powder, being mixed with four times its'weight of carbonate of foda, and expofed for two hours:to a red heat, in a platinum crucible, and afterwards heated with muriatic acid, was all diflolved, except about a grain of blackifh matter, which was fet afide. 5. THE muriatic folution being concentrated by evaporation, alittle white matter was feparated. It weighed ith of a grain; and poflefled the characters of oxide of titanium. 6. WHEN as6 0S ANALYSIS of a BLACK SAND 6. WHEN evaporated to drynefs, and rediffolved in water, a white powder remained, which proved to be filica, and Pass. after being heated to rednefs, weighed one grain. 4. THE watery folution being fuperfaturated with potath, and boiled for a few minutes, was thrown upon a filter, to fe-. parate a reddifh-brown matter, which had been precipitated. The clear liquid which paffed through the filter, was mixed with a folution of fal ammoniac. A foft white matter flowly fubfided. It was alumina, and, after being heated to rednefs, weighed half a grain. 8. THe brown-coloured matter which had been precipitated by the potafh, when dried upon the fteam-bath, weighed 20:2 grains. It diffolved with effervefcence in muriatic acid. The folution had the appearance of the yolk of an egg. . When boiled for fome time, and then diluted with water, it became white, and let fall a curdy precipitate, which weighed, when dry, 4.6 grains, and poflefied the properties of oxide of tita- nium. g. THE refidual liquor being mixed with an excefs of ammo- nia, let fall.a brown matter, which, after being dried, drench- ed in oil, and heated to rednefs, weighed 6 grains. . It was ftrongly attracted by the magnet, but was of too light a colour to be pure oxide of iron. I therefore: diffolved it im muriatic acid, and placed it on the fand-bath, in a porcelain capfule. When very much concentrated by evaporation, finall white’ needles began to make their appearance in it... The addition of hot-water made them difappear; but they were again form- ed when the liquor became fufficiently concentrated. Thefe eryftals, when feparated, weighed 1.3 grains, and proved, on examination, to be white oxide of arfenic. During the folution of from the RIVER DEE. 2577 of the 6 grains in muriatic acid, a portion of black matter fepa- rated. It weighed 0.2 grains, and was totally diffipated before the blow-pipe in a white fmoke. Hence, it muft have been ar- fenic. Thefe 1.5 gr. are equivalent to rather more than 1 grain of metallic arfenic. Thus, it appears, that the 6 grains contained 1 grain of arfenic, which explains the whitenefs of their colour. The reft was iron. It can {carcely be doubted, that the proportion of arfenic prefent was originally greater. Some of it muft have been driven off when the iron oxide was heated with oil. 10, THE infoluble refidue, (No. 4.), was with great difficul- ty diffolved in fulphuric acid. When the folution was mixed with ammonia, a white powder fell, which weighed 0.8 grains. It was accidentally loft, before I examined its properties. But I have no doubt, from its appearance, that it was oxide of tita- nium. 11, Tuus, from the 100 grains of iron-fand, the following conftituents have been extracted by analyfis : Black oxide of iron, - 98.70 White oxide of titanium, 12.65 Arfenic, - - I.00 Silica and alumina, _— - 1.50 Total, 113.85 Here there is an excefs of néarly 14 grains, owing, without _ doubt, to the combination of oxygen with the iron and the titanium during the analyfis. Hap the iron in the ore been in the metallic ftate, the ex: cefs of weight, inftead of 14, could not have been lefs than 30. For the black oxide is known to be a compound of r00 metal and 258 ANALYSIS of a BLACK SAND and 37 oxygen. Hence, I think, it follows, that the iron in our ore muft have been in the ftate of an oxide, and that it’ muft have contained lefs oxygen than black oxide of iron. A good many trials, both on iron-fand, and on fome of the other mag- netic ores of iron, induce me to conclude, that the iron in moft of them is combined with between 17 and 18 per cent. of oxy- gen. This compound, hitherto almoft overlooked, by che- roifts, I confider as the real protoxide of iron. THENARD has lately demonftrated, the exiftence of an oxide intermediate be- tween the black and the red; fo that we are now acquainted with four oxides of this metal. But the protoxide, I prefume, does riot combine with acids like the others. Analogy leads us to prefume the exiftence of afifth oxide, between the green and the red. Ry | * As to the titanium, it is impoffible to know what increafe of weight it has fuftained, becanfe we are neither acquainted with it in the metallic ftate, nor know how much oxygen its differ: ent oxides contain. It is highly improbable, that, in iron-fand, the titanium is in the metallic ftate, if it be made out that the iron is in that of an oxide. The experiments of VAUQUELIN and HEecuT, compared with thofe of Ktaprotn, have taught us that there are three oxides of titanium, namely, the blue, the red, and the white. From an experiment of VAUQUELIN and Hecut, and from fome of my own, I am difpofed to con- fider thefe oxides as compofed of the following proportions of metal and oxygen: METAL. OXYGEN. r. Blue,. © too 16 acIeed,.. “16S ee 3. White, 100 49 I find, that when the white oxide of titanium is reduced to the {tate of red oxide, it lofes.one-fourth of its weight; and that | red from the RIVER DEE. 259 red oxide, when raifed to the ftate of white oxide, increafes ex~- actly one-third of its weight. It was the knowledge of thefe facts, that led me to the preceding numbers. And I think they may be ufed, till fome more direct experiment lead us to pre- cife conclufions. _ Rep oxide being the only ftate in which this metal has yet occurred feparate, we may conclude that it combines, in this ftate, with metallic oxides, and that the titanium in iron-fand, is moft probably in this ftate. But white oxide, diminifhed by one-fourth, gives us the equivalent quantity of red oxide. On that fuppofition, the titanium prefent, before the analyfis, in the 100 grains of ore, weighed 9.5 grains. THE appearance of the arfenic furprifed me a good deal, as it was altogether unexpected. I am difpofed to afcribe it to fome particles of arfenic pyrites which might have been acci- dentally prefent. This conjecture will appear the more pro- bable, when we reflect, that arfenic pyrites very frequently ac- companies iron-fand. Before the microfcope, the iron-fand ap- pears to contain fome white fhining particles, which, probably, are arfenic pyrites. Tue fmall quantity of filica and alumina, I afcribe, without hefitation, to grains of quartz and felfpar, which had adhered to the iron-fand, and been analyfed along with it. Some fuch grains were actually obferved and feparated. But others, pro- bably, efcaped detection. 12. Ir thefe fuppofitions be admitted as well founded, the iron-fand was compofed of Protoxide of iron, 85.3 Red oxide of titanium, 9.5 Arfenic, - - 1.0 Silica and alumina, - 1.5 Lofs, - - 27 100.0 Vou. VI.—P. II. Kk The 260- ANALYSIS of a BLACK SAND The lofs will not appear exceflive, if we confider, that a por- tion of the arfenic muft have been fublimed, before the pre- fence of that metal was fufpected. Uron the whole, I think we may confider the fpecimen of iron-fand examined, as compofed of g parts protoxide of iron, and 1 of red oxide of titanium. The prefence of titanium in this ore had been already detected by Lampapius, though, as. I have not feen his analyfis, I cannot fay in what proportion. Il. ISERINE. TuHE colour of this ore is iron-black, with a fhade of brown. It confifts of {mall angular grains, rather larger than thofe of the iron-fand, but very fimilar to them in their appearance. Their edges are blunt ; they are fmoother, and have a ftronger glimmering luftre than thofe of the iron-fand. Luftre femi- metallic, inclining to metallic. The fracture could not be di- ftinctly obferved, but it feemed to be conchoidal; at leaft no- thing refembling a foliated fracture could be perceived. Opake, femihard, brittle, eafily reduced to powder; colour of the - powder unaltered; fpecific gravity 4.491 *; {carcely attract- ed by the magnet. ~ 1. A HUNDRED grains of the powdered ore were mixed with fix times their weight of carbonate of foda, and expofed for two hours to a red heat, in a platinum crucible. The mafs obtain- ed being foftened with water, diffolved completely in muriatic acid. When the folution was concentrated, it affumed the ap- pearance * Ir, as the following analyfis would lead us to expect, the fpecimen exami- ned was a mixture of four parts.iferine, and one part quartz and felfpar, the {pe- cific gravity of pure iferine fhould be 4.964. from the RIVER DEE. 261 pearance of the yolk of an egg. It was boiled, diluted with water, and fet afide for fome time. A white matter gradually depofited, which, when dried on the fteam-bath, weighed 53 grains, and poffefied the properties of oxide of titanium. 2. THE liquid thus freed from titanium, was evaporated to drynefs, and the refidue rediflolved in water, acidulated with muriatic acid. A white powder remained, which, after being heated to rednefs, weighed 16.8 grains, and poffeffed the pro- perties of filica. 3. THE folution was precipitated by ammonia, and the brown matter which had feparated, boiled for fome time in liquid pot- afh. The whole was then thrown on a filter, to feparate the undiffolved part, and the liquid which came through, was mix- ed with a folution of fal ammoniac. A white powder fell, which, after being heated to rednefs, weighed 3.2 grains. It was alumina. 4. THE brown fubftance collected on the filter, was dried, drenched in oil, and heated to rednefs. It was ftrongly attract- - ed by the magnet, and weighed 52 grains. _ s. Ir was digefted in diluted fulphuric acid; but not being rapidly acted upon, a quantity of muriatic acid was added, and the digeftion continued. - The whole flowly diffolved, except a blackifh matter, which became white when expofed to a red heat, and, as far as I could judge from its properties, was oxide of titanium, flightly contaminated with iron. It weighed 1.8 grains. 6. Tue acid folution being concentrated by gentle evaporation, a number of fmall yellowifh-coloured needles made their ap- Kk 2 . pearance / 262 ANALYSIS of a BLACK SAND pearance init. By repeated evaporations, all the cryftals that would form were feparated. They weighed 6 grains. I redif- folved them in water, and added fome ammonia to the folu- tion. A fine yellow powder fell, which I foon recognifed to be oxide of uranium. It weighed 4.2 grains. 4. Tuus it appears, that the 52 grains (No. 4.), attracted by the magnet, contained 46 grains of iron, and 6 grains of ura- nium and titanium. 8. Tue following are the fubftances feparated from 100 grains of iferine, by the preceding analyfis : Oxide of titanium, 54.8 Oxide of iron, - 46.0 Oxide of uranium, 4.2 Silica, - - 16.8 Alumina, - ae Total, 125.0 Here is an excefs of no lefs than 25 grains, to be accounted for by oxygen, which muft have united to the three metals during the procefs. As to the filica and alumina, there can be little hefitation in afcribing them to grains of fand, which had been mixed with the ore. The pure iferine, in all probability, was compofed of iron, titanium, and uranium. If we fuppofe that each of thefe metals exifted in the ftate of protoxide, we muft diminifh the titanium by one-fourth, the iron by one-feventh nearly, and the uranium, according to BucHouz’s experiments, by one-fifth. This would give us, Titanium, from the RIVER DEE. 263 Titanium, - 41.1 Iron, - FieemiGOah Uranium, - 3:4 Silica and alumina, 20.0 103.9 Here, then, is ftill an excefs of nearly 4 per cent. But this I am difpofed to afcribe to the oxides of titanium and uranium, having been only dried upon the fteam-bath. Upon the whole, it appears, that, in the fpecimens of iferine analyfed, the pro- portions of titanium and iron were nearly equal, and that the uranium did not exceed 4 per cent. The appearance of uranium furprifed me a good deal. I perceive, however, that it has al- ready been detected in this ore, from an analyfis publifhed by Profeffor JamEson, in the fecond volume of his Mineralogy, which, I underftand, was made by Lampapivus. The fpeci- men examined by Lampapius yielded very nearly 60 parts of titanium, 30 of iron, and 10 of uranium. Whereas, in mine, if the foreign matter be removed, there was obtained, very nearly, — - - 48 titanium, 48 iron, 4 uranium, y ) Cole) But, there can be no doubt, that the iferine which I analyfed was ftill contaminated with a good deal of iron-fand ; for it was impoflible to remove the whole. 13th 2 264. ANALYSIS of a COPPER ORE et RB ST a nn eae Analysis of the Grey Coprer Ore, from Airthrey. THE copper mine of Airthrey, near Stirling, confifts of a thin vein, which runs. through the weft corner of the Ochils. It has been twice wrought, by two different companies. But, in both cafes, was abandoned, after a few years trial. I went to it fome years ago, and examined the ore, at the requeft of one of the proprietors... The fpecimens which were employed for the fubfequent analyfis, were the pureft that I could feled, - out of a confiderable quantity. I was told, however, that from the lower level, which was at that: time full of water, much richer ore had been extracted. But, afterwards, when the lower level was freed from its water, I went down to it myfelf; and found the ore precifely of the fame kind’as in the upper, with this difference, that it was more mixed with calcareous fpar, and perhaps, on that account, more eafily {melted. Tue veinftones in the Airthrey mine are fulphate of barytes, and carbonate of lime, and with thefe the ore is almoft always more or lefs mixed. Tue colour is at firft light fteel-grey ; but the furface foon tarnifhes, and becomes of a dark dull leaden-grey, and in fome places aflumes a beautiful tempered fteel tarnifh. .Maflive and difleminated. In fome {pecimens, it exhibits the appearance of imperfect cryftals. Internal furface fhining and metallic ; but, by expofure, it foon becomes dull. Fracture fmall-grained, inclining to even. Fragments indeterminate, and rather blunt- edged. Semihard, the degree being almoft the fame as that of calcareous fpar; for thefe two minerals reciprocally fcratch each from AIRY HREY. 265 each other. Streak fimilar, opake, brittle, eafily frangible ; {fpecific gravity 4.878. 1. To free the ore as completely as poflible from foreign matter, it was reduced toa coarfe powder, and carefully pick- ed. It was then digefted in diluted muriatic acid, which dif- folved a quantity of carbonate of lime, amounting to 13 per cent. of the original weight of the ore. 2. THUS purified, it was dried on the fteam-bath, and 100 grains of it were reduced to a fine powder, and digefted in di- luted nitric acid, till every thing foluble in that menftruum was takensup. The refidue was digefted in the fame manner, in muriatic acid; and when that acid ceafed to act, the refidue was treated with nitro-muriatic acid till no farther folution could be produced. . The infoluble matter was of a white co- lour; it weighed 6.9 grains, and was almoft entirely fulphate of barytes. No traces of fulphate of lead, nor of oxide of anti- mony, could be detected in it by the blow-pipe. 3. THE three acid folutions being mixed together, no cloudi- nefs appeared, nor was any change produced; a proof that the ore contained no filver. | .4. Tue folution being evaporated nearly to drynefs, was di- luted with water, and precipitated by muriate of barytes. By this means, the fulphuric and arfenic acids, which had been, formed during the long-continued action of the nitric acid on the ore, and the prefence of which had been indicated by re- - agents, were thrown down ; for nitrate of lead, added to the re- fidual liquid, occafioned no precipitate; a proof that no arfe- nic acid was prefent. 5- THE 266 ANALYSIS of a COPPER ORE §. Tue liquid, thus freed from arfenic acid, was mixed with an excefs of ammonia. It affumed a deep blue colour, while a brown matter precipitated. It was feparated by the filter, and - being dried, drenched in oil, and heated to rednefs, it was to- tally attracted by the magnet. It weighed 45.5 grains, and was iron. 6. THE ammoniacal liquid was neutralifed by fulphuric acid, and the copper thrown down by means of an iron plate. It weighed 17.2 grains. 47. To afcertain the quantity of fulphur and arfenic, 100 grains of the purified ore, in the ftate of a fine powder, were put into the bottom of a coated glafs-tube, and expofed for two hours to a red heat. When the whole was cold, and the bot- tom of the tube cut off, the ore was found ina round folid mafs, having the metallic luftre, a conchoidal fracture, and the colour and appearance of variegated copper-ore. It had loft 16 grains of its weight. 8. The upper part of the tube was coated with a yellowith- brown fubftance, like melted fulphur. It weighed 12.6 grains. Thus, there was a lofs of 3.4 grains. As the tube was long, this lofs can fcarcely be afcribed to fulphur driven off. I ra- ther confider it as water. For towards the beginning of the procefs, drops of water were very perceptible in the tube. Whether this water was a conftituent of the ore, or derived from the previous digeftion in muriatic acid, cannot be deter- mined, 9. WHEN the 12.6 grains of yellowifh brown matter de- tached from the tube, were digefted in hot potafh-ley, the whole was diffolved, except a fine blackifh powder, which weighed from AIRTHREY. 267 weighed 1 grain, and was arfenic. The diflolved portion I confidered as fulphur. 10. THE potath folution, being mixed with nitric acid, 4 grains of fulphur fell. The remaining 7.6 grains muft have been converted into fulphuric acid, by the ation of the nitric acid. Accordingly, muriate of barytes occafioned a copious precipitate. 11. THE 84 grains of roafted ore being reduced to a fine powder, mixed with half their weight of pounded charcoal, and roafted a fecond time in a. glafs-tube, one grain of fulphur fu- blimed. But the tube breaking before the roafting had been continued long enough, the procefs was completed in a cru- cible. The roafted ore weighed 70 grains. 12. From the preceding analyfis, we learn that the confti- tuents of the Airthrey ore, are as follows: Iron, - 45-5 Copper, - 17.2 Arfenic, - 14.0 Sulphur, - 12.6 Water, - 3-4. Foreign bodies, 6.9 99.6 Lofs, - ey. 100.0 If we fuppofe the water and the earthy refidue to be only acci- dentally prefent, then the only eflential conftituents are the firft four, and the ore would be a compound of Kron,*- "51.0 Copper, 19.2 Arfenic, 15.7 Sulphur, 14.1 100.6 Vou. VI.—P. II. Ll If 268 ANALYSIS of a COPPER ORE, tse. If we compare this analyfis with feveral analyfes of grey cop- per ore, lately publifhed by Kiaprorn, we fhall find, that the conftituents are the fame in both but the proportions of the two firft ingredients are very nearly reverfed. KLAPROTH ob- tained from 0.4 to 0.5 of copper, and from 0.22 to 0.27 of iron, This renders, it obvious, that the two ores were not in the fame ftate. I have little doubt, that the difference, how- ever, is merely apparent, and that it arofe, altogether, from. a quantity of iron pyrites, and perhaps alfo of arfenic pyrites, which I could not feparate from the grey copper ore which I examined. Both of thefe minerals could be diftin@ly {een in many of the fpecimens, intimately mixed with the grey cop- per; and I have no doubt that the fame mixture exifted, even in thofe fpecimens which’ were felected as pureft. The differ- ence in the proportions of copper and arfenic, obtained by Kx APROTH * in his various analyfes, is fo confiderable, as to lead to a fufpicion, that even his fpecimens, in all probability, contained a mixture of foreign matter. ® GEHLEN’s Jour, vol. y. p. 9. 21.13. VIII. VII. New Series for the QuapRATURE of the Conic SEc- Trons, and the Computation of LocGaRiTHMS. By Wiiu1am Wautace, one of the Professors of Ma- thematics in the Royal Military College at.Great Mar- low, and F.R.S. Evry. | Read 27th June 1808. | HE Quadrature of the ane Sections, and the Compu- tation of Logarithms, | are problems of confiderable importance, not only in the elements of Mathematics, but alfo in the higher branches of that fcience. On this account, every fuccefsful attempt to fimplify their refolution, as well as any new formule which may be found applicable to that purpofe, muft always be interefling, and muft in fome meafure contri- bute to the improvement of mathematical knowledge. _ 2, THE object of this Paper, is to give folutions of thefe problems, which fhall be at once fimple and elementary, with- out employing the fluxional or other equivalent calculus ; and it is prefumed, that thofe which follow, will be found to par- take fo much of both thefe properties, that they may even ad- ;mit of being incorporated with the elements of Geometry and Analyfis. Befides, the formule which refult from the invefti- Bion, are, as far as I know, entirely new, while each is appli- Eihe2 cable 2/70 NEW SERIES for the cable to every poflible cafe of the problem to be refolved. Now this laft circumftance is the more remarkable, as it generally happens, that a feries which applies very well to the quadra- ture of a curve within certain limits, is quite inapplicable be- yond them. 3. ALTHOUGH, in a general way, this Paper may be faid to treat of the quadrature of the Conic Sections, yet there is one of them, namely, the Parabola, which I fhall not at all notice ; becaufe, although its area may be found in a way analogous to - that which is here employed in the cafe of the other two, yet the formula which would thence refult, muft, from its nature, be the fame as would be found by any other mode of proceed- ing. As the quadratures of the ellipfe, and any hyperbola may be deduced from thofe of the circle and equilateral hyperbola, I fhall, in the following Paper, treat only of the two laft; and as the quadrature of a fector of a circle, and the rectification of its bounding arch, are reducible the one to the other, it is a matter of indifference which of thefe we confider. I fhall, however, confine myfelf to the latter. 4. In treating of logarithms, I might, after the example of the earlier writers on this fubject, deduce the formule for their computation from thofe which we fhall find for the quadrature of the equilateral hyperbola. I prefer, however, treating this fub- ject in a manner purely analytical, without adverting at all to the hyperbola, being of opinion, that every branch of mathematics ought, as much as poflible, to be deduced from its own pecu- liar principles; and therefore, that it would be contrary to good method, to have recourfe to the properties of geometrical figure, when treating of a fubject entirely arithmetical. | 5. To QUADRATURE of the CONIC SECTIONS, &c. 271 5. To proceed now in the inveftigation of the different fe- ries, for the rectification of an arch of a circle, let A denote any arch, the radius being fuppofed unity. Then, from the arithmetic of fines, we have PL ETE ERNE heat tanA 2tantA 2 2 In this formula let each term of the feries of arches I I I I a,-4, -4,=- Aes» ——A4, 274 8 21—2 a Qn—1 (which is a geometrical progreflion, having the number of its terms #, and its common ratio 4,) be fucceflively fubftituted for A, and let the refults be multiplied by the terms of the corre- fponding feries of fractions I I T 79 °° °° gn—2? Qa—1? then we fhall obtain the following feries of equations : I I I 1 a - — — tana, tana 2tanta 2 I I - = eee — -tan74, 2tanta 4.tant+a 4 I I I —_—_—_—__—_—__ = s————_-_ — -tania 4tan+ta 8 tania ae ss a T tan 1 Hleo A’ tan; A. tanzA fam. A |. tan A tanzA therefore tan A ; and tan 2A S * tantA. In this expreffion, let $a, +a, ga, &c. be fubftituted for A, and let the refults be divided by 8, 16, 32, &c.; then we get Se it a? poi 4a TX tan : ; 4tania ¢ 5 ORs 1 t tal a v5 tan y's a> +——— X tanga, jtanta &e. from which i hat in the feries, ? =? 4.2 tan? rom witich it appears, that 1n the leries, - =——_ + _ tan-a a tana 2 2 I IT I : i I I me or Pes eat ae tan re 7 t &e. each term after the third (that 4 after {tan 4a), is greater than a third proportional to the two terms immediately before it, taken in their order; and this is, another limit to the rate of convergency of the feries. to. Tue, limits which we have found to the rate of conver- gency of the feries, enable us alfo to affign limits to the fum of all the terms after any given term. Let. the feries be put under this form,, | ail I I I I I : ar ia a pigs Awe Ft Tm) + TV (m4) “tr T(m+2) ex" &e. Vou. VI.—P. IT. M m. where 246 “NEW SERIES for the where Tm), T(m41), T(m+2),. &c. denote the terms whofe places in the feries are exprefled “) the numbers m, m-+1, m+2, &c. Then, becaufe T(m+2)< : Tem4n)s T(m+3) < “ Tom42), T(m+4) < 4 Tinta) Sci. We have Tom+a) + T (m+ 3) + Tem44), &e. < 3 (Tet) cu + T (m42) + Tim+3) +, &e.) ‘That is, putting S for Tim42) + T(m+3) + T(m44) +, &e. or for the fum ofall the terms after T(m4 1), s <2 (Tet) + S), and hence 2 S = A E TAPED TR S a T(m+1)- Thus it appears, that the fum of all the terms of the feries fol- lowing any term after the firft, is lefs than the third part of that term. 11. AGAIN, from what has been faid in Article 9., we have Tm +2)> Tet) T(m+1), and therefore pie > Timet) (m) (m+1) bs (m) Tosa) 5 Timt2) ang Teta T(m43) and fimilarly Tusaey. Hi} => Tian 7 an Tee > yas i ay and ®UADRATORE of the CONIC SECTIONS, &c. 277 and fo on, From which it follows, that ge v4: T(m42) > —pet) T(m+1)s (m) T(m+1) Dm) T(m+4) > a Tn 43)s T (m4 3) > Tom42); | &e. HENCE, taking the fum of the quantities on each fide of the. fign >, and putting S for T(m42) + T(m43) + Tata) +, &e. we get S > a (Tents) +S). — Set) T'(m+1) Therefore s so sr Dial? and ,confequently by re= | dudtion, cf, Sn) all okt geht 12300 Rymiea omg) ofy fo Ta) — Tints) ino Gan. ‘which it appears, that the fum of all the terms following any- affligned | term. after the third, is greater than a third pro- portional to. the difference, of the, two terms immediately be- fore it and the latter of the two, But fince this limit will not differ much from: the former, which is ; “tas ib it may be “yes —4 1 Tewe | 3 (Tim — Tim+ 1) OT ee ag fi T(m+1)3 Mm 2 which 278 NEW SERIES for the which formula, by reduction, will be found to be the very fame as the other. 12, THE refult, then, of the whole inveftigation, may be briefly ftated as follows: Let @ denote any arch of a circle of which the radius is unity, then fhall I ~tan ~ 4 pis 4 ; tan a tan mp 2 1B 4. +E Ske 6¢ to + T(m) + Tomtr) + S53 sb Ae Tim) and ‘T ero denote ‘any two fucceeding terms of the feries — ~ tan : a +e tan 54 +, &c., their places in it being Sox by the 2 he mand m+13; and where S is put for the fum of all the remaining terms); and the limits of S are the two Oe a £ Tees) ahd F Tay (Tora Fern) Tey a 3 (T (m) — T (m+ 9) is, S is lefs than the former, but greater than at latter. tie expreflions tan = ray, tan a a, tan 54, &e, are cern de- 2 duced from tan a, and from one another, by a eral! known fos mula in og arithmetic of fines, which may be ea hg ai gia I Year WS eel! aren tan At.) 13. | Now proceed to the inveftigation of a fecond formula for the rectification of the circle; and for this purpofe refume the QUADRATURE of the CONIG SECTIONS, &c. AOI O13 19 ig I MESH Pop! oI : the equation ee Ee tan = A WhICh q tan A 2tantA.~ 2 2g ti y king the fquare of each fide, is transformed to Meo ere t) Gens ® r fer (fo tan A. # 2 2 In this formula, let each term of the feries of arches okt eat, ane? a Ce ee: si, AM cs is 272. by ta- “sof which the number of terms is 7, be fubftituted fucceflively for a, and let the refults be multiplied by the cc en terms of the feries of Paetiogs? a ee ete ae A 4p dO) BARE get nee there will be forthed the feries of equations : Bthieve os —otsche saeqh “tani ag tan’ a Wee eee tan 4 2 I oe ae I aT =e are = AT) tan’ = ai al Daf =i = tan -@ 2° tan’+a 4 tan’ + 4 4 I | I at Hoh tien ee Op ot etna aes etal 4 4 tania 8* tan’ +4 43 9 I I I I —— SOs — tan’? — a 8° tan’ 3a 16° tan =", 2 43 16 ho“ ) I I a CFO 22n—4 tan—“ 22n—2 tan” x 2 ! 7. ie ga eat Qr—1 : | I a I ith : fe tan a 22n tan’ — Qn fs a Q227a—2 tan Qn—ti & 280 NEW SERIES for the Let the fum of thefe equations be taken, as in the inveftigation of the firft formula, and obferving that the feries I I I I ele TL ies z aE 2 Tae” 3° S eT ere ae 2tan’$a 4 tanya 8’ tanga 5 2gd ae Q7—! is found on both fides of the refulting equation, let it be reject- ed from both; then we obtain Now it appears, that one part of this exprefflion, viz. I I I I I st potato et 24 2.43 2.42—=V" ~ ~ is a geometrical feries, the firft term of which is =; the laft I Pee! f ; term ———, and common ratio - 3 therefore its fum is QA ie 4. é “om ' re I 2 (r+). 3 4 Alfo, fince 2" tan = is the expreffion for the perimeter of a polygon, formed by dividing the arch a into 2—t equal parts, by drawing tangents at the points of divifion, and producing | she ee [ I v1 | 22" tan’? ¢ hic wh tote 2” ee T tan’? a I tan? La J tanzla Be pan? 4 < ne om ies 8 rr a I 12 I eC I I wie TOW) ee Ee a ee ag eh 4” 3” Mer ariarvarse aR T= eo ). QUADRATORE of the CONIC SECTIONS, &c. 281 the adjacent tangents until they meet, (Art. 6.); therefore 22” tan? < will be the fquare of that perimeter. Let the pe- rimeter itfelf be denoted by P, then, fubftituting P* in the equation inftead of 22” tan “, and = (i— +) inftead of the EON: nee feries to which it is equivalent, and bringing 5 ~ to one fide, we get | aeaigeet zo g tan cat © tans Bs sgt ae L 15. THIs is true, whatever be the value of 2, the number of terms of the feries in the parenthefis. Let us now conceive feries to be continued indefinitely, then, as upon this hypo- thefis, n may be confidered as indefinitely great, — « will become 4r lefs than ie) affignable quantity, and therefore 2 (x — =) 5 4” ill become fimply 2 5 ; moreover, P will in this cafe become a, (Art. 7.), and P* will become a. Thus, upon the whole, we fhall 282 | NEW SERIES for the fhall have agar. “Hz RAT Get 3 ss aoe I I : hos == — (7 tan! sat ans eet ~ tan’ gat ~ tan’ <3 4 : | 4 &) and a may be confidered as a fecond formula for the reétiff- eation of any arch of a circle; for the procefs by which an arch is found from the fquare of its reciprocal is fo fimple, that the latter being known, the former may alfo be regarded as known. 16. INstEAD of exprefling the fquare of the reciprocal of the arch in this manner, by the fquares of the tangents of its fub-multiples, we may exprefs it otherwife by the -fquares of i — 1% f their fecants. For fince tan* : a= fect’ g— I, and tan’-@ 2 — fec? + a — 1, and fo on, therefore the feries 4 pian Sat tan’ gat paw pat 5 tam Fat, ee, 4 2 16 is hh oe to | E focr I ug 2 So Z 2 eee. ris me as fec* ae = fec’ g Ot. 7, et yg at Be ri I I ae ee en But the latter part of this expreffion is evidently an infinite geometrical QUADRATURE of the GONIG SECTIONS, &c. 283 geometrical feries, of which the firft term is ? and ratio : ; therefore its fum will be : 3 hence, by fubftitution, and put- , I I ting -—— for Ear he I, we have pes (Lipide sat... vt, a =a (tee redharayae Zatz fee's gets - fec rhe +, &e.), _ which is the feries to be inveftigated. 17. THERE is, however, another form, under which the fe- ries brought out in Article 15. may be given, and which I con- fider as the beft adapted of any to the actual calculation of the length of an arch. This transformation will be effected, 1—cof2A "at, in” the well-known ee tale, SORE ah inftead of A, we fubftitute fucceflively a, Para &c.3; we iq A doitlw 2911 1033) I—cof ,4 eee +, Sc} and this is the fecond feries which I propofed to. inveftigate, reduced to its moft convenient form. 18. We may determine two limits to the rate of conyergen- cy of the feries juft now found, in the fame manner-as we have found the limits of that of our firft feries ; and, indeed, the rea- foning employed in the one cafe is immediately appacaly to the other. For if the firft feries, which is a co 2 it + 2 tan 2 a Stan? ga! tan ~.4-++, &c. a tana 2 2 4 Fe ial | 8° be QUADRATURE of the CONIC SECTIONS, &c. 285 be put under this form 7 + Ta) + Te) + _ + Ty) +, &e. a tana where T(:) is put for + : tan + 5% and T (2) for z tans a, and T(3} Ep AT | ee for g tanga, &c., then, as the formula given at the conclufion th l ° ° D —cof+ = of the laft article, becomes by fubftituting tan’ rs for : TF eofta = aT 1—cofta and tan <4 for ata ae? and fo on, (i 1+cofa cofa , 1 4 1-—cola “ 6 S| = ‘ey ay eae 7 So ; { C an ato ant a+ i qitn at, &c.) it may be otherwife expreffed thus, [2 11+ cofa t= ) 41—cola pO FT otTedt Mandy Xe) where it is to be obferved, that the fymbols T(2), T(3), T (4), denote the very fame quantities in both feries. Now, as we have found (Art. 8. and g.), that each term of We the feries of quantities T (2), T(3), T(4), &c. is lefs than i of the term immediately before, but greater than a third proportional to the two terms immediately before it, taken in their order, it Nn 2 is 286 ~NEW SERIES for the manifeft, that each term of the feries in our fecond formula muft be lefs than = of the term before it, but greater than a third proportional to the two terms immediately preceding it ; and thefe are the limits to the rates of convergency of our fe- cond feries. 19. WE may alfo affign limits to the fum of all terms, after any propofed term: for putting it under this form L — (Ti + Tee Ta) + Tat) + T(n42) +, &e.)s where T(1), T(2), + «+ T(u), &c. now denote merely the terms of the feries taken in their order, then becaufe I T(m+2) < 74 T(m41)s i | T(m+3) < 7A T(m+2); Te ea Tie (m4) < =e T(m+3), &e. Therefore, ‘Tm+a) FT (m4 3) + Tim 44) by &e. < Z (Te + T(m42) + Tim+3) +, &c.) That QUADRATURE of the CONIC SECTIONS, &c. 287 That is, putting S for Tom+2) + Tam+3) + T(m+4) +y &e. : I S< *, § Tom+1) ++ S}, and hence S < Se T (m+ x)» Thus it appears, that the fum of all the terms following any . I term, is lefs than 7 of that term. 20. As to the other limit, it muft be the fame:as the like li- mit of our firft feries, on account of their having the fame li- mit to their correfponding rates of convergency. That is, putting S to denote as above, then T(m+r) ae T(m) — T(m+1) Fema) 5 T(m) — 16 T orS > J ak te ie) er Py rie Se . 5 py pent (Lm) aE a) 21. Ir yet remains for us to confider how the feries of quan- i 6 E— cold’ F-— col lA ! ; tities T+ cota’ I+ cofta’ &c. are to be found. Now this may be done, either by computing the cofines of the feries of I 3% &c. one from another by means of the . ts. ao arches 4, — 4, - 4, 2. 4 Ecol 3 - formula cof 5 A= 1+ col A, and thence computing the fe- 2 1—cofta 1—cofta 1+colta’ 1+ cofia compute each fraction at once from that which precedes it, by a formula which may be thus inveftigated.: ries of fractions &c. Or we may Put 288 NEW SERIES for the P 1—cofA_, 1—cofsA __, _I—t iets =” ae eo ee . q TtcofA _ i. i, _1—? ee eee oe aD CON oe ae ae ‘ _ ay te col © I —?’ ! = 29 Cheren a} ay d erefore rae” Vik? and hence Lie oe Sr Tong ee Pi sor 22. Upon the whole, then, the refult of the inveftigation of the fecond feries may be ffated briefly as follows. Let a de- note any arch of a circle, its radius being unity, then (i 1-feofa 4 &, 41—cola 6 (1 t—cofta, ri—cofya, ti~cofya 2.) | fF rFeothat p rfeot ya" ger eorg at | t + Tm) + Ta@+1) +8 vis where T(m) and T(m+1) denote any two fucceflive terms of the feries in the parenthefis, and S denotes the fum of all the fol- lowing terms 5 and here $ will always be between the limits I I (Tay — 16 To 41) Tons) 15°" sate i it 15 (T(m) — T(m+1)) that is, it will be lefs than the former, but greater than the lat- ter quantity. | Tue feries of cofines are to be deduced one from another by means of the formula | fA cof LA= Ni Ms inh ee 2 2 Or, | = a 34 I ‘tan? A 8tan}iA 8 LtanzA QUADRATURE of the CONIC SECTIONS, &c. 289 Or, compute the feries of quantities ¢, 7’, t’, 7”, &c., one from another by means of the formule be Scoble.) pe Vit) 2) es Se, 6 fe te 1+ cola Vata S Ve eee Then will (11+cofa *.J 4 1—cola Pl TS bed fos + EOL O AT VAT ty +5); where T(m), ae 1), and S denote the fame as before. I consIDER this fecond feries for the rectification of the circle (under either of its forms), as preferable to the other fe- ries given at Article 12. for two reafons ; firft, becaufe of its ereater rate of convergency, and, next, becaufe the quantities TG T . ° cof a, cof a a, cof y a, &c., alfo the quantities ¢, z’, ¢”, &c. can 2 be deduced more eafily one from. another than the feries of I I I tangents tan 4, tan = qa, tan r 4, tan g &c. 1023. Tuat we may, inveftigate another’ feries, let us I I I eee, ee tap 2 A: the chbes tanA 2tanz+A 2 nr ad refume the formula of both fides of the equation being now taken, the refult.is ok I _ Heewtiety Bis. I tan 2 at g tan’ 5 Af To 290 ~NEW SERIES for the To the fides of this equation let tlie Si oa fides of the equation t= I I + ¥ =—S5 an A tan A Saar -2 j be added; then we get I x ‘ait zy tan? A + tanA 8 aa +A aaa, ae tT .. r+tan’A___ fec”A "ys Oy Sigg ae canseA? 204 Similarly, I I I ee —q (tan 5 A+ tan?— A). I | fee; I ao Biss as, BaD -A tan? —- A= atts Bot tan + Ashitam 2 AJ at 3S I ae i =a I 2. df ° tan= A (1+ tan 5 A) Stan : A fec =A: therefore, by fubiti- tuting, we get fevtA — fecrtA I I Pi tan?> A” 8tan?iA 8 i 2 a Ee 24. From this formula, by fubftituting a, 2 a, f Qi = for A, and multiplying the fucceflive refults by the fraGions I ‘ ! r; 7 = +++ gy we deduce the following feries of equations, the number of which is 2: fec’a tans a QUADRATURE of the CONIC SECTIONS, &c. 294 2 : 2 L fect a = Gelce ts 4 — I tan! a fec’ la, tan? a 2'tans’ ia 8 2 Z ics fec ia I I I | ae 48s im) SO So = tan a 2 et oer! 4 eaten precios ie ~ tan 2 afec’ L gs 8 8 VoL. i II. Oo and 292: 3 > NEW SERIES for the» and here the number of terms amPeRPE the feries i in, ‘the ‘pa- renthefis is 7. Let us now conceive the feries to go on ad infinitum, fo that n may. be confidered as indefinitely great, then it is manifeft, ; ee that feg* = will become equal to, rad’ ;, now 2» tan = will -be- come 4, (Art. 6. and 7.) therefore 23*tan3 ss will become a? ; fec* — hence, fubftituting ~ = *. for ee: in our equation, and’ tran- pp 32tans & a , , + aa {pofing, we get at laft fec*a | 1 [ S + Ftan Fa fect 2 a+% tan @ fec = at E aS. 4 : I I fec? 7 pe bate @2 tans ~/G@1EC gashs &e. 3613 and this is the third feries which I propofed to ge for the rectification ofan arch of a circle. 25. THE feries we have juft now frase is evidently of a very fimple form; it alfo converges pretty faft, each term be. ing lefs.than the 16th of that which precedes it. As, however, to apply it to actual calculation, it will be neceflary to ex- tract the cube-root of a number, which is an operation of con- fiderable labour when the root is to be found to feveral figures, perhaps, confidered as a practical rule, this third formula is in- ferior to the two:former... But if, on’ the other hand, we re- gard it merely as an elegant analytical theorem, it does not feem. lefs. deferving of notice tham either, ef them.. 26. THE QUADRATURE of the CONIC SECTIONS, 8c. . 293 ., 26.'Txe mode. of ‘reafoning by which we have.found feries Jexprefling the three finft, powers of the: reciprocal.of an.arch, » will-apply equally to any higher power, but the feries will be- ‘\come more and more complex as we proceed, befides requiring in their application the extraction of high roots. In the cafe-of the fourth power, however, the feries is fufficiently fimple, and converges fafter than.any we have yet inveftigated, while, at the fame time, in its application we have only extractions of “the {quare root. On thele accounts, I fhall here give its invef- siesta. " I I I etre the elias ee eae Gates AS tan — ye let the fourth power, and alfo the {quare of each fide of the equation be ‘taken, the refult will be re, ee Bifvte, bY f = La — - tan 6 re tan¢A 16 tan* $A erat g a BA p= tama A, : | i pea pA tan’ tan’ A ~ 4tan?+A | BET, the firft of thefe equations * multiplied by “ and the fecond by, 3, and. let the refults be added ; then, reducing the fractions to a common denominator, we get a+4tan’A_ sHaten Ai Ir An fie cat “tan? Ar 536 .c tan? tA 5 © 36. j4tan P2as an a LET us, for the fake of brevity, exprefs the nee quanti- ty eater e by the Fymbek yA Git is not to be under- ; ftood : as the ‘product of two quantities f and A, but as a charac- ter denoting a particular funétion of the arch A ;) and, fimilar- Ty, let pe 2 be denoted by f +A, and foon. Alfo / Oo 2 let 204 NEW SERIES for the let the other complex expreffion 4tan’ + A + 3 tan*+/A be de- noted by F 4 A; and if there were others like it, that is, which only differed by having + A, 3 A, &c. inftead of + A, they would be denoted by F4 A, F; A, &c.; then our laft en wall ftand thus, wes ots py cd oF he ye SA = freA 6 berg Fa and fimilarly, putting +A, +A, 7A, &c. fucceflively for A, and ee : I -Cpishgy multiplying the refults by the feries of fraGtions 76’ 16” i6% &ec. By continuing this feries of equations to m terms, and then taking their fum, and rejecting what is common to each fide of the refult, exactly as in the BPEAL SS a of the three preced- ing formule, we gee get ek EAE deren A - +iF3 a+ Sr: A+Z,Fi A... bE S fA 7 = ua and this equation holds true, 7 pele any whole pofitive num- ber whatever. 27. Lar. QUADRATURE of the CONIC SECTIONS, &c. 295 my LET us how, however, sa tr n aceite great, then : a tr -A 3+ 4tan the quantity aa f=, or Sp becomes fimply 3 % be- . (2" tan =) pd A ; . A caufe tan a and confequently 4 tan*—, vanifhes, and 2” tan — = Qn P Qn ah becomes A, as we have already had occafion to obferve (Art. 6. and 7.).Alfo the geometrical feries I I I eae having the numberof its terms indefinitely great, and their common ratio ve be cae Therefore, by fubftitution and. ‘ranfpofition we have | | oe. & =fA+i : a5 2Fs A+ =cFSA+ 2 FSA4, &e. 2 or, a eee for fA, and, F3)A; &c.: the quantities which, - thefe fymbols exprefs, [ 3+4tan’ A 14 f° tant A TY 5 y | mn, 1 2a Se = =~ Si (4tan’>A+3tan++A)+ (4 tan] A-++3tan*+A) bd t 5g Catan Ps ES ae and’ thid 4 is one form of es pce which we propofed to invef- tigate, » Yb 7 28. Tuis.. 298 NEW SERIES: for they. 28. Tuts feries, however, ile of being expreffed _ under another form, better adapted to calculation, and to effect this transformation, let us begin Bors the term oo In —cof2A this quantity let 2 ‘TH ofa At fubftiuted for tan’ A; it then a a Popa, 090: rae becomes, afer Mado redudion, aii 10 soli PEaGEE Gill A- gain, in this expreflion let es be fubftituted for cof* 2 A, we then get tiods bas gebig tahhAriiol 13cof 4 A + 72 of'2 A tan* A 73 3+ col4 A— 4 cola A” The remaining terms of the féries, which‘are Scalar to one another, and of the form 4 tan’ — an 3 tan A, ceva of a é, pices A like» transformation ; - >for “ ae fubftituting®’ ea for a anny ofa ee cof = A ‘ tan’ A, and again ——— for cof"! TA in the renuhe we iL) : ; ts3 h) = gett i. — SS, /.>3 Le 4 F360 ae col 4. Tatsum bas | beloqorq 9% " gab cof Bactopeot “Al | bai 3383 ee By . QUADRATURE of ‘the CONIC SECTIONS, &c. 294 By; fubftituting: thefe: Regevoonneen nia in the b denice it becomes: | A= Pu esreer@ nies 3 core A — 4 col 2A ag _ Fe Rg cof 12 600A Ty 13.—s0f A — ra.cof A 16; 3-+-col2zA+4colA, 16".3-+ cof A+ 4coft A, be le 13 —‘cof+ A —12cof+ ea &e. t 16° pi ed A'+ 4 cof+ wee tert + 2 8 8 ee Slane wi ae a ee 3 3 — tan’ — n And this is true, 2 being any number whatever. Now, if we confider that 3” tan = exprefles the fum of the fides of a figure formed by dividing the arch into 3” equal parts, and drawing tangents at the points of divifion, whofe orders, reckoned from one end of the arch, are indicated by even numbers, (that end itfelf being reckoned one of them), and producing each to meet thofe adjoining to it, and the laft to meet a radius of the circle produced through the other end of the arch, it will be obvious, that 2 being fuppofed to increafe indefinitely, the expreffion 37 tan will have for its limit the arch a, and in this cafe the feries will go on ad infinitum. ‘Thus we fhall have 8 tania tanta tan a tan a4 =" 9 tan’: DY net = 2? —tan2a += 2 Dig PKC 6 | 3 3 3° 3 3 3-—tan’;54 and by sf atiaeGtion) 1 Pipa Niue 8 tania tanta oy Cain tg <= gt aya tame +8 lar paaering ie @ tana, 3 3—tan'}a 3° 3—tan‘$a 3° 3—tan’, and this is the feries which I propofed to inveftigate. Vo. VI.—P. IL. Pp 31. THE 300 NEW SERIES for the 31. THE feries we have juft now found, may be prefented under various forms. Thus, by confidering that Too « COLA oo s2du COlNS Sith Ik tan At. mA” “2in* A ~ 1 col 2A’ and that fin A tam ATS Sega Sued 2 A COL ee ay 5 Sit ees 3—tan AS fm’ A” 24col°A—1™~ 21--2cei2e cof’ A it will appear that by due fubftitution the feries may be other- wife exprelfled as follows : be fin: a 2 finave cl (hae fin; 4 2 fina rl, 21—cola 33+2colta 3° 1+2c0la) 331+2c0l.4 And other forms might be given to it, but they would all con- verge with the fame quicknefs, and each term would be lefs than ;th of the term before it. The feries, however, under whatever form it be given, and all others which like it require for their application the trifection of an arch, are, when com- pared with thofe we formerly inveftigated, of little ufe as prac- tical rules ; becaufe it is well known that to determine the fine, or other fuch function of an arch from a function of its triple, is a problem which produces a cubic equation ofa form which does not admit of being refolved otherwife: than by trials, or by infinite feries, both of which procefles are fufficiently labo- rious, and only to be employed where the object in view can- not be attained by eafier means. 32. As from the different feries we have found for the reéti- fication of an arch of a circle, the {pirit of our method muft be fufficiently obvious, 1 fhall not inveftigate any others at pre- fent. QUADRATURE of the CONIC SECTIONS, &c. 30% fent. Before leaving this part of our fubject, however, it may -be proper to obferve, that the fecond feries may be deduced from the firft, and the third from the fecond, and fo on with ref{pect to innumerable others of the fame kind, by the fluxional or differential calculus. _ For refuming the firft feries I I Be = tan < j tana -tamard &e. @ tana s he a5 8 8 Ts if we take the fluxion of oe term, confidering @ as a variable quantity, we have —da __ —dafec’a ae a meee *Son''a tig T da fec’ = a2 he , dafec’ = iat pda fec"s at, &c. and hence, changing the figns, and rejecting da from each term, and putting 1-+ tan” {a for fec*= a, we find 1 hae he fee ) tan’ a =) \ sd PtP +, &e. — ( - tan’ - = 4 F tan’ +a tan = L G ‘a an’ 5 +9 an gat, &c.) In this expreiflion, anes of the numeral feries * zit ai, pe &c, (which is a geometrical progreflion having its common ratio *) fubftitute its value, viz. ; and the refult is 1g a tan” a sat? ft tan” 5+ Gta’? gat Guta t, &e.2 which is identical with the formula found at Art. 15. Ppa FRom 302 NEW SERIES for the From this feries, by a like mode of proceeding, we may de- duce our third feries, and thence, again, our fourth, and fo on: but this mode of inveftigation, although very fimple, is certain- ly lefs elementary than that which we have followed. And it muft be kept in mind, that one principal objeét of this paper is to employ only the firft principles of geometry and analyfis in treating of the fubjects announced in its title. 7 33- By a mode of deduction differing but little from that employed in the laft article, we may even derive our firft feries from a known formula, the invention of which is attributed to EuLER. It is this, : . a = finafec= a fect afec! at, &c. * 2 4 8 From this expreflion, by the theory of logarithms, we get log a= log fin a + log fec ; a-+ log feos a-+ log fec 34 +, &c, we have now only to take the fluxions of all the terms, and re- ject da, which is found in each, and the refult is I I I I I I I I Ao —— + = tan = at Stan - @-+ —tan=2¢4t, &c. pa 7 oma 2 ia 4 Ts 8 T: which is the feries in queftion. 34. I now proceed to the inveftigation of formule for the quadrature of the hyperbola, and as the principles from which they. * Tus formula, although very elegant as an analytical transformation, does not feem to admit of being applied with advantage to the reGtification.of an arch, on account of the great number of faétors of the- product which would be requi- red to give a refult tolerably correct. QUADRATURE of the CONIC SECTIONS, &c. 303 they are to’ be deduced are in effect the fame as we have had occafion to employ when treating of the circle, it will be pro- per to ule the fame form of reafoning, and the fame mode of notation, in the one cafe as in the other. THEREFOR, in the equilateral hyperbola ABB’, of which C is the centre, (Plate IX. Fig. 1.), and CA the femitran{verfe axis ; let CB be drawn to any point B of the curve, and BD perpen- dicular to CA; then, in imitation of the notation commonly ufed in the arithmetic of fines, which we have followed in the former part of this paper, we fhall confider the co-ordinates CD, DB, as functions of the hyperbolic feGor ACB, and put- ting S to denote its area, we fhall denote the abfciffa CD by ab S, and the ordinate BD by ord S. Draw AE touching the curve at its vertex, and eae CB in E; then, from fimilar triangles, we have AE = == x CA; therefore fuppofing the femitranfverfe axis AC to be unity, ord S ab S° AE = Now this expreflion for the tangent correfpond- fin A cof A’ preffion for the tangent of an angle A, we may fimilarly de- note AE by the abbreviation tanS. In like manner, if CB’ be drawn to a point B’ of the curve, bifecting the fector ACB, and meeting AE in E’, and B/D’ be drawn perpendicular to CA; = to a hyperbolic fector S, being analogous to. the ex- then, ‘as the fecor ACB’ will be ~S, it follows, that CD’ = ab i S, B'D’ = ord : S, and AE’ = tan - S; and fo on. 35. From: 304. NEW SERIES for the 35. From the nature of the hyperbola, we have ab S = ab?2 $+ ord*+S, ordS = 2ab -SordiS; re 2 ee therefore, by divifion, a a r ord +3 ordS ~ 2ordtS ' 2 ab+tS’ I : I I I that is rgb oto ~ tan-S. ? haa eee fe 2 Tuts laft formula exprefles a property of the hyperbola per- fectly analogous to that of the circle (Art. 5.), from which we have deduced our firft four feries for the rectification of an arch. Therefore fimilarly, putting s, = 5, ra &c. fucceflively inftead of S, and multiplying by the feries of numbers 1, = ? &c. we have as in that article igo I I I —~ = ———-+-=- tan -s, tan 5 atantgs . 2 2 I ill I I Saha: Swen Hie oR 3 2 tan; Ss 4 tan zs 4 4 I I I I ee a = ee 7 = a = tan ed 5; 4tan7s 8 tan ys 8 8 &e. Tus feries of equations being fuppofed continued until their number be ”, by proceeding exactly as in Art. 5. when treating of the circle, we obtain 4} 2” tan — 3" QUADRATURE of the CONIC SECTIONS, &c. 305 ¢ I +(Ftanis = tant s+ Ttant s+ tants... Beat he ie hs tS Cu oS tans 16 16 36. Let us now fuppofe the hyperbolic fector ACB to be di- vided into 2” equal parts, by lines drawn from the centre to the points I, 2, 3, 4,...-7 in the curve, and tangents to be drawn at the extremities of the hyperbolic arch AB, and at the alter- nate intermediate points of divifion 2, 4, 6, &c. fo as to form the polygon AFF’ F” F’” BC. Then, by a known property of the hyperbola, the triangles ACF, FC 2, 2 CF’, FC 4,... F” CB are all equal, and as their number is 2”, the whole polygon bounded by the tangents, and by the ftraight lines AC, CB will be equal to the triangle ACF taken 2” times. But the area of this triangle is >ACx AF =~ tan = (becaufe AF = tan ra ; 2 2 2 2 \ therefore 2” tan = exprefles twice the area of the polygon AFF’ F’ F’” BC. Let Q denote this area, then, fubftituting : Q. for 2” tan = and multiplying all the terms of the feries by 2, we have bind I Bu 2 Ir Bi siy I —j| tan —-s —-tan- §s — tan-— Ss ~tan—S... f ( 2 ats 4 i 3) Hi 8 16 I Ss ’ — tan — }. hi q é Sic ge =) Now 6 let 306 NEW SERIES for the Now, the rectilineal fpace Q_is evidently lefs than the hy- perbolic feétor s ; but 2 may be conceived fo great that the dif- ference between Q and s fhall be lefs than any aflignable f{pace, as it is eafy to demonftrate upon principles ftridly geometrical ; therefore, if we fuppofe ” indefinitely great, then Q becomes s; and as, upon this hypothefis, the feries goes on ad infinitum, we have } he —(tan bs Stents ctang s+ 3 tone Le rs akg tan s which is our firft feries for the quadrature of an hyperbolic E I fector. And as ans si cites te tan - TS, by refolving this equation in refpect of tan ; S, we get the formula iy ake uP by which the feries of quantities tan? s, tan s+ord a¥ ab s—ords +I Let us now put f for the fraction soir ores then, re-- or : ord = marking that — = tan =, we have ab - p tans = ivan? ay Pr. Vo. VI.—P. II. Qq an (398 NEW SERIES for the an equation which exprefles the property we > propeled to invef- tigate. 2 38. WE have now only to fuppofe z in this formula to have thefe values, 2, 4, 8, &c. heared and to fubftitute inftead of the terms of our feries ds) 4 Se ee G : 2 4 tan's cheie ass as given by the Fite! putting ait’ * inftead os of tan s, and the feries becomes I : T ss : ie i = ae. r_ 2a Ss Deh By FE ich B Ed) — ‘+, fel S ord s dQ: 2 43 ptt opicka op Hit and this is the new form under which we propofed to exhibit it. = 39. LeT us now inquire what are the limits of the rate of ‘convergency of this feries.; and in doing’ this, it will be moft convenient to refer to the firft of its two forms. Now, from 3 Ti I XL treee toi : ie «Cférmka———— OL oe) as td ES, “we pet the ‘formula tan'S 2tan+S 2 27 i 2 * Ieee 1 ei tan} S= 2 tan$(1-+ tan - S). But 1-4 tan’ 5 S> 1, and/ 2 2 “ar MH | I s therefore Titan S (1+ tan’ at S) > > tan S, hence it follows, that 2 2 tan £S>tanS. Thus it appears, that each term of the fe- 2 2 | ries QUADRATURE of the CONIC SECTIONS, &c. 309 “ee ee st . I . ries of quantities gun 5, tan gs, &c. is greater-than half the : ‘ f e ° e ° I T term before it; and as thefe, multiplied by the fractions 5 &e. refpectively conftitute the terms of the feries, each term of the feries, under either of its forms, is greater than one-fourth ofthe'term before it. ‘40. AGAIN, from the formula tan = S = “tan Ss e + tan’ S) adam 5S i139 atten. S°s_. fan 5; ds | we find nas Soh | + tant - aS and fimilarly, 1 -+- tan’ 7 S:.. Bat. from the nature of the hyperbola 2tan;S 2tanis a 22 and Tak tan’ ~ Igey+tan’ <8; ‘therefore ag 4 a tan>S tans tan’?+S hence tan Ige at 4 tanS Therefore, putting ~ s inftead of S, and multiplying by war we have TI ~ 7 ae a I — tang $< = 4 x 5 fan —s, 1 J tan—s 4 nN 2n from which it appears, that each term of the feries, following the fecond, is lefs than a third proportional to the two terms immediately before it. So that, upon the whole, it appears, that the limits of the rate of convergency of our firft feries for an hyperbolic fector, are the fame as thofe of our firft for an arch Qq2 . of . 61H cae NEW SERIES for the of a circle, (fee Art. 8. and g.), only the greater limit in the one cafe correfponds to the lefler limit in the other, and vice versa. 41. We might now, from thefe limits to the rate of conver- gency, determine two limits to the fum of all the terms of the feries following any given term, by the mode of inveftigation employed at Art. ro. and Art. 11. in the cafe of the circle; but the refult in both cafes would be found to be the fame, with the difference of the fign , and> for <; that is, we would find the fum of all the terms following any term of the feries, to be greater than one-third of that term, but lefs than a third proportional to the difference between the two terms immediately before it and the latter of the two. 42. Upon the whole, then, our firft formula, for the quadra- ture of an hyperbolic fector, may be exprefled as follows. abs + ords Let s denote the area of the fector, and put f for ahi. aa Then, p— I gle a p—1 ee I 2abs s I I I arch oy g wee weg ee + Tim) + Tings) +R} where T(m) and T(m4-1) denote any two fucceeding terms of the feries, and R the fum of all the following terms *. And : here * Tue fame feries may alfo be put under another form, which it may not be improper to notice briefly, on account of the facility with which the terms may be QUADRATURE of the CONIC SECTIONS, &c. 313 + here R > - T m+1) i Mee Riese ete). Ba ar : i T(m) — T(m+1) Soiehnie As thefe limits to R differ but little when the terms T(m),. T(m1) are confiderably advanced in the feries, the latter may be exprefled more conveniently for calculation thus R <+ Timp) + ; 3. 3 (T (m) — Ter+0)) 43. LET us next inveftigate a feries for the quadrature of the hyperbola, which may be analogous to our fecond {feries for the rectification of the circle. For this purpofe, proceeding as at I I I rete. we sefume the’ formula —— = ——_—_— = a - 3 tanS Bran S aot aM 2 and taking the {quare of each fide of the equation, get PEily BobQe I 21 T tan’S ‘4 tan go pees Biot I 2 Inftead be deduced one from another by the help of the common trigonometrical tables, It is this, Pam ‘gabs Sr cord ¢ I $ U I a” "Pr : — (fing + = fino + = fina” a 3 fin ai”... T(my + Tn$1) + Re The arches a’, a”, al”, al’, &c. are to be deduced one from another as follows. Ord sat ate i , then, fina! = tan = a, fing” = tan 2 @! fin a” = ab s 2 2 Take a fuch that ina= i yw tan © a’, fin ai” = tan athe &c. The fymbols Tim), T(m+1) and R, denote: the fame things as in the other form of the feries. 312 NEW SERIES for the 4 Inftead of S, we now fubftitute in this expreffion s, + 5, Ls, —S, oS, q°rs &c. fucceflively, and multiply the refults by the terms of the feries I, ? rar rE &c. fo as to form the following feries of equations, the number of which is 2. I I [ = => 2 tan’-~ 5 4+ — tan’s 2 tans Se pak 2 2 T a 1 I 2 tan’ + tants. 60} ca" 2d. I I I =o os * tan? os — \ 4 tan’+s 8’ tan’ zs uo 4° fi i 2.4° I I I so ees t+ = tan’ as + — S*tan*ss 6*tan’ys 4? 2.42 ece. From thefe, by proceeding in all refpedcts as in the article above quoted, that is, by adding,. and rejecting what is common to each fide of the fum, we get fi I rl tan a u ase: I I I = =~ tan — i = tan? — ig cat 7 + i ak +e ~ tan’ + $90 tan™= ¢ 8 4r Qn Now, QUADRATURE of the CONIC SECTIONS, &c. 313 Now, as we have found (Art. 36.), that 2" tan = exprefles twice the area of the polygon AFF’F’F” (Plate IX.), the numerical yalue of which we have there denoted by Q, it follows, that 22" tan’ = —! Q:. Moreover, the geometrical fe- 2' I I I I ries ~ ea aC Bei ican hai A saat therefore, by Gsbihisudil and tranfpofition, we get I 2 I . | aarp — 3 (#3) G4 Hire — (3 tan’ 53 “fe Ztam’s 5 aa = tan’s Rhb oi +s, tan?+s, &c. Thefe being fubftituted in the feries, and afterwards s put inftead of 25, +5 inftead of s, ts inftead of 45, &c. (fo as to produce a refult involving only the abfciff correfponding to the fector s, and its fub-multiples) ; and, finally, the whole being divided by 4, we fhall get (abs*+ i c2 bab nisa Be (fe a/t abst tabis—i1‘) 1 abts—-3 L (eax TG abies oe es piean vee and this expreffion is analogous to our fecond feries for an arch of a-circle, as given at Art. 17. tan Ss = | Ki 46. Wr may now inveftigate what are the limits to the rate of convergency of this feries, as alfo the limits to the fum of all its terms following any afligned term. With refpect to the firft of QUADRATURE of the CONIC SECTIONS, &c. 315 of thefe inquiries, it appears, that the terms of the feries, un- der its firft form, (Art. 3) are exactly the fquares of the cor- refponding terms of the former feries, under its firft form (Art. 36.), fo that the one being written thus, == P— (Ter) + T(2)- +++ T(m) + T (mos) + T(m42) +, &e-) the other will be Sher SAG, HT (2) «e+ Tm) + T2m41) + T (m42) +, &-), and here P and P’ are put for the parts of the. two expreflions which do not follow the law of the remaining terms, but T(1), T (2), &c. denote the fame quantities in both. Now, as each term in the former feries has been proved to be greater than one-fourth of the term immediately before it (Art. 39 ), each term of the latter muft be greater than one-fixteenth of the term immediately before it ; and this is one limit to the rate of convergency. ~ AGarn, as it has been proved (Art. 4o.), that in the firft {eries T(a4+2) < =, therefore, fquaring, we have T" To" Now this quantity is a third proportional if (at2) <>; to T” (n) and e (n+1)3 hence it follows, that the greater limit of the rate of convergency in-the two feries is the very fame; _ that is, each term is lefs than a third proportional to the two terms immediately before it. As thefe limits to the rate of convergency differ from thofe of our fecond feries for an arch of a emcie (Age. rer}, only by the leffer limit in the one cafe correfponding to the greater in the other, and the contrary, it is fufficiently evident, Vou. VI.—P. II. Ror that 316 NEW SERIES for the that by proceeding, as in the cafe of the circle, to determine limits to the fum of all the terms following any afligned term, we would obtain an analogous refult, namely, that the fum of all the terms following any afligned term is greater than ;'-th of that term, but lefs than a third proportional to the difference of the two terms immediately before it, and the latter of the two. 47. Ir now only remains to be confidered, how the numeri- cal values of the terms of the feries are to be found. Now, this may evidently be done by computing the values of the quanti- ties abts, ab+s, abis, &c. from the abfciffa correfponding to the whole fector, and from one another by the known for- mula S ab i$ =f Pett abis—1 abis—tI &e and thence the values of the quanties ——2—~—_—-. [4 "= q ~ abes+ 1 abis+r’ Or we may deduce each of thefe from that which precedes it, by a formula analogous to that found at Art. 21. in the cafe of the circle, and which may be inveftigated as follows. Let abS—1 ab+S—1 ; : rhe ee te eae Bs abs as 1 Att hie Seat » then we have ab $ —y and ab Sate — —1_;-we have alfo ati sesies Bada, and 2 I—f I i fince by the nature of the hyperbola abi Sa a oP therefore QUADRATURE of the CONIC SECTIONS, &c. 317 theref pene = Itt ererore " : Sarg and hence I — ipa I—/Y/1I —? SR eee ie a eee 1+V1—t which is the formula required. 48. Tue refult of the whole inveftigation of this fecond fe- ries, for the area of an hyperbolic fector, may now be collected into one point of view, as follows. Puttinc s for the area of the fector, let its correfponding ab{cifla be denoted by the abbreviated expreflion abs; alfo let the abfciffe correfponding to the other fectors which are its fub-multiples be denoted fimilarly. CompuTe the feries of quantities ab>s, abis, abs, &c. from abs, and one another, by the formula abtg—VabS+1 kip 2 Then fhall 62 0a te-imininne, abs—1I 3 = rabjs—T ar rabjs—r, t abys—t1) — Jqgabis+r' Pabis+i ee teed where R denotes the fum of all the terms following the term T(m+1), and this fum is always contained between the limits T*(m+1) I iy Hons qq’ e terba) = Tm+1), an ream: aR, Rr 2 é being 318 NEW SERIES for the being greater than the former, but lefs than the latter. This laft limit may alfo be otherwife expreffed thus, (16 Tim+1) —T m) T(m+1) I — 2 Gay + 15 CTiGg 2 Tint é Or compute the feries of TENS t, t', t’, &c. one from another by thefe formule oy ee oe TLL ~PL an t-— abs—TI ims ee fT > &c, —~ abs +1’ r+NVi-+?z 1+Vi+7 Then fhall Ls Sea 62 =, t aia ; Obs eT Fe gets igeh nacho Deol ae ends alte the fymbols Tym), T(m41), and R, being put to denote the fame as before. 49. WE might now inveftigate other feries for the quadra- ture of an hyperbolic fedtor, fimilar to the third and fourth fe- ries we have found for the rectification of an arch of a circle; but this inquiry would extend the Paper to too great a length. For this reafon, and alfo becaufe the manner of proceeding in the one cafe is exactly the fame as has been followed in the other, it feems unneceflary, in the cafe of the hyperbola, to ex- tend our inquiries farther. I fhall therefore now proceed to the third and laft object propofed in this Paper, namely, the in- veftigation of formule for the calculation of logarithms, be- ginning with a few remarks that may ferve to connect thefe formule with the common theory. 50. Ir is ufually fhewn by writers on this fubje@, that all numbers whatever are confidered as equal, or nearly equal, to one SUADRATURE of the CONIC SECTIONS, &c. 319 one or other of the terms of a geometrical feries whofe firft term is unity and common ratio, a number very nearly equal to unity, but a little greater; and any quantities proportional to the exponents of the terms of the feries, are the logarithms of the numbers to which the terms are equal. Locaritums, then, being not abfolute but relative quanti- ties, we may aflume any number whatever as that whofe loga- rithm is unity ; but a particular number being once chofen, the logarithms of all other numbers are thereby fixed. Hence it follows, that there may be different fyftems, ac- cording as unity is made the logarithm of one or another num- ber; the logarithms of two given numbers, however, will al- ways have the fame ratio to each other in every fyftem what- ever; thefe properties which are commonly known, are men- tioned here only for the fake of what is to follow, as we have already premifed. 51. TAKING this view of the theory of logarithms as the foundation of our inveftigations, LET us put 7 for the common ratio of the geometrical feries, x for any number or term of the {eries, 6 for the number whofe logarithm is unity, y for the exponent of that power of 7 which is equal to x, m for the exponent of the power of r which is equal to 4. ‘Then we have x =7’, and =r”, and becaufe by the nature of logarithms log x: log4::y:m, therefore log x =~ x logd; . ‘ Mm but by hypothefis log 6 = 1, therefore log x = z 52. Let 320 NEW SERIES for the 52. Ler v denote any number greater than unity, and p and n any two. whole pofitive numbers ; then, by a known formula O— Ff Deas 1, fote+utor... to} VU n ese files iL 2 v —r= tt ho+e +ue+ut... +o}, therefore, dividing each fide of the firft of thefe equations by the correfponding fide of the fecond, we get v—1r vtvu+t+vui+ut...+v? v—i vtvtuwtut... pu Now, v being by hypothefis greater than unity, the fraction on the right hand fide of this equation is lefs than this other fraction wi deg? oh o Ut oo lteter Idi +i+is..+ 1 (tom terms)” 7%? becaufe it has manifeftly a lefs numerator, and at the fame time a greater denominator. The fame fraction is, however, greater than this fraction dyitk: Toh spa lnodly Eedtoye tatmeyr op vty tevu'+u"...+ 0” (ton terms) 20” becaufe it has a greater numerator, and a lefs denominator Therefore, UP — TF vey? — : rae —_ v" : = = — ‘7 — I] n VU" _and hence, dividing the firft of thefe expreflions by v*, and mul- tiplying the fecond by v”, © p ve—1r p uv" (ve — 1) ner vw (u"—1) 2 SRE pt t (a). / ; 53. PuTTine QUADRATURE of the CONIC SECTIONS, &c. 32%. 53. Purtine v and p to denote, as in laft article, it is mani- feft that the feries 2 a ae ae ls + uv % is greater than this other feries tp ct) +s). ct & (fo piterms) = Z, but lefs than this feries v+u+u-+u...+v (topterms) = pv; but by a known formula, the fum of the firft of thefe three ie- ries is / v—tI , therefore, I U—*tI U-——I ee | i > Pp; I < pu, p ra v—I V——I and hence it follows, that vp t + 0% RS! I ? Pb Gia oE} > Yad 54. Ler us now recur to the fymbols 7, x, d, y and m, whofe values are affigned in Art. 51. and let us aflume y = /, and r= v"; then, from the two expreflions (w) in Art. 52, we have Js n 322 NEW SERIES for the fui’ 7 at eee a and hence, multiplying by 2, and dividing by m, 2 aL =e a(r—1) "—1) yy n(* + T) mem m(r—t)’ m TO peor r But. 7” =+y and 7” = 8, (Art. §1.), from which it follows, that - I I ry =x, andr—=—b' ; moreover, 7 = log «; therefore, fubftitu- ting, we get 3 n to x) I I Le (x” maty lop Ree ws > Soret ies ey m(b"— 1) = [pF r) and in thefe expreflions 7 denotes any whole, pofitive number whatever. 55. By fubtracting the leffer of thefe limits to the logarithm of x from the greater, we find their difference to be n(x —1) esr, eo Beri ee x m(b —t) syn[ > Now QUADRATURE of the GONIG SECTIONS, &c. 323 Now we have found, that one factor of this expreflion, viz. 2 io 2) cannot exceed the logarithm of x; with re- I x om es 2) aj-[4 T I {pect to the other factor b° x” —1, fince it appears from the firft of the four formule (8), (Art. 53.), that a 1, and therefore that 27’ < 21, ’ I : and z¢’ ¢, and t’> ae Hence it appears, in BGO (a NEW SERIES for the in the firft place, that each term of our feries, taking its co-ef- ficient into account, is greater than one-fourth of the term be- fore it. AGAIN, becaute - -=- =(2 +2”); and, fimilarly, 7=5 “Gz +72”), and it having been proved that ¢’ <7, fo that fimilarly, t” (—-) ot : x x4 1 2 I 2 x?—TI ' x 4—]T EEN, at =X .. weave | = = 2S , and ff PG: q > oe ; XE 7? er ihe HFT &c. Again, becaufe X’= : (2am , therefore 2X* — x2 2 («+2) +1, but . («+2) = X, therefere-21%"* = Kee 2 and X’ =/Stt. In like manner, it will appear, that x” — ee! &e. 63. From the preceding inveftigation it appears, upon the whole, that our fecond feries for the calculation of a logarithm may be exprefled as follows. Purtine « for any number, let a feries of quantities X, X’, X’, X”, &c. be found fuch that eat I tt fea DS ye X= = 4+ = xX Nee X Nese Sue. Then will log’ x QUADRATURE of the CONIC SECTIONS, &c. — 335 (igre Papen. athe: (~—1)* ag 12 I Ht eX 4 KET X’—I #) X’—I I “—T 2 iz Ts = ge oe oo i Ae ‘ Lo + Tim) + Tents) + RY and here Tm), T(m+1), are put for any two fucceflive terms of the feries,:and R for the fum of all the following terms: And in every cafe’R is greater than re T(m+1), but lefs than I 16 Tom+1) — T(m) peal yo IP ee feo eed (Tm) —Tom+1)) ae 64. From the analogy of the two formule from which we have deduced the feries for the rectification of an arch of a circle, and for the calculation of logarithms, it is eafy to infer _ that there will be correfponding feries for the refolution of each of thefe problems. And as the two preceding feries for a logarithm have been inveftigated in the very fame way as the firft two feries for an arch of a circle, fo, by proceeding exact- ly as in the inveftigation of the third and fourth feries for the circle, we may obtain a third and fourth feries for a logarithm. The mode of deduction, then, being the fame in both cafes, and alfo fufficiently evident, I fhall fimply ftate the refult of the inveftigation of a feries for logarithms which is analogous to our fourth feries for an arch of a circle, (Art. 28.). Ler «x be any number, and X, X, X’, X”, &c. a feries of quantities formed from x, and one another, as f{pecified in the beginning of the laft article. Then I — NO ais 336 NEW SERIES for the OP d Core a ee Mele | 6(x—1)! “8i9.20 I ' r Mids oo rt X’'+12X’—13 } log?x. 4 + nt Os ae EL 2 050" X' + 4 X’+ 3 | tr X%+12X"+ 13 L + 516 ERT Te Be The terms of this feries approach continually to thofe of a geometrical feries, of which the common ratio is om and hence it follows, that the fum of all the terms after any affign- ed term, approaches the nearer to e of that term, according as it is more advanced in the feries. 65. Brsipes the foregoing, our method furnifhes yet another kind of expreflion for the logarithm of a number, namely, a produé confifting of an infinite number of factors, which ap- proach continually to unity. Such an expreflion may be invef- tigated as follows. From the identical equation X—I= (x? — 1) (X? + I). Let there be formed the feries of equations K——I1= 2 (x? 1 pi 1) “* +t, 2 - a(x? —1) 4 (x? — 1) TS fata t as = e 41) =s@h@— PF, WL =. bs wy = (x"—1)=m(x —r1) SB here - QUADRATURE of the CONIC SECTIONS, &c. 337 here m is put for any integer power of 2. Let the product of the correfponding fides of thefe equations be now taken, and the common factors rejected, and the refult will be I u a. al ee 7 w?—+rw4+r x8§ +r n+ ee it eg) EDN : ear) 2 2 2 2 and hence E 2 2 2 2 m (x” —1)=(—1) L 1 t) eo; wu I x? +7 w*+r x8 +1 mm if, Ta si eet This equation holds true, m being any power of 2 whatever. Let us, however, conceive it indefinitely great. Then the I number of factors will become infinite, and m (Gs 1) will become Nap. log x (Art. 57.). Therefore, 2 2 2 2 oc Repos t= Ge ny) oe sama oi fo doit at?+tyarttyr vtty witty ad infinitum. Tue product of any finite number of thefe factors being al- ways a function of this form m (2 —1) will of courfe be great- I I a er than log z, (Art. 54.). However, the function —m («° —1) Wm I or m G — — }, being in like manner expanded into an infi- ™ wv nite 338 NEW SERIES for the nite product, we get from it logr=(a—~1) 4 é —-, & ao are Sti +0 5% ad infinitum. and the product of any finite number of factors of this expref- fion will always be lefs than log «. TueEseE formule, which are analogous to that given by Ev- LER for an arch of a circle, (fee Art. 33.), are not inelegant, confidered as analytical transformations. It does not feem, however, that without fome analytical artifice, they can be ap- plied with advantage to the actual calculation of logarithms, by reafon of the great labour which would be neceflary to ob- tain a refult tolerably accurate. 66. I sHaLL now conclude this Paper, with fome examples of the application of the formule to the computation of the length of one-fourth the circumference of a circle whofe radius is unity, (which is the extreme and the moft unfavourable cafe), and to the computation of a logarithm; as alfo of the modulus of the common fyftem of logarithms, which is the re- ciprocal of NariEeR’s logarithm of ro. EXAMPLE SUADRATURE of the CONIC SECTIONS, &c. 339 Exampxel. The length of an arch of 90°, computed to 12 places of decimals, by means of the firft feries, (Art. 12.). Here a= 90°. = COL a =O tan da tan > a=!f tans a = 0. jigat s56a499a- tan + a= 0.1989123673796 tan y's 4= 0.0984914033571 tan '5 4 = 0.0491268497694 S < .000000 1248357 S > .0000001 248356 Vou. VI. P. Il. tan zz 4 = 0.0245486221089 tan 2, a ='0:01227246 2379. tan =4;5 @ = 0.006136000157. tan _1. 4.= 0.003067971201. 512 tan +cy_4 4 = 0,001533081094.. tan >s'55 4 = 0.00076699054.. “Litan + = Ce tan $4 = tae a ihe: ? _—_ zz tan=54= I Uu -500 000 000 000 o 2°93) 553 399 :593 5 024 864 045 9225 — .006 155 712 7098 7O0T $35 214 055 3 7 4 = .000 383 572 2205 000 095 878 612 3 -000 023 968 7506 -000 005 992 131 3 2000 OOI 498 029 3 -000 000 374 5071 -000 000 124 835 7 = 636 619 772 367 7 Arch of 90°, or 4= 1.570 796 326 795. EXAMPLE 340 . NEW SERIES for the EXAMPLE II. The length of an arch of 90°, computed by — the fecond feries, (Art. 22.). col 4 6 ess a = 0.995 18472667. a cof+ a= 0.7071067811865 cof ="; 4 = 0.99879545621.. cof 4 = 0.9238795325113 - cof z'; 4= 0.9996988187... cof + a = 0.980785280403. ore | Amount of pofitive 1 1+ cof a rte aE — .416 666 666 6667 terms, 4 4—cofa’ 8 ~ SR She ae sie: thee 4° 1+ coltia ry 1—cof{ta 451+ coft ae I 1 —cof + a 41+ cola t 1— cola 4°>1+colysa = ,010 723 304.703 4 = -000 618 220 ee 6 = .000 age 892 7990 = .000 002 356 882 2 1 100554 — 490 000 147 1276 4°1+cof 4 1 1—cof74 471+ cofs,4 i oo 000 612 8 } a nt a 4a 643 “ Hence S = .000 000 000 612 7 > .000 900 000" Amount of negative terms, .OI1 381 932097 2. Difference between the cesses or = = "405 284 734 5695 and negative terms, = .000 000 009 1927 a= .636 619 772 3677 Arch of 90°, or 4 = 1.570 796 326:795- EXAMPLE QUADRATORE of the CONIC SECTIONS, &c. — 34% Examece III. The length of an arch of 90°, ee from the fourth feries, ee 28.). cofa =o cof 7a = 0.980 785 280... cols @= 0.907 £06481 1865 cofi-a= 0.995 184 7.. Cole nck Chgad oh ¥55 Se. = cof s'74 = 0.998 80. . Sateen I 13—cofa+12cofta 2 -.3-16° 3+4+cola—4colfla = .163 053 go2 o108 -OOT 215 277 777 8 7 8.8.9.10 Amount of pofitive terms, 4164 269 179 788 6 I 13—col+a—t12cofta _ Be ea ee BAO? ° 3ePcoltw 4 acolfa — °909 O83 201 799 5 I 13—cofi{a—t2colja oN NCB TOF cape ids Ss I 13—cof~a—12col+, 35% 16>" Pat ca yer ice dete: 13 — cof,a— 12cofj,4 3-166 3+ coft,a+ 4 cof, 4 Each of the remaining terms, being near- ly ,';th of the term before it, their fam -009 000 000 000 8 will be nearly ¥; of the laft term, or +000 000 198 794 2 -000 000 003 O744 = 000 000 000 047 8 & Amount of negative terms, .000 013 463 7137 Difference between the eae i Da seal tlh a and negative terms, or os 7a 255716 0749 = -405 284 734 569 3 = .636 619 772 3676 Sl Se ci Arch of 90°, or 4 =.1.570 796 326 795. Uu 2 EXAMPLE 2 NEW SERIES for the ExampLe IV. The reciprocal of Naprer’s logarithm of 10, (which is the modulus of the common fyftem), calculated by the fecond feries for logarithms. (See Art. 63.). « = 19, and ‘ence X=, 5.05. A0k'} = 12080 8734-54-20. XK’ = 1-739 252 713 092 7 X= 1.002 589 934 6... X’ = 1.170 310 367 614 6 X" = 1.000 649 274 05 & X" = 1.041 707 820 748. A.’ 1.064 (61 Bona eee —————— x ay = 4123 4596 790 1295 1 i i 2 08 3 333 333 333 3 Sum of pofitive terms, .206 790 123 4508 - 2 ype = 010 867 116 4758 = a = .001 226 137 760 6 in SS = .000 079 796 one ° , = ie = .000 005 038 882 6 a = .000 000 315 7452 re oes! = .000 000 019 746 g Z ae = .000 000 001 234.4 R > .0000000000822,9) R <.0000000000823,1§ ‘ Sum of negative terms, 0018 178 426 445 8 Difference of the apa, I__ — 4788611697 0110 ; and negative terms, log* 10 R = .000 000 000 082 3 I et ’ exe ees 481 903. EXAMPLE QUADRATURE of the CONIC SECTIONS, &c. 343 ExampLe V.. Napier’s logarithm of 10, calculated by the third feries for logarithms. ig = To, X = 5.05 Reo se 1-7 3025271 2093: X’= 1.17031036761.. v(a +4e+1) 6 (a —1)* 1 X+12X' —13 3.167 X+4X'+ 3 TRE ye KA Ps sR te Ok ae 8 rt X" 412 X”— 13. 3.16' X’+4X"4+3 T --X% +12 X"" — 13 yr XMv+12X*°—73 gs of laft term = fum of the a maining terms nearly, From fum of pofitive terms, Subtract ~=—— 8.9.10 : . T There remains ——— log* 10 I log* 10 I log Lo (See Art.64.). jn Gages eT TS a a T.0417078207. . . PC LOS? ES wee os EVOOZSOOU.@ swe = .035 817 =e 714.8 -OOI 121 093 421 4 000 024 041 1229 = .000 000 409 2394 = .000 000 ee 5357 -000 000 000 102 7 -000 000 000 OO! 6 = .001 388 888 888 9 = 1635 574 372249 6 -188 O11 697 O11 3 = +434 294 481 903 . EXAMPLE 344 NEW SERIES for the ExampLe VI. To fhew that the feries inveftigated in this Paper are applicable in every cafe, whether the number whofe logarithm is required be large or {mall, let it be required to calculate the common logarithm of the large prime number 1243 to feven decimal places, by the fecond feries, (Art. 63.). KS Ea X” = 1.42356148 MX = 621.50040225 X'Y = 1.10080913 X’'= 17.64228446 A= 1.0248925.. a’ Se SOR F054 52 a = £.00020K5. acai — -000 805 8oI | Tz = -083 333 333. Sum of pofitive terms, -084 139134 Xx’ —1 2 Mar = 1055 794 813 5 | OX = = 1007 914 766 4 x XE = .000 682 688 e So = .000 046 861 a CS = .000 003 oo! Lea = .000 000 189 R > .000000012,6 R = .000 000 013 R_ < .000000012,7 Sum of negative terms, -004 442 331 I Pu ae = .o19 696 803 I ree Nap. log x ~ Sais: Conan al 1434294482 = 3.094 471. ¥. Common log of 1243 = 0140345300 : No. TX. Trans. R.S Edin Vol Up. 3b Ae L A TE I Ne . _ | or. WOM he Pein tana | ; Cow naa Be df 78k “p aed int ce : 2 y nike Hey he A AR Cae 498 | i} i , i oe a ede mae Fo cr ae a ee ' alll + : aT ett ¥ ney’ F ms ‘ ew! i page ern al = , 0 ah a digstl r hh alaceyls meal este pete ratenwe a cwnctmatnnysinmne — se minimise a Om tain ti Te teat te tine aa pay J ‘ ‘ A * Ny sie y . : ra a he ; Rooks ; - Rate 2 = x ms y aay 5 rt » 5 oe ae . a * { a 5. a “ 2 es ” 4 M4 ~? — if ee , i ‘ 1 ’ s ‘gp * x . IX. Remarks on a Minera from GREENLAND, supposed to be Crystatitisep GapoLinite. By THomas Autan, Esa. F.R.S. En. [ Read 21st November 1808]. MONG a parcel of minerals which I procured laft fpring, there are {pecimens of two very rare foflils ; one of them, the Cryolite, the other I believe a variety of the Gadolinite. _ The former, is accurately defcribed in the different mineralo-- gical works, and I have little to add to the information con- tained inthem. But the Gadolinite appears to be very imper- feétly known, and has never yet been defcribed as a crystallifed: foffil. | THE minerals in queftion were found on board a Danith: prize, captured on her paflage from Iceland to Copenhagen, and were fold with the reft of her cargo at Leith. On examination, I was furprifed to find they correfponded fo little with the fof- fils which are ufually brought from that ifland, and confe- quently endeavoured to trace from the fhip’s papers, any parti- culars that might lead to the knowledge of their geographic - origin. All I could learn was, that they were fent. from Davis’ Straits by a Miffionary. . I consi1pEr this limited information, however, fufficient to fix on the coaft of Greenland as the place from whence they had- BAGG «, On @ MINERAL supposed to be had been brought; the only.Cryolite known in Europe having been fent by a Miflionary from Greenland to-Copenhagen. Tue Gadolinite, from its extreme fcarcity, is a mineral to be found in very few cabinets; and when this collection fell into my hands, was one of ‘thofe I knew only by de- {cription. I was led to fufpeét that fome of the minerals in this parcel belonged to that fpecies, by obferving, im- bedded in a piece of granite, fome fmall. fhapelefs maffes, ' whofe external characters appeared to correfpond entirely with thofe affigned to the gadolinite ; but on reference to the mi- neralogical works which treat of this ftone, I found more difh- culty than could have been fuppofed in afcertaining whether they did fo or not. The inveftigation, however, furnifhed a ftrong proof of the fuperiority of chemical teft over external character ; for although the fhape, luftre, fracture, and geogno- ftic relations, left me fcarcely any room to doubt, yet on apply- ing the blow-pipe and acids, it was quite evident, that the ftone I firft tried could not be gadolinite. I examined with great care the reft of the parcel, and picked out feveral, which, though very different, refembled in various refpects the one that originally at- tracted attention; and with a view to fatisfy myfelf, I fent duplicates to a friend in London, from whom I learnt, that one of thofe which I fuppofed to be gadolinite was certain- ly that mineral. Notwithftanding the very refpectable autho- | rity I had obtained, to which I was inclined to pay the utmoft deference, it was not till after minute and repeated inveftiga- tions that I found myfelf difpoted to fubmit to it; the phyfical characters of the fpecimen in queftion Set fo very widely from thofe I was taught to expect. Ir is more than twenty years fince the gadolinite was firft obferved by M. ARRHENIUS, in an old quarry at Roflagie, near Ytterby in Sweden. It was defcribed by Mr Geyer, and by him confidered as a black zeolite. In / CRYSTALLISED GADOLINITE. 347. in 1794, M. Gapotrn analyfed it, and found that it con- tained 38 per cent. of an unknown earth, whofe properties approached to alumine in fome refpects, and to calcare- Gus earth in others; but that it eflentially differed from both, as well as from every other known earth. In 1797 M. Exeserc repeated the analyfis of M. Gano- LIN, and obtained 474 per cent. of the new earth. This in- creafe of quantity he attributed to the greater purity of the fpecimens he fubmitted to experiment, and in confequence of having confirmed the difcovery of Gapouin, he called the ftone after him, and gave the name of Yttria to the earth. AnaLyses by VAUQUEL:N and Kiaproru have fince ap- peared. The quantity of yttria obferved by the former amounted only to 35 per cent. ; ‘but the latter ftates 593 per cent. Tue {mall portions of this mineral, which, from its rarity, it is natural to conclude were at the difpofal of thefe celebrated chemifts, may in fome meéafure account for the diverfity of their refults ; but it is likewife by no means impoffible, that the mineral itfelf may have varied 1 in the pela wiy of its chemi- cal ingredients. Tue difference which we find in the mineralogical defcrip- tions of this foffil; hitherto only found in one fpot, is much more difficult toaccount for. If the information I have other- wife obtained be correé, of which I have not the flighteft doubt, we cannot, help attributing a certain degree of carelefi- nefs to fome of the authors, particularly the French writers, who have fuch opportunities at command *, of inveftigating every point relative to natural hiftory. The great veneration Vou. VI. P.II. | X xX | they * Lucas notes the Gadolinite as one of the minerals in the colle&tion at the Fardin de Plantes, 348 On a MINERAL supposed to be they entertain for the talents and accuracy of the celebrated Haiiy, may induce them to think his obfervations require no concurring teftimony ; and, on the other hand,.the pupils of the German School, confider no mineral deferving a place in their fyftem, till it has been examined and clafled by their illu- ftrious mafter, whofe authority will be handed down by them with equal refpect to pofterity. . Ir is unneceflary to occupy the time of the a te in gi- ving a comparative view of the different defcriptions of the Gadolinite. I fhall only notice a few prominent features. Ir is defcribed by every one of the authors, as poflefling a fpecific gravity of upwards of 4, and as acting powerfully upon the magnet. This laft character is noticed by Profeflor JameE- son, in the firft account he gives of the gadolinite; but in the fecond it is omitted, along with fome others. KLaprotu takes no notice of its magnetic power, but ftates the f{pecific gravity at 4.237. Tue French writers defcribe the colour as black and-reddith black. The German as raven or greenifh black. Thele varia- tions, with feveral others which may be obferved on referring to the different authors, fhew that fome incorrectnefs muft ex- ift. But the moft remarkable of all is, that the gadolinite, if ever magnetic, is not always fo; for the fpecimens in the poffef- fion of the CounT DE Bournow are not, nor, as he informs me, _ are any that he has ever feen. It is therefore reafonable to _ conclude, that magnetifm in the gadolinite.may depend on accidental caulés. Tue following is the defcription of the foffil, which I fup- pofe to be that fubftance in a cryftallifed ftate; although no- thing fhort of analyfis can afford indifputable teftimony of the identity of any mineral fo little known. SPECIFIC CRYST ALLISED GADOLINITE. 349 Petia Gravity, 3.4802. The {pecimen “— 1136.39 grains. Its furface is a little decompofed, and it has alio fome minute particles of telfpar intermixed with it; both of which would affect the refult in fome degree ;_ but nei- ther were of fuch amount as to do fo in any confiderable degree. | Harpness: fufficient to refift fteel, and {cratch glafs, but not quartz. | Lustre: fhining, approaching to refinous. _ FRACTURE: uneven, verging to flat conchoidal. Cotour: pitch black which I confider velvet black with a fhade of brown; when pounded, of a greenilh grey co- Jour. Figure: it occurs cryftallifed. The fimpleft figure, and ~ perhaps the primitive form, is a rhomboidal priim, whofe planes meet under angles of 120° and 60°. In tome of the fpecimens, the acute angle is replaced by one face, in others by two, thereby forming fix and eight fided prifms. All the fpecimens I pofleis are only frag- ments of cryftals, none of which retain any part of a ter- mination. ‘They occur imbedded in felfpar, probably gra- nite. CueMIcAL CHARACTERS: before the blow-pipe froths up, and melts but only partially, leaving a brown {coria ; with borax it melts mto a black glafs. When pounded, and heated in diluted nitric acid, it tinges the liquid of a ftraw colour ; and, fome time after cooling, gélatinates. Tue principal diftinguifhing character of the gadolinite, is its forming a jelly with acid, a charaéter belonging to few other minerals. The Mezotype Lazulite, Apophilite, Adelite, and Oxide of Zinc, fo far as I know, alone poffeis the fame qua- lity ; aud it cannot eafily be miftaken for any of them. Sx 2 Ir 350 On a MINERAL supposed to be Ir has not the fmalleft attraction for the magnet; it does not decrepitate and difperfe when expofed to the aoe pares 3 it is not in any fhape tran{parent. Tue Swedith foflit occurs in roundifh amorphous maffes, im- bedded and diileminated in a granitic rock, having the external furfaces covered with a flight whitith coating, perhaps from the attachment of micaceous particles. ‘There is no fuch ap- pearance on the furface of the cryftallifed gadolinite. Tue fituation which this mineral fhould hoid in the fyftem has been a matter of difficulty among mineralagifts. Haiy has placed it in the clafs of Earthy Foflils, immediately after his Anatafe and Dioptafe,—rather an unfortunate fituation, both thefe having been recognifed as ores of known metals, titanium and copper, fince the publication of his admirable treatitfe. WERNER, on account of its weight, has clafled it among the metals; and from its natural alliances, and chemical com- pofition, has given it a place among the irons *. If weight en- titled it to be clafled among the metals, feveral other minerals have an equal claim to the fame fituation. Of its natural alli- ances we know very little, farther than that the Swedith di- {trict where it is found abounds in iron; and as to its chemi- cal compofition, if 174 per cent. of iron be fufficient to counter- balance 592ths of a new earth, it would be right to arrange it accordingly. The analyfes of fo many chemifts of known ce- lebrity, are certainly fufficient to juftify the conftitution of a new {pecies for its reception. WERNER, however, may feel himfelf licenfed in this arrangement, as he does not confider it neceflary that a mineral compound fhall preferve the charac- ters of its components ; but that any of the components may give to the compofition characters fufficiently marked, to de- termine its relations. It is upon this diftinétion that he founds the difference between-the predominant and charaéteriftic prin- ciples t. | THE * JAMESON, Vol. ii. p. 613. + BrocHant, vol, i. p. 44. CRYST ALLISED GADOLINITE. . 351 THE arrangement of BRONGNIART appears much more ju- dicious; he has placed it at the commencement of the Earthy Minerals, and affighs as a reafon, that it is unique in its com- pofition ; and if placed in any other fituation, it would interrupt. the feries, either in refpect to its compofition or external cha- racters. Or the Cryolite I have very little to obferve, in addition to the - defcriptions given in the different mineralogical works. The {pecific gravity I found to be 2.961; Haiiy ftates it at 2.949. Among the various mafles I examined, there was no trace of cryftallization, farther than the cleavage, which is threefold, and nearly at right angles. The maffes broke in two direc- tions, (which may be fuppofed the fides of the prifm), with. great facility, leaving a very {mooth furface ; but the tranfverfe cleavage was more difficult, and by no means fo fmooth. Se- ‘veral of the fpecimens being mixed with galena, pyrites, and: cryftals of {parry iron-ore, it would appear that the cryolite is. a vein-ftone ; but I was not fo fortunate as to find any of it at- tached to a rock fpecimen, fo as to throw light on its geo- gnoftic relations. 4 - 3) . , oe Pas a he: ig - F an Fo , ; « a8 : . - ent. Wer ae Fe: s AF isl petpent efit ; rea: 4 ‘i ~* nel Sag ¥ wag we aby Ae, SPE i I. X. On the Progress of Heat when communicated to Spheri- cal Bodies from their Centres. By Joun Piayratr, F.R.S. Lonp. Sect. R.S. Evin. and Professor of Natural Philosophy in the University of Edinburgh. { Read March 6. 1809. ] i N argument againft the hypothefis of central heat has been ftated by an ingenious author as carrying with it the evidence of demonftration. (9 “a n “ THe effential and characteriftic property of the power producing heat, is its tendency to exift every where in a ftate of equilibrium, and it cannot hence be preferved without lofs or without diffufion, in an accumulated ftate. In the theory of Hutton, the exiftence of an intenfe local heat, acting for a long period of time, is affumed. But it is impoflible to pro- cure caloric in an infulated ftate. Waving every objection to its production, and fuppofing it to be generated to any ex- tent, it cannot be continued, but muft be propagated to the contiguous matter. If a heat, therefore, exifted in the cen- tral region ofthe earth, it muft be diffufed over the whole mafs; nor can any arrangement effectually counteract this diffufion. It may take place flowly, but it muft always con- tinue progreflive, and muft be utterly fubverfive of that fy- {tem of indefinitely renewed operations which is repre- Vou. VL. 'P. If wy “ fented 354 On the PROGRESS of HEAT “-fented as the grand excellence of the Huttonian Theory *.”’ “ Again, he obferves, in giving what he fays appears to him a demonftration of the fallacy of the firft principles of the Hut- - tonian Syftem, “ it will not be difputed, that the tendency of “ caloric is to diffufe itfelf over matter, till a common tempe- “ rature is eftablifhed. Nor will it probably be denied, that a “ power conftantly diffufing itfelf from the centre of any mafs “ of matter, cannot remain for an indefinite time locally accu- “* mulated in that mafs, but muft at length become equal or “¢ nearly fo over the whole f.’’ 2. I muft confefs, notwithftanding the refpeét I entertain for the acutenefs and accuracy of the author of this reafoning, that it does not appear to me to poffefs the force which he afcribes to it; nor to be confiftent with many facts that fall every day un- der our obfervation. . A fire foon heats a room to a certain de- gree, and though kept up ever fo long, if its intenfity, and all other circumftances remain the fame, the heat continues very unequally diftributed through the room ; but the temperature of every part continues invariable. If a bar of iron has one end of it thruft into the fire, the other end will not in any length of time become red-hot; but the whole bar will quick- ly come into fuch a ftate, that every point will have a fixed temperature, lower as it is farther from the fire, but remain- ing invariable while the condition of the fire, and of the Haeek that furrounds the bar, continues the fame.. The reafon indeed is plam: the equilibrium of heat is not fo much 2 primary law in the diftribution of that fluid, as the limitation of another law which is general and ultimate, confifting in the tendency of heat to pafs with a greater or a lefs velocity, according to circumftances, from bodies where the temperature ae * Murray’s Syffem of Chemistry, vol. 111. Appendix, p. 4a. +) Page st. im SRRPLERIC AD BOD TES. 355 is higher, to thofe where it is lower, or from thofe which con- tain more heat, according to the indication of the thermome- ter, to thofe which contain le‘s. It is of this genera! tendency, that the equilibrium or uniform diftribution of heat is a confe- quence,—but a confequence only contingent, requiring the pre- fence of another condition, which may be wanting, and actually is wanting, in many inftances. This condition is no other, than that the quantity of heat in the fyfiem fhould be given, and fhould not admit of continual increate from one quarter, nor diminution from another. When fuch increafe and diminution take place, what is ufually called “ the equilibrium of heat’’ no longer exifts. Thus, if we expofe a thermometer to the fun’s rays, it immediately rifes, and continues to fland above the temperature of the furrounding air. The way in which this happens is perfectly underftood: the mercury in the thermometer receives more heat from the folar rays than the air does; it begins therefore to rife as ‘foon as thofe rays fall on it; at the fame time, it gives out a portion of its heat to the air, and always the more, the higher it rifes. It continues to rife, therefore, till the heat which it gives out every inftant to the air, be equal to that which it receives eve- ry inftant from the folar rays. When this happens, its tempe- ature becomes ftationary ; the momentary increment and de- crement of the heat are the fame, and the total, of courfe, con- tinues conftant. The thermometer, therefore, in fuch circum- ftances, never acquires the temperature of the furrounding air ; and the only equilibrium of the heat, is that which fub- fifts between the increments and the decrements juft mention- ed: thefe indeed are, ftrictly fpeaking, zn equilibrio, as they ac- curately balance one another. This f{pecies of equilibrium, however, is quite different from what is implied in the uni- form diffufion of heat. Yy 2 3. In 355 «On the PROGRESS of HEAT 2 In order to ftate the argument more generally, let A, B, C, D, &c. be a feries of contiguous bodies ; or let them be parts of the fame body; and let us fuppofe that A receives, from fome caufe, into the nature of which we are not here to inquire, a conftant and uniform fupply of heat. It is plain, that heat will flow continually from A to B, from B to C, &c.; and in order that this may take place, A muft be hotter than B, B than C, and fo on; fo that no uniform diftribution of heat can ever take place. The ftate, however, to which the fyftem will tend, and at which, after a certain time, it muft arrive, is one in which the momentary increafe of the heat of each body is juit equal to its momentary decreafe ;.fo that the temperature of each individual body becomes fixed, all thefe temperatures together forming a feries decreafing from A downwards. To be convinced that this is the ftate which the fyftem muft af fume, fuppefe any body D, by fome means or other, to get more heat than that which is required to make the portion of heat which it receives every moment from C, juft equal to that which it gives out every moment to E; as its excefs of tempe- rature above E is increafed, it will give out more heat to E, and as the excefs of the temperature of C above that of D is diminifhed, D will receive lefs heat from C; therefore, for both reafons, D muft become colder, and there will be no top to the reduction of its temperature, till the increments. and decrements become equal as before. ) 4. Ir, therefore, heat be communicated to a folid mafs, like the earth, from fome fource or refervoir in its interior, it muft go_ off from the centre on all fides, toward the circumfer- ence. On arriving at the circumference, if it were hindered from proceeding farther, and if {pace or vacuity prefented to heat an impenetrable barrier, then an accumulation of it at the furface, and at laft a uniform diftribution of it through the whole mafs, would inevitably be the confequence. But if. heat i@SPAE REOAL) BOD PES. 357 heat may be loft and diffipated in the boundlefs fields of va- . cuity, or of ether, which furround the earth, no fuch equili- brium can be eftablifhed. The temperature of the earth will then continue to augment only, till the heat which iffues from it every moment into the furrounding medium, become equal to the increafe which it receives every moment from the fup- pofed central refervoir. When this happens, the temperature at the fuperficies can undergo no farther change, and a fimilar effect muft take place with refpedt to every one of the {phe- rical and concentric {trata into which we may conceive the ~ folid mafs of the globe to be divided. Each of thefe muft in time come to a temperature, at which it will give out as much heat to the contiguous ftratum on the outfide, as it receives from the contiguous ftratum on the infide ; and, when this hap- pens, its temperature will remain invariable. 5- THAT we may trace this progrefs with more accuracy, let us fuppofe a fpherical body to be heated from a fource of heat at its centre ; and iet 4, h’, b”, be the temperatures at the furfaces of two contiguous and concentric ftrata, the diftances from the centre being x, «’, 2”; and let it alfo be fuppofed, that the thicknefs of each of the ftrata, to wit, x/—x, and «”—z’, is very {mall. THEN fuppofing the body to be homogeneous, the quantity of heat that flows from the inner ftratum into the outward, in a given time, will be proportional to the excefs of its tempera- ture above that of the outward ftratum multiplied into its quan- tity of matter, that is, to (A—h’) («3 —x°). §. In 358 On the PROGRESS of HEAT - 6. In the fame manner, the heat which goes off from the fecond ftratum in the fame time, is proportional to (h'— bh’) (*’> —x’'2) ; and thefe two quantities, when the temperature of the fecond ftratum becomes conftant, muft be equal to one another, or (b— h’) («’° —«°*) = (b'— b”) (#38 — x’). Bur becaufe 4—d’, and «’—x are indefinitely {mall, b—b'=b, and *3—x3= 307? 3 therefore bx 3a°x =a given quantity ; which quantity, fince * is given, we may re- pire fe a x a prefent by a’x'; fo that} = — = ae or, becaufe 4 is EE ME ( negative in refpect of %, being a deccrement, while the latter is an as : 1 ax a increment, = — sa and therefore h=C-++ a 7. To determine the conftant quantity C, let us fuppofe that the temperature at the furface of the internal nucleus of ignited matter is = H, and r= radius of that nucleus. Then, in the particular cafe, when x= 7 and 4=H, the preceding 2 2 ; : a a equation gives H=C -++ =, fo that C= H— = and confe- Piiy Cae ick wou V1 . h, —— eee — * cee Sur = ae e quently =H Ah Gg? Sees (; 2) 8, Ir is evident, from this formula, that for every value of | y there is a determinate value of 4, or that for every diftance from the centre there is a fixed temperature, which, after a certain time, muft be acquired, and will remain invariable as long ‘Yi, SRE RIAL BODIES: 456 long as the intenfity and magnitude of the central fire conti- nue the fame. g. Ir remains for us to determine the value of a’, which, though conftant, is not yet ete, or known from obferva- tion. At the furface of the globe we may fapipots the mean tem- perature to be known: let T be that temperature, and let R = the radius of the globe. Then, when x =R, b= ey and by fubftituting in the general formula, we have T =H+— ae —-); Bal ——H) a Ren i 8) ad. a SERENE nos R r Taus b= H+ — G- *) sn G=D G-) HEncE alfo by reduction RT—rH , Rr(H—T) Mimi aed) el Ae R74 — TL or bag 2 hie bua ae From this equation, it is evident, that / ph H , Or —r the excefs of the temperature at any diftance x from the centre, above a certain given temperature, is inverfely as x. But the conftruction of the hyperbola which is the locus of the preceding 360 On the PROGRESS of HEAT preceding equation, will exhibit the relation between the tem- perature and the diftance, in the way of all others leaft fubjed& to mifapprehenfion. Let the circle (Plate X. fig. 3.) defcribed with the radius AB, reprefent the globe of the earth ; and the circle defcribed with the radius AH an ignited mafs at the centre. Let HK, perpendicular to AB, be the temperature at H, the furface of the ignited mafs; and let F D be the temperature at any point whatever, in the interior of the earth, BM reprefenting that at | the furface. Then AB being =R in the preceding equation, AH=r, HK=H, BM=T; AF=x, and FD—d, thefe two laft being variable quantities ; fince G-— REE)» = RE we have, (taking AE = — and drawing E L parallel to AB, meeting H K in Ny dad PDO) ox OE l= which is a given quantity. -Tuererore D isina re@angular hyperbola, of which the centre is E, the affymptotes E G and E L, and the rectangle of the co- HK—BM ordinates, equal to BA.AH X BH oD which amounts to the fame, to KN.NE. Ir is evident from this, that if the {phere were indefinitely extended, the temperature at the point B and all other thmgs remaining the fame, the temperature at its fuperfices would not be lefs than A E, or than the quantity a ea THE in SPHERICAL BODIES. 361 RT—rH . ee fuppofed here to - be fubtraéted ; if RT be lefs than 7H, it will change its fign, and muift be taken on the other fide of the centre A. THE quantity AE, or _ to. Tue refults of thefe deductions may be eafily reprefent- ed numerically, and reduced into tables, for any particular ' values that may be affigned to the conftant quantities. Thus, if the radius of the globe, or R = 100, that of the ignited nu- cleus or y= 1; the temperature of the nucleus, or H = 1000, and T the temperature at the furface = 60, the formula be- comes b= 50.505 -+ ae Values of x Values of 4 fe) 145-454 20 98 .423 3° 82 .599 40 74 .686 5° 69 .938 60 66.330 70 63°.92 80 62 .361 go 61.055 100 60. Vot. VI. P. II. (hy 4 Iz. OTHER 362 On the PROGRESS of HEAT 1x. OTHER things remaining as before, if we now make r= 10, then b= — 44.444-+ poe oii x b 20 477°.550 30 303 556 - 40 226 .556 50° 164-556 60 145 556 70 104 .556 80 85 .056 go 64.556 100 60.000 12. ix: R= 30; °o's= 3,0 Als tecoe, ai 1. = 60; y A ETS, ° b= — 1044.44 + ~—Eb4t Values of x Values of 4 I0000°.00 4477 -7° — 2637 .04 1716.67 1164.44 796 «30 53333 346.16 182 .72 60.00 i O00 ON AN PW Db 13. THE in SPHERICAL BODIES. 363 £3. I. THE general conclufions which refult from all this are, that when we fuppofe an ignited nucleus of a given mag- nitude, and a given intenfity of heat, there is in the {phere to which it communicates heat a fixed temperature for each par- ticular ftratum, or for each fpherical fhell, at a given diftance from the centre; and that a great intenfity of heat in the inte- rior, is compatible with a very moderate temperature at the furface. 2. HowEver great the {phere may be, the heat at its furface _ cannot be lefs than a given quantity ; R, 7, H and T remain- ing the fame. It muft-be obferved, that though R is put for the radius of the globe; it fignifies in fact nothing, but the dif- tance at which the temperature is T, as r does the diftance at which the temperature is H. THEREFORE were the fphere indefinitely extended, the tem- perature at its fuperficies would not be lefs than the quantity SS that is, not lefs than 50.5 in the firft of the~pre- ceding examples, than — 44.4 in the fecond, or — 1044.4 in the third. 4 14. In all this the {phere is fuppofed homogeneous ; but if it be otherwife, and vary in denfity, in the capacity of the parts for heat, or in their power to condudt heat, providing it do fo as any function of the diftance from the centre, the calculus may be inftituted as above. For example, let the denfi- b. b+ x’ then we have as before ty be fuppofed to vary as 3 b+ x heat in a ftratum placed at the diftance x from the centre, LZ 2 or (b—2') (x — x’) for the momentary increment of 364. On the PROGRESS of HEAT . 5 e b ; are e . or 5X 3x2xxX bop = © 4 Siven quantity, or to a x, and BE 2 Nag therefore 4 = — a : 20% ax 2 « a x —-—. Hence b= bx a =F “2 ; Log x, Suppofe that when x= 7, the radius ef the heated nucleus, 4 =H; then H= Tara ego and “C'= H— — bie ae r; therefore b= a 3r In this expreffion a’ will be determined, if the temperature at any other diftance R from the centre is known. Let this be _ T >; then by fubftitution we have : a a r To H— cs 4545 Log => ands vz" => : i. a ae ee T—H Hence 4= H+ a Aah x I | I x (a Las a 15. THis * im Site RECALL OBO D DE S. 365 15. THIs is given merely as an example of the method of conducting the calculus when the variation of the denfity is taken into account, and not becaufe there is reafon to believe that the law which that variation actually follows, is the fame that has now been hypothetically affumed. 16. THE principle on which we have proceeded, applies not only to folids, fuch as we fuppofe the interior of the earth, but _ it applies alfo to fluids like the atmofphere, provided they are fuppofed to have reached a fteady temperature. The propaga- tion of heat through fluids is indeed carried on by a law very different from that which takes place with refpect to folids ; it is not by the motion of heat, but by the motion of the parts of the fluid itfelf. Yet, when we are feeking only the mean re- fult, we may fuppofe the heat to be fo diffufed, that it does not accumulate in any particular ftratum, but is limited by the equality of the momentary increments and decrements of tem- perature which that ftratum receives. This is conformable to experience ; for we know that a conftancy, not of temperature, but of difference between the temperature of each point in the ‘atmofphere and on the furface, actually takes place. Thus, near the furface, an elevation of 280 feet produces, in this country, a diminution of one degree. The ftrata of our atmo- fphere, however, differ in their capacity of heat, or in the quantity of heat contained in a given fpace, at a given tempe- rature. Concerning the law which the change of capacity follows, we have no certain information to guide us; and we haye no refource, therefore, but to aflume a hypothetical law, agreeing with fuch facts as are known, and, after deducing the refults of this law, to compare them with the obfervations made on the temperature of the air, at different heights above the furface of the earth. 17, LEP 366 On the PROGRESS of HEAT 17. Ler us then fuppofe, that the ftrata of the atmofphere have a capacity for heat, which increafes as the air becomes r rarer, fo as to be proportional to mb *, x denoting, as be- fore, the diftance from the centre of the earth, 7 the radius of the earth, m and 4 determinate, but unknown quan- iis —i um ; ; tities, fuch that mb ors, exprefles the capacity of air for heat, when of its ordinary denfity, at the furface of the earth. The formula thus aflumed, agrees with the extreme cafes ; | ; wm for, when. x=~7, the capacity of heat = Pa! finite . S EF . . ° quantity ; when + increafes, ; diminifhes, and fo alfo does r ~ . . is ™ . . b”, if b is greater than unity, and therefore — increafes conti- bF nually. It does not, however, increafe beyond a certain limit, ES I m for when x is infinite — becomes =, OF Mm bx 18. HENCE, in SPHERICAL BODIES. 367 18. Hencz, by reafoning as in § 6. the momentary incre- ment of the temperature, or fenfible heat, of any ftratum, is as -S directly, and its capacity for heat, or m 6b * inverfely, that is, b= —S x b* mm 5 a" id 3 m 31% r Pg ths LEMOS eee nay | LET ~ =) then — 3 Hh fo that ei cee and | MUS gr Bee ar a tigi therefore pis waa 4, Hence 6 = C-- Boo b = a. = th nse ae ne 19. To determine C, if T be the temperature of the air at y) a b the furface, when r=7r, T= O + 3mr Loge? and: C = a b T — JmrLogs 5 | ab Le Sis HENCE ates Ae "a aartgee Saadeh 7 jie Be) 3mr Log db T— Tus formula, when x= 7 gives = T, and when w is in- a (b—1) amr e In all mtermediate finite, it gives 2= T— cafes, 368 On the PROGRESS of HEAT = r - cafes, as x is greater than r, 6* is lefs than 4, (6 being a ‘ id number greater than 1) and therefore 4—4= is pofitive, fo that 4 is lefs than T, as it ought to be. 20. WE may obtain an approximate value of this formula, without exponential quantities, that will apply to all the cafes in which « and r differ but little in refpe@ of 7, that is, in all the cafes to which our obfervations on the atmofphere can pof- fibly extend. Ir, in the term 4* we write r+z for x, z being the height of any ftratum of air above the furface of the earth, r r _ a « = hrt+s, we have J ‘21. But, from the nature of exponentials, we know 47 = r r* (Log 6)’ r> (Log 5) id ty ee hae ee r r* (Log 6)’ i+ ag oe 2(r+2) +, &c. Nowe e2e= pee op es &c. And if we leave ag See f Pag We out the higher powers of x, we have nearly r rad + R in SPHERICAL BODIES. 369 wT™ 8 + R ww & | ce | © a r THEREFORE, by fubftitution, we have 6’+* = 1+(1-) Logs + (x—22) Mee 4, ee = if + logs + E285 4 COED? 4, a ar “] t — 2 Logb —* (Log py 2 EY. ae. ( Now, fons the nature of exponentials, 08 b° ae 53 Pai age bass — a+ +, &c. And = = Log += = (Log b)° ++ = ord a , &e. = = Log b x fogs + ee +, &c.) on = Log 6 ; rhormteee a 2 is very fmall, a ae Vox. VI. P. IL. ce. 370 On the PROGRESS of HEAT, €e. , (bre) 6— — ? Log 5, and therefore (§ 19. te “3mr Loeb = (s— —iglztenty a Ls 3 hence when 2 is very 3mr Log b = om fmall, b=T—2 a 3mr 22. THEREFORE when 2, or the height above the furface is {fmall, ) diminifhes in the fame proportion that the height in- creafes, which is conformable to experience. In our climate, when. z= 280 feet, a4 X 280 —1°; fo 3m r that the co-efficient . = = a and therefore b=T— — Wuen the conflant quantities are thus determined, the for- mula agrees nearly with obfervation. In the rule for barome- trical meafurements, it is implied, that the heat of the atmo- {phere decreafes uniformly; but the rate for each particular cafe is determined by actual obfervation, or by thermometers — obferved at the top and bottom of the height to be meafured. _ XI. XI. Experiments on Allanite, a new Mineral from Green- land. By Tuomas Tuomson, M.D. F.R.S.E. Fellow of, the Imperial Chirurgo-Medical Academy of Peters- burgh. [Read Nov. 5. 1810.] BOUT three years ago, a Danith veffel * was brought into Leith as a prize. Among other articles, fhe contained a fmall collection of minerals, which were purchafed by Toomas ALLAN, Efq; and Colonel Imriz, both members of this Society. The country from which thefe minerals had been brought was not known for certain; but as the collection abounded in Cry- olite, it was conjectured, with very confiderable oi lnc that they had been collected in Greenland. Amonc the remarkable minerals in this collection, there was one, which, from its correfpondence with Gadolinite, as defcri- bed in the different mineralogical works, particularly attracted the attention of Mr ALLan. Confirmed in the idea of its being a variety of that mineral, by the opinion of Count Bournon, added to fome experiments made by Dr WoLLasTon, he was in- duced to give the defcription which has fincé been publifhed in a preceding part of the prefent volume. Axsout a year ago, Mr ALLAN, who has greatly diftinguifhed himfelf by his ardent zeal for the progre{s of mineralogy in all A2 its * DER FruuLine, Captain Jacoz KETELSON, captured, on her _ Passage from Iceland to oper 372 On ALLANITE, a new its branches, favoured me with fome fpecimens of this curious mineral, and requefted me to examine its compofition,—a re- queft which I agreed to with pleafure, becaufe I expected to obtain from it a quantity of yttria, an earth which I had been long anxious to examine, but had not been able to procure a_ fufficient quantity of the Swedith Gadolinite for my purpofe. The object of this paper, is to communicate the refult of my experiments to the Royal Society,—experiments which cannot appear with fuch propriety any where as in their Tranfac- tions, as they already contain a paper by Mr ALLAN on the mi- neral in queftion. . | I. DEscRIPTION. I am fortunately enabled to give a fuller and more accurate defcription of this mineral than that which formerly appeared, Mr Atran having, fince that time, difcovered an additional quantity of it, among which, he not only found frefher and better characterifed fragments, but alfo fome entire cryftals. In its compofition, it approaches moft nearly to Cerite, but it dif- fers from it fo much in its external characters, that it muft be confidered as a diftiné fpecies. I have therefore taken the li- — berty to give it the name of Allanite, in honour of Mr ALian, to whom we are in reality indebted for the difcovery of its pe- culiar nature. ALLANITE occurs maflive and difleminated, in irregular mafles, mixed with black mica and felfpar; alfo cryftallifed ; the varieties obferved are, 1. A four-fided oblique piifm, meafuring 117° and 63°. 2. A fix-fided prifm, acuminated with pyramids of four fides, fet on the two adjoining oppofite planes. Thefe laft are fo minute as to be incapable of meafurement. But, as nearly as the eye can determine, the form refembles Fig. 1.3; the prifm of which has two right ks and. four meafuring 135°. 3. A e ‘ Transactions ht. S Tdin’ VL VIP373. SS = a SEAN AE ae No SS = Y L Yy J wy | " hire ae mt} i it apa § . 4 Sate Dh, Ther Perey bt es MINERAL from GREENLAND. : 373 3. A flat prifm, with the acute angle of 63° replaced by one plane, and terminated by an acumination, having three principal facettes fet on the larger lateral planes, with which the centre one meafures 125° and 55°. Ofthis * fpecimen, an engraving is given in the annexed Plate, Pie: 2. SPECIFIC gravity, according to my experiments, 3.533. The fpecimen appears to be nearly, though not abfolutely, pure. This fubftance, however, is fo very much mixed with mica, that no reliance can be placed on any of the trials which have been made. Count Bournon, furprifed at the low fpecific gra- vity noted by Mr ALLAN, which was 3.480,broke down one of the fpecimens which had been fent him, in order to procure the fubftance in the pureft ftate poffible, and the refult of four experiments was as follows, 4.001 3197 3-654 a ‘In a fubfequent experiment of Mr ALLAn’s, he found it 3. 66 ie From thefe it appears, that the fubftance is not in a pure {tate. Its colour is fo entirely the fame with the mica, with which it is accompanied, that it is only by mechanical attrition that they can be feparated. Coxiour, brownifh-black. EXTERNAL luftre, dull; internal, fhining and refinous, flight- ly inclining to metallic. FRACTURE, {mall conchoidal. FRAGMENTS, indeterminate, fharp-edged. — OPAKE. SEMI-HARD in a high degree. Does not fcratch quartz nor felfpar, but feratches hornblende and crown-glafs, BRITTLE. EASILY 374 On ALLANITE, a new Eastiy frangible. Powner, dark greenifh-grey. Berore the blow-pipe it froths, and melts imperfe@ly into a brown {coria. GELATINISEs in nitric acid. In a ftrong red heat it lofes 3.98 per cent. of its weight. II. ExpERIMENTS TO ASCERTAIN ITS COMPOSITION. My firft experiments were made, on the fuppofition that the mineral was a variety of gadolinite, and were pretty much in the ftyle of thofe previoufly made on that fubftance by Exr- BERG, KLAPROTH, and VAUQUELIN. I. 100 grains of the mineral, previoufly reduced to a fine powder in an agate mortar, were digefted repeatedly on a fand ' bath in muriatic acid, till the liquid ceafed to have any action on it. The undiffolved refidue was filica, mixed with fome frag- ments of mica. When heated to rednefs, it weighed 33.4 grains. ' 2. THE muriatic acid folution was evaporatel almoft to dry- nefs, to get rid of the excefs of acid, diflolved in a large quan- tity of water, mixed with a confiderable excefs of carbonate of ammonia, and boiled for a few minutes. By this treatment, the whole contents of the mineral were precipitated in the {tate of a yellowith powder, which was feparated by the filtre, and boiled, while ftill moift, in potafh-ley. A fmall portion of | it only was diflolved. ~The potafh-ley was feparated from the undiffolved portion by the filtre, and mixed with a folution of {al ammoniac, by means of which a white powder precipitated — from it. This white matter being heated to rednefs, weighed 7.9 grains. It was digefted in fulphuric acid, but 3.76 grains refufed to diffolve. This portion poffeffed the properties of fi- lica. The diffolved portion being mixed with a few drops of fulphate MINERAL from GREENLAND. 395 fulphate of potath, fhot into cryftals of alum. It was therefore alumina, and amounted to 4.14 grains. 3. THE yellow matter which refufed to diffolve in the pot- afh-ley, was mixed with nitric acid. An effervefcence took place, but the liquid remained muddy, till it was expofed to heat, when a clear reddifh-brown folution was effected. This folution was evaporated to drynefs, and kept for a few minutes in the temperature of about 400°, to peroxidize the iron, and render it infoluble. A fufficient quantity of water-was then poured on it, and digefted on it for half-an-hour, on the fand- bath. The whole was then thrown upon ‘a filtre. The dark red matter which remained on the filtre, was drenched in oil, and heated to rednefs, in a covered crucible. It was then black, and attracted by the magnet; but had not exadtly the appearance of oxide of iron. It weighed 42.4 grains. 4. Tue liquid which paffed through the filtre, had not the {weet tafte which I expected, but a flightly bitter one, fimilar. to a weak folution of nitrate of lime. Hence it was clear, that no yttria was prefent, as there ought to have been, had the mineral contained that earth. This liquid being mixed with carbonate of ammonia, a white powder precipitated, which, after being dried in a red heat, weighed 17 grains. It diffolved in acids with effervefcence ;. the folution was precipi- tated white by oxalate of ammonia, but not by pure ammonia. When diffolved in fulphuric acid, and evaporated to drynefs, a light white matter remained, taftelefs, and hardly foluble in. water. Thefe properties indicate carbonate of lime. Now, 17 grains of carbonate of lime are equivalent to about 9.23 grains. of lime. 5- From 376 On ALLANITE, a new s. From the preceding analyfis, fuppofing it accurate, it fol- lowed, that the mineral was compofed of Silica, - oS ei - 37.16 Lime, ~ - - 9-23 Alumina, - « - - 4.14 Oxide of iron, - « - 42.40 Volatile matter, sie - 3-98 96.91 Lofs, - - . 3-09 100.00 But the appearance of the fuppofed oxide of iron, induced me to fufped, that it did not confift wholly of that metal. I thought it even conceivable, that the yttria which the mineral contained, might have been rendered infoluble by the applica- tion of too much heat, and might have been concealed by the iron with which it was mixed. A number of experiments, which it is needlefs to {pecify, foon convinced me, that, befides iron, there was likewife another fubftance prefent, which pof- fefled properties different from any that I had been in the ha- bit of examining. It poffefled one property at leaft in com- mon with yttria ; its folution in acids had a {weet tafte ; but few of its other properties had any refemblance to thofe which the chemifts to whom we are indebted for our knowledge. of yttria, have particularifed. But as I had never myfelf made any ex- periments on yttria, I was rather at a lofs what conclufion to draw. From this uncertainty, I was relieved by Mr Atian, who had the goodnefs to give me a {mall fragment of gadoli- nite, which had been received dire@tly from Mr EKEBErRG. From this I extracted about 10 grains of yttria; and upon com- paring its properties with thofe of the fubftance in queftion, I found MINERAL from GREENLAND. ~ 377 found them quite different. Convinced by thefe experiments, that the mineral contained no yttria, but that one of its confti- tuents was a fubftance with which I was {till unacquainted, I had recourfe to the followi ing mode of analyfis, in order to ob- tain this fubftance in a pure ftate. toe Ill. ANALYsIS OF ALLANITE. 1. 100 grains of the mineral, previoufly reduced to a fine powder, were digefted in hot nitric acid till nothing more could be diffolved. The undiffolved refidue, which was filica, mixed with fome {cales of mica, weighed, after being heated to rednefs, 35.4 grains. 2, Tue nitric acid folution was tranfparent, and of a light- brown colour. When ftrongly concentrated by evaporation, to get rid of the excefs of acid, and fet afide in an open capfule, it concreted into a whitifh folid matter, confifting chiefly of foft cryftals, nearly colourlefs, having only a flight tinge of yellow. Thefe cryftals being left expofed to the air, became gradually moift, but did not fpeedily deliquefce. The whole was therefore diffolved in water, and the excefs of acid, which _ was ftill prefent, carefully neutralifed with ammonia. By this treatment, the folution acquired a much deeper brown colour ; but ftill continued tranfparent. Succinate of ammonia was then dropped in with caution. A copious reddifh-brown pre- cipitate fell, which being wathed, dried, and heated to rednefs in a covered crucible, weighed 25.4 grains. It poffeffed all the characters of black oxide of iron. For it was attracted by the magnet, completely foluble in muriatic acid, and the folu- tion was not precipitated by oxalate of ammonia. 3. THE liquid being ftill of a brown colour, I conceived it not to be completely free from iron. On this account, an ad- Vou. VI. P. II. 3 B . ditional 378 _ On ALLANITE, a new ditional quantity of fuccinate of ammonia was adde. And we precipitate fell; but inftead of the dark reddifh-brown colour, _ which characterizes fuccinate of iron, it had a beautiful flefh- red colour, which it retained after being dried in the open air. | ‘When heated to rednefs in a covered crucible, it became black, and had fome refemblance to pag ioe It weighed 7.2 grains. 4. Tus fubflance attracted my peculiar attention, in confe- quence of its appearance. I found it to poflefs the following characters : _ a. Ir was taftelefs, and not in the leaft attracted by the mag- net, except a few atoms, which were eafily feparated from the reft. ty ot ‘ b. Ir was infoluble in water, and not fenfibly acted on when boiled in fulphuric, nitric, muriatic, or nitro-muriatic acid. c. Brerore the blow-pipe it melted with borax and microcof- mic falt, and formed with both a colourlefs bead. With car- bonate of foda it formed a dark-red opake bead. d. WHEN heated to rednefs with potafh, and digefted in wa- ter, fnuff-coloured flocks remained undiffolved, which gradual- ly fubfided to the bottom. The liquid being feparated, and exa- mined, was found to contain nothing but potafh. When mu- riatic acid was poured upon the fnuff-coloured flocks, a flight effervefcence took place, and when heat was applied, the whole diflolved. The folution was tranfparent, and of a yellow co- lour, with a flight tint of green. When evaporated to drynefs, to get rid of the excefs of acid, a beautiful yellow matter gra- dually feparated. Water boiled upon this matter diffolved the whole. The tafte of the folution was aftringent, with a flight metallic flavour, by no means unpleafant, and no cea was perceptible. e A MINERAL from GREENLAND. 399 wits A portion of the black powder being expofed to a red heat for an hour, in an open crucible, became reddifh-brown, and loft fomewhat of its weight. In this altered ftate, it was foluble by means of heat, though with difficulty, both in nitric and fulphuric acids. The folutions had a reddifh-brown co- lour, a flight metallic aftringent tafte, but no fweetnefs. f. Tue folution of this matter in nitric and muriatic acid, when examined by re-agents, exhibited the following pheno- mena : (1.) With pruffiate of potafh, it threw down a white precipi- tate in flocks. It foon fubfided ; readily diffolved in nitric acid ; the folution was green. (2.) Pruffiate of mercury. A light yellow precipitate, fo- luble in nitric acid. (3.) Infufion of nut galls. No change. (4.) Gallic acid. No change. (5.) Oxalate of ammonia. No‘change. (6.) Tartrate of potafh. No change. (7.) Phofphate of foda. No change. (8.) Hydro-fulphuret of ammonia. Copious black flocks. Liquor remains tranfparent. (9.) Arfeniate of potath. A white precipitate. (1o.) Potath. - - ~ Copious yellow-coloured (11.) Carbonate of foda. - - piss 3 readily eee in (12.) Carbonate of ammonia. J nitric acid. (13.) Succinate of ammonia. A white precipitate. (14.) Benzoate of potafh. . A white precipitate. * (15.) A plate of zinc being put into the folution in muriatic acid, became black, and threw down a black powder, which was infoluble in fulphuric, nitric, muriatic, ni- tro-muriatic, acetic, and phofphoric acids, in every 3 B2 ~ temperature, + ~ On ALLANITE, a new temperature, whether thefe acids were concentrated or diluted. (16.) A plate of tin put into the nitric folution, occafioned no change. ea (17.) A portion being inclofed in a charcoal crucible, and ex- poied for an hour to the heat of a forge, was not re- duced to a metallic button, nor could any trace of it be detected when the crucible was examined. THESE properties were all that the fmall quantity of the matter in my pofleffion enabled me to afcertain. They une- quivocally point out a metallic oxide. Upon comparing them with the properties of all the metallic oxides known, none will be found with which this matter exactly agrees. Cerium is the metal, the oxides of which approach the neareft. The colour is nearly the fame, and both are precipitated white by pruffiate of potafh, fuccinate of ammonia, and benzoate of © potafh. But, in other refpects, the two fubftances differ entire- ly. Oxide of cerium is precipitated white by oxalate of am- monia and tartrate of potafh; our oxide is not precipitated at all: Oxide of cerium is precipitated white by hydro-fulphuret of ammonia; while our oxide is precipitated black : Oxide of ce- rium is not precipitated by zinc, while our oxide is thrown down black. There are other différences between the two, but thofe which I have juft mentioned are the moft ftriking. THESE properties induced me to confider the fubftance which I had obtained from the Greenland mineral as the oxide of a metal hitherto unknown ; and I propofed to diftinguith it by the name of Junonium. In the experiments above detailed, I had expended almoft all the oxide of Junonium which I had in my pofleffion, taking it for granted that I could eafily procure more of it from the Green- land MINERAL from GREENLAND. 38% land mineral. But, foon after, l was informed by Dr Wotza- ston, to whom I had fent a fpecimen of the mineral, that he had not been able to obtain any of my fuppofed Junonium in. his trials. This induced me to repeat the analyfis no lefs than three times, and in neither cafe was I able to procure any more of the fubftance which I have defcribed above. Thus, it has been out of my power, to verify the preceding details, and to put the exiftence of a new metal in the mineral beyond doubt. At the fame time, I may be allowed to fay, that the above ex- periments were made with every poflible attention on my part, and moft of them were repeated, at leaft a dozen times. I have no doubt myfelf of their accuracy ; but think that the exift- ence of a new metal can hardly be admitted, without ftronger proofs than the folitary analyfis which I have performed. 5. Tue liquid, thus freed from iron and junonium, was fuper- faturated with pure ammonia. - 4.1 Oxide of iron, - - 25.4 Oxide of cerium, - - 33-9 _ Volatile matter, - - Ae II2.0 Wot. Vi. Pell. 3C I 386 (On ALLANITE, &e. I omit the 7 grains of junonium, becaufe I only detected it in one fpecimen of allanite. The excefs of weight in the preceding numbers, is to be afcribed chiefly to the carbonic acid combi- ned with the oxide of cerium, from which it was not complete- ly freed by a red heat. I have reafon to believe, too, that the proportion of iron is not quite fo much as 25.5 grains. For, in another analyfis, I obtained only 18 grains, and in a third 20 grains. Some of the cerium was perhaps precipitated along _ with it in the preceding analyfis, and thus its weight was appa- rently increafed. | . XII. A Chemical Analysis of Sodalite, a new Mineral from Greenland.. By Tuomas Tuomson, M.D. F.R.S. E. Fellow of the Imperial Chirurgo-Medical Academy of Pe- tersburgh. [Read Nov. 5. 1810.] HE mineral to which I have given the name of Sodalite, ~was alfo put into my hands by Mr Arran. In the - Greenland collection which he purchafed, there were feveral fpecimens of a rock, obvioufly primitive. In the compofition of thefe, the fubftance of which I am about to treat, formed a conftituent, and, at firft appearance, was taken for felfpar, to which it bears‘a very ftriking refemblance. Tuts rock is compofed of no lefs than five different foffils, — namely, garnet, hornblende, augite, and two others, which form the pafte of the mafs. Thefe are evidently different minerals ; but in fome fpecimens, are fo intimately blended, that it requi- red the fkill of Count BourNnow to make the difcrimination, and afcertain their real nature. Even this diftinguifhed mine- ralogift was at firft deceived by the external afpedt, and confi- dered the pafte as common lamellated felfpar, of a greenith co- ~ lour. But a peculiarity which prefented itfelf to Mr ALLAN, in one of the minerals, induced him to call the attention of Count BourNnon more particularly to its conftraction. On. a clofer examination of the mineral, M. de Bournom found that fome fmall fragments, which he had detached, pre- 3C2 fented 388 | On SODALITE, a new fented rectangular prifms, terminated by planes, meafuring, with the fides of the prifm, 110° and 70° or nearly fo,—a “form which belongs to a rare mineral, known by the name of Sahlite, from Sweden. He further obferved, intermixed along with this, another material ; and after fome trouble, fucceeded in detaching a mafs, prefenting a regular rhomboidal dodecahe- dron. It was to this form that Mr Attan had previoutfly re- quefted his attention. Some time before this inveftigation, M. de Bournon had examined a mineral from Sweden, of a lamellated ftructure, and a greenifh colour, which, he found, indicated the fame form. From this circumftance, together with fome external refemblance, which ftruck him, he was induced to conclude, that our mineral was a variety of that fubftance. To that fubftance the name of Swedith natrolite had been gi- ven, in confequence of the inveftigation of Dr Wot.LasTon, who found that it contained a large proportion of foda. THERE are few minerals, however, that are fo totally diftiné& in their external charaéters as the natrolite of KLaprotu, and the fub{tance we are now treating of. The mineral examined by KLAPROTH occurs at Roegan*, on the Lake of Conftance, in porphyry-flate, coating the fides of veins and cavities in a ma- mellated form, the texture of which is compa@, fibrous, and ra- diated; the colour pale yellow, in fome places pafling into white, and marked with brown zones. Hitherto it had never been found in a ftate fufficiently perfect to afford any indi- cations of form. Lately, however, M.:de Bournon was fo fortunate as to procure fome of it, prefenting very delicate needleform cryftals, which, by means of a ftrong magnifier, he was able to afcertain, prefented flat rectangular prifms, ter- minated by planes, which, he thought, might form angles of 60° * Ir has been obferved alfo by Profeflor JamEsoy, in the fletz-trap rocks behind Burntifland. MINERAL from GREENLAND. 389 60° and 120 with the fides of the prifm. With this, neither our mineral nor the Swedifh can have any connection, farther than fome analogy which may exift in their compofition. CoNCERNING the Swedifh mineral, I have not been able to obtain much {fatisfactory information. There is a fpecimen of rm in Mr Atyan’s cabinet, which he received directly from Sweden, fent by a gentleman who had juft before been in Lon- don, and was well acquainted with the collections of that city, from which it is inferred, that the {pecimen in queftion is the fame as that examined by Count Bournon and Dr WoLLa- STON. . WERNER has lately admitted into his fyftem a new mineral {pecies, which he diftinguifhes by the name of Fetéstezn. Of this I have feen two defcriptions ; one by Hai, in his Tableau Comparatif, publifhed laft year ; and another by Count DunIn Borkowski, publifhed in the 69th volume of the Journal de Physique, and tranflated in Nicholson’s Journal, (vol. 26. p. 384). - The fpecimen, called Swedish Natrolite, in Mr ALuan’s pof- . feffion, agrees with thefe defcriptions in every particular, ex- cepting that its fpecific gravity is a little higher. Bor- KOwsKI ftates the {pecific gravity of fettsteim at 2.563; Haiiy at 2.6138; while I found the fpecific gravity of Mr ALLAN’s fpecimen to be 2.779, and, when in fmall fragments, to be as high as 2.790. This very near agreement in the properties of the Swedifh natrolite, with the characters of the fettftein, leads me to fuppofe it the fubftance to which WERNER has given that name. This opinion is ftrengthened, by a fact mentioned by Haiiy, that fettftein had been at firft confidered as a varie- ty of Wernerite. For the {pecimen fent to Mr ALLAN, under the name of Compact Wernerite, is obvioufly the very fame with the fuppofed natrolite of Sweden. Now, if this identity be ad- mitted, it will follow, that our mineral conftitutes a {pecies apart. It bears, indeed, a confiderable refemblance to it; but neither the cryftalline form, nor the conftituents of fettftein, as 390 On SODALITE, a new as ftated by Havy, are fimilar to thofe of the mineral to which I have given the name of Sodalite. The conftituents of fett- ftein, as afcertained by VAUQUELIN, are as follows : Sires’ = ee - 2 44.00 Alumina, : - 34-00 Oxide of iron, - - 4.00 Lime, -. - - OE? Potath and foda, - - 16.50 Lofs, sai: - - 1.38 100.00 Il. DESCRIPTION OF SODALITE. SoDALITE, as has been already mentioned, occurs in a pri- mitive rock, mixed with fahlite, augite *, hornblende, and gar- net }. TT Ir occurs maflive ; and cryftallifed, in rhomboidal dodecahe- - drons, which, in fome cafes, are lengthened, forming fix-fided prifms, terminated by trihedral pyramids. Its colour is intermediate between celandine and mountain - ereen, varying in intenfity in different fpecimens. In fome cafes it feems intimately mixed with particles BS fahlite, which doubtlefs modify the colour. External luftre glimmering, internal fhining, in one direc- tion vitreous, in another refinous. : Fracture foliated, with at leaft a double cleavage ; prot fracture conchoidal. FRAGMENTS indeterminate; ufually fharp-edged. TRANSLUCENT. * Turs fituation of the augite deferves attention. Hitherto it has been, with a few exceptions, found only in fletz trap rocks. + Tue particular colour and appearance of this garnet, fhews that the rock came from Greenland: For fimilar garnet has never been obferved, except in {pecimens from Greenland. MINERAL from GREENLAND. 391 . TRANSLUCENT. HarDNEss equal to that of felfpar. Iron fcratches it with difficulty. BRITTLE. _ Easizy frangible. SPECIFIC gravity, at the temperature of Goxtevege: ) The fpecimen was not abfolutely free from fahlite. WHEN heated to rednefs, does not decrepitate, nor fall to powder, but becomes dark-grey, and aflumes very nearly the appearance of the Swedifh natrolite of Mr Attan, which I confider as fettftein. If any particles of fahlite be mixed with it, they become very confpicuous, by acquiring a white colour, and the opacity and appearance of chalk. The lofs of weight was 2.1 percent. I was not able to melt it before the blow- pipe. II. CHEMICAL ANALYSIS. 1, A HUNDRED grains of the mineral, reduced to a fine pow- der, were mixed with 200 grains of pure foda, and expofed for ‘an hour to a ftrong red heat, in a platinum crucible. The mix- ture melted, and: affumed, when cold, a beautiful grafs-green colour. When foftened with water, the portion adhering to the fides of the crucible acquired a fine brownith-yellow. Ni- tric acid being poured upon it, a epaiplcte {olution was ob- tained. - 2. SusPEcTING, from the appearance which the fufed mafs affumed, that it might contain chromium, I neutralifed the fo- lution, as nearly as pofflible, with ammonia, and then poured into it a recently prepared nitrate of mercury. A white preci- pitate fell, which being dried, and expofed to a heat rather un- der rednefs, was all diflipated, except a {mall portion of grey matter, 2g On SODALITE, a new matter, not weighing quite 0.1 grain. This matter was info- luble in acids; but. became white. With potath it fufed into a colourlefs glafs. Hence I confider it as filica. This experi- ment fhews that no chromium was prefent. I was at a lofs to account for the precipitate thrown down by the nitrate of mer- cury. But Mr Arian having fhown me a letter from Exe- BERG, in which he mentions, that he had detected muriatic acid in fodalite, it was eafy to fee that the whole precipitate was calomel. The white powder weighed 26 grains, indicating, according to the analyfis of CHENEVIX, about 3 grains of mu- riatic acid. 3. Tue folution, thus freed from muriatic acid, being con- centrated by evaporation, gelatinifed. It was evaporated near- ly to drynefs; the dry mafs, digefted in hot water acidulated with nitric acid, and poured upon the filter. The powder re- tained upon the filter was wafhed, dried, and heated to rednefs. It weighed 37.2 grains, and was filica. 4. THE liquor which had paffed through the filter, was fu- perfaturated with carbonate of potafh, and the copious white precipitate which fell, collected by the filter, and boiled while yet moift in potafh-ley. The bulk diminifhed greatly, and the undiffolved portion aflumed a black colour, owing to fome oxide of mercury with which it was contaminated. 5. THE potath-ley being pafled through the filter, to free it from the undiffolved matter, was mixed with a fufficient quan- tity of fal-ammoniac. A copious white precipitate fell, which being collected, wafhed, dried, and heated to rednefs, weighed 27.7 grains. This powder being digefted in fulphuric acid, diffolved, except 0.22 grain of filica. Sulphate of potafh being added, and the folution fet afide, it yielded alum cryftals to the very laft drop. Hence the 27.48 grains of diffolved powder were’ alumina. , 6. THE MINERAL from GREENLAND. 393 6. Tue black refidue which the potafh-ley had not taken up, was diffolved in diluted fulphuric acid. The folution being evaporated to drynefs, and the refidue digefted in hot water, a white foft powder remained, which, heated to rednefs, weighed 3.6 grains, and was fulphate of lime, equivalent to about: 2 grains of lime. 7. THE liquid from welrichi the fulphate of lime was fe: rated, being exactly neutralifed by ammonia, fuccinate of am- monia was dropped in; a brownifh-red precipitate fell, which, being heated to rednefs in a covered crucible, weighed 1 grain, and was black oxide of iron. 8. THE refidual liquor being now examined by different re- “agents, nothing farther could be precipitated from it. ‘g. Tue liquid (No. 4.) from which the alumina, lime, and iron had been feparatéd by ‘carbonate of potafh, being boiled for fome time, let fall a fmall quantity of yellow-coloured mat- ter. This matter being digefted in diluted fulphuric acid, part- ly diffolved with effervefcence ; but a portion remained undif- folved, weighing 1 grain. It was infoluble in acids, and with potafh melted into a colourlefs glafs. It was therefore filica. The fulphuric acid folution being evaporated to drynefs, left a refidue, which poflefled the properties of fulphate of lime, and. which weighed 1.2 grains, equivalent to about 0.7 grains of lime. 10. THE conftituents obtained by the preceding analyfis be-. ing obvioufly defective, it remained to éxamine whether the mineral, according to the conjecture of BourNon, contained an alkali. For this purpofe, 100 grains of it, reduced to a fine. powder, and mixed with 500 grains of nitrate of barytes, were expofed for an hour to a red heat, in a porcelain crucible. The fufed mafs was foftened with water, and treated with muriatic acid. The whole diffolved, except 25 ovis of a white pow- Vor. VI..P. TE 3D der, . 304, On SODALITE, a new ee) der, which proved on examination to be filica. The muriatic acid folution was mixed with fulphuric acid, evaporated to dry- nefs; the refidue, digefted in hot water, and filtered, to fepa- rate the fulphate of barytes. The liquid was now mixed with an excefs of carbonate of ammonia, boiled for an inftant or two, and then filtered, to feparate the earth and iron precipita- ted by the ammonia. The liquid was evaporated to drynefs, and the dry mafs obtained expofed to a red heat m a filver cru - cible. The refidue was diflolved in water, and expofed in the open air to {fpontaneous evaporation. The whole gradually fhot into regular cryftals of fulphate of foda. This falt being expofed to a {trong red heat, weighed 50 grains, indicating, ac- cording to BERTHOLLET’s late analyfis, 23.5 grains of pure fo- da. It deferves to be mentioned, that during this procefs, the filver crucible was acted on, and a fmall portion of it was af- terwards found among the fulphate of foda. This portion: was feparated before the fulphate of foda was weighed. TueE preceding analyfis gives us the conftituents of fodalite as follows : Silica, - - ~ Ae (o Pene Alumina, - - 27.48 Lime, - - - 2.70 Oxide of iron, - - 1.00 Soda, - - - 23-50 Muriatic acid, - = 3.00 Volatile matter, - - 2.10 Lofs, - - - ph (0) 100.00 MINERAL from GREENLAND. 395 Mr Atuuan fent a fpecimen of this mineral to Mr ExeE- BERG, who analifed it in the courfe of laft fummer. The con-> f{tituents which he obtained, as he ftates them in a letter to Mr ALLAN, are as follows : Silica, ~ = = nya: Alumina, - - se SP BBs : Soda, - - - Aig Muriatic acid, - - 6.75 Oxide of iron, . - 0125 100.00 ‘Tuts refult does not differ much from mine. The quantity of muriatic acid is much greater than mine. The lime and the volatile matter which I obtained, efcaped his notice altogether. If we were to add them to the alumina, it would make the two analyfes almoft the fame. No mineral has hitherto been found containing nearly fo much soda as this. Hence the reafon of the name by which I have diftinguifhed it. 3D 2 ITE. RE sate ~. * x “ip PE IDI EDIE DALE IEE OEE IPO EDEL P ILL EET LE OLE ILLES BOLE LSA LALLA L IEEE ALES Oe XIII. Demonstration of the Fundamental Property of the Lever. By Davip Brewster, LL. D. F. R.S. Evin. [Read December 3. 1810.} 4 T is a fingular fact in the hiftory of fcience, that, after all the attempts of the moft eminent modern mathematicians, to obtain a fimple and fatisfaGtory demonftration of the funda- mental property of the lever, the folution of this problem gi- ven by ArcuimeDEs, fhould ftill be confidered as the moft legi- timate and elementary. Ga.iteo, Huycrens, DE ta Hire, . Sir Isaac NeEwron, Maciaurin, Lanpen, and HAMILTON, have directed their attention to this important part of mecha- nics ; but their demonftrations are in general either tedious and abftrufe, or founded on affumptions too arbitrary to be recog- nifed as a proper bafis for mathematical reafoning. Even the demonftration given by ArcuimeEpsEs is not free from objec- tions, and is applicable only to the lever, confidered as a phy- fical body. GatiLzo, though his demonftration is fuperior in point of fimplicity to that of ARCHIMEDEs, reforts to the ine- legant contrivance, of fufpending a folid prifm from a mathe- matical lever, and of dividing the prifm into two unequal parts, which act as the power and the weight. The demonftration given by HuyceEns, affumes as an axiom, that a given weight removed 308 DEMONSTATION of the removed from the fulcrum, has a greater tendency to turn the lever round its centre of motion, and i is, befides, applicable on- ly toa commenfurable proportion of the arms. The founda- ~ tion of Sir Isaac Newron’s demonftration is ftill more inad- miffible. He affumes, that if a given power at in any direc- tion upon a lever, and if lines be drawn from the fulcrum te the line of direction, the mechanical effort of the power will be the fame when it is applied to the extremity of any of thefe lines ; but it is obvious, that this axiom is as difficult to be pro- ved as the property of the lever itfelf. M. Dr 1a Hire has given a demonftration which is remarkable for its want of ele- gance. He employs the reductio ad absurdum, and thus deduces the propofition from the cafe where the arms are commenfu- rable. The demonftration given by Macraurin has been highly praifed ; but if it does not involve a fetitio principit, it has at leaft the radical defeat, of extending only to a commen- furable proportion of the arms. The folutions of LaNDEN and Hami ton are peculiarly long and complicated, and refemble more the demonftration of fome of the abftrufeft points of mechanics, than of one of its fimpleft and moft elementary truths. In attempting to give a new demonftration of the fundamen- tal property of the lever, which fhall be at the fame time fimple and legitimate, we fhall aflume only one principle, which has been univerfally admitted as axiomatic, namely, that equal and opposite forces, acting at the extremities of the equal arms of a lever, and at equal angles to these arms, will be in equilibrio. With the aid of this axiom, the fundamental property of the lever may be eftablifhed by the three following propofitions. In Prop. E. the property is deduced in a very fimple manner, | when the arms of the lever are commenfurable. In. Fundamental Property of thee LEVER. 399 In Prop. II., which is totally independent of the firft, the de- monftration is general, and extends to any proportion between the arms. - ‘In Prop. III. the property is eftablifhed, when the forces act in an oblique direction, and when the lever is either rectilineal, angular, or curvilineal. In the demonftrations which have ge- nerally been given of this laft propofition, the oblique force has been refolved into two, one of which is direted to the fulcrum, while the other is perpendicular to that direction. It is then affumed, that the force directed to the fulcrum has no tendency to di- sturb the equilibrium, even though it atts at the extremity of a bent arm; and hence it is eafy to demonftrate, that the remaining force is proportional to the perpendicular drawn from the ful- crum to the line of direction in which the original force was applied. As the principle thus aflumed, however, is totally in- admiffible as an intuitive truth, we have attempted to demon- ftrate the propofition without its affiftance. Prop. 1.—Jf one arm of a straight lever is any multiple of the other, a force acting at the extremity of the one will be in equilibrio with a force acting at the extremity of the other, when these forces are reciprocally proportioned to the length of the arms to which they are applied, Ler AB (PuatTe XI. fig. 1.) be a lever fupported on the two fulcra F,f, fo that Af=/fF =FB. Then, if two equal weights C, D, of 1 pound each, be fufpended from the extremities A, B, they will be in equilibrio, fince they act at the end of equal arms Af, BF ; and each of the fulcra f, F, will fupport an equal part of the whole weight, or r pound. Let the fulcrum / be now remo- ved, and let a weight E, of 1 pound, a& upwards at the point fs the equilibrium will ftill continue ; but the weight E, of 1 pound, acting upwards at f, is equivalent to a weight G of 1 pound, acting downwards at B. Remove, therefore, the weight E, 400 DEMONSTRATION of the E, and fufpend the weight G from B; then, fince the equili- brium is ftill preferved after thefe two fubftitutions, we have a weight C, of one pound, acting at the extremity of the arm AF, in-equilibrio with the weights D and G, which together make two pounds, acting at the extremity of the arm FB. But FA is to FB as 2 is to 1; therefore an equilibrium takes place, when the weights are reciprocally proportional to the arms, in the particular cafe when the arms are as 2 to 1. By making F f fucceffively double, triple, &c. of FB, it may in like manner be fhewn, that, in thefe cafes, the propofition holds true. LEMMA. If any weight BC cb, (fig. a. No. 1.), of uniform shape and density, is placed on a lever A, whose fulcrum is @, it has the same ten- dency to turn the lever round 9, as if it were suspended from a point G, so taken that bG=Ge. Ir a weight W, of the fame magnitude with BC, acts upwards at the point G, it will be in equilibrium with the weight BC, and will therefore deftroy the tendency of that weight to turn the lever round g. But the weight W, acting upwards at the point G, has the fame power to turn the lever round @, as an equal weight w, acting downwards at G. Confequently the tendency of the weight BC to turn the lever round @, is the fare as the tendency of an equal weight w, acting downwards at G. Prop. II. If two forces applied to a lever, and acting at right angles to it, have the same tendency to turn the lever round its centre of mo- tion, they are reciprocally proportional to the distances of the points at which they are applied from the centre of motion. Let A od, (fig. 2. No. 2.) be a lever whofe fulcrum is 9, and let it be loaded with a weight BD dé of uniform fhape and den- fity. Fundamental Property of the LEVER. 401 fity.. Then by the lemma, this weight has the fame tendency to turn the lever round, as if it were fufpended from the point n, fo taken that bu =dn. Make oc = od, and let the weight BD dé be divided at the points C and F, by the lines Ce, F 9. The weights CF @c, D F od, being in equilibrio, by the axiom, have no tendency to turn the lever round ¢, confequently the remaining weight BC ed, has the fame tendency to turn the le- ver round @ as the whole weight BD dé. Hence if bm = cm, the weight BCcé acting at the point m, will have the fame tendency to turn the lever round ¢, as the weight BD dd acting dt fot Now, DD ao: BGCh=0 : bc and: mec; and “fince bec=bd—cd, we haveme=4bd—icd=nd—ycd=n¢, and ad=neticdzme+icdamg. Confequently, BDdd:BCcb=mo:n9@. ~ LEMMA. Two equal forces acting at the same point of the arm of a lever, and in directions which form equal angles with a perpendicular drawn through that point of the arm, will have equal tendencies to turn the lever round its centre of motion. Let AB (fig. 3.) be alever with equal arms AF, FB. Through the points A, B, draw AD, BE, perpendicular to AB, and AP, Ap, BW, Bw, forming equal angles with the lines AD, BE. Produce PA to M. Then, equal forces acting in the directions AP, Bw, will be in equilibrio. Buta force M equal to P, and acting in the direction AM, will counteract the force P, ading in the direQiion AB, or will have the fame tendency to turn the lever round F; and the force W, acting in the direction BW, will have the fame tendency to turn the lever round F as the 402 DEMONSTRATION of the the force M: Confequently the force W will have the fame ten: dency to turn the lever round F as the force w; and this will hold true, whether the arms AF, FB, are ftraight or curvili- neal, provided that they are both of the fame form. Prop. III.—Jf a force acts in different directions at the same point in the arm of a lever, its tendency to turn the lever round its centre of motion, will be proportional to the perpendiculars let fall from that centre on the lines of direction in which the force is applied. Let AB, (fig. 4.) be the lever, and let the two equal forces BM, Bm, act upon it at the point B, in the direction of the lines BM, Bm. Draw BN, Ba, refpectively equal to BM, Bm, and forming the fame angles with the line PB w perpendicular to AB. To BM, Bm, BN, Bz, produced, draw the perpendiculars AY, Ay, AX, Ax. Now, the fide AX = AY,and Ax = Ay, on account of the equality of the triangles ABX, ABY; and if M/, Ma, be drawn perpendicular to Ba, the triangles ABY, BM /, will be fimilar, and alfo the triangles ABy, Bm2: Hence we ob- tain : AB: AY = BM: Bd, and AB: Ay= BM: Ba | Therefore, ex equ, AY: Ay= Bl: Ba. ~ Complete the parallelograms BM oN, Bmwn, and B/, Ba will be refpectively one-half of the diagonals Bo, Bo. Now let two equal forces BM, BN, act in thefe direGions upon the lever at B, their joint force will be reprefented by the diagonal Bo, and confequently one of the forces BM will be :. 2" Transactions R.S Edin? Vol. WLP 278 rf IP MibCASA eI. ; So Fig.2.42. " 1 . i ~ A es ay hig f ; we - 4 bs. "¥ yh ay ly ‘ t , ; ie ‘ is Pree, ea , - ' a ’ 744 - ee ee y a 2 a - A = = - Fundamental Property of the LEV E R.. 403; be reprefented by BJ = 4Bo. In the fame manner, if the two equal forces Bm, Bn, act upon the lever at B, their joint force will be reprefented by Bw, and one of them, Bm, will be repre- fented by BA=4Bwo. Confequently the power of the two: forces BM; Bm, to turn the lever round its centre of motion, is reprefented by B/, Ba, refpectively ; that is, the force BM. is to the force Bm as B/ is to Ba; that is, as AY is to Ay, the perpendiculars let fall upon the lines of their direction. 3E 2 | XIV XIV. On the Rocks in the vicinity of Edinburgh. By Tuomas Atzan, Esq; F..R.S. Evin. [Read March 4. 1811. | LTHOUGH fcience has only within thefe few years ac- knowledged the importance of Geology, the eagernefs with which it has been cultivated, affords fufficient proof of the intereft it is capable of creating. Of this we have a recent ex- ample in the laborious undertaking of Sir Gzornce MacKxeEn- z1E and his friends, who, notwithftanding all the dangers pre- fented by a voyage through the moft tempeftuous ocean, and the deprivations to which they were expofed, in a journey through a country deftitute of the flighteft trace to guide the route of the traveller, were not deterred from exploring the inhofpitable fhores of Iceland. - Thefe, and other travellers, have extended our knowledge of various diftri€s on the furface of the globe; but we have ftill to lament the extreme imper- fection of the fcience, which, as yet, has affumed no decided cha- racter or form. This appears principally owing to the want of - fome fimple method, grounded on clear and intelligible princi- ples : - 406 On the ROCKS in the ples; perhaps alfo, to its having always been the object of thofe who have treated the fubject, to accommodate their ob- fervations to a particular theory ; and when this is the cafe, it is obvious, that the mind cannot refufe itfelf the fatisfaction, of dwelling with comparative enthufiafm on facts which appear fa- vourable to the adopted fyftem ; while others of a different ten- dency, are either reluctantly, and therefore fuperficially confi- dered, or what is yet worfe, even ftudioufly avoided. In the prefent ftate of our knowledge, to diveft geology of theory, would be to deprive it of all its intereft. We muft not defpair, however, that by the multiplication of particular fads, and the expofition of others, with which we are ftill unac- quainted, a fyftem of geology may yet. be formed, founded ex- clufively on the phenomena of nature, or at leaft on reafoning much lefs hypothetical than is now required. Tue moft obvious means of attaining this object, feems to be a careful, minute, and candid examination of every circum- ftance which appears to convey an explanation of itfelf, with- out reference to any theory; and from thefe we may ultimate- ly hope to obtain fome data, equally certain and comprehen- five. Ir is with this view, that I have always formed my collec-. tions of geological fpecimens ; and although it will appear, that the arguments I have deduced are favourable to one fet of opinions, yet I can aflert with confidence, that the diftri@ which it is the object of the prefent paper to examine, has been faith- fully explored, and, I hope, candidly defcribed.. - 2 VIGINIT YH of EDINS URG H." 407 Ir is generally admitted, that no city in Europe is more favour- ably fituated than the metropolis of Scotland, for the ftudy and purfuit of geology: even the ground which it occupies, when laid open for the erection of buildings, has occafionally prefented fome very interefting phenomena. ‘The hills in the immediate neighbourhood, always at command, afford a never-failing fource of refearch; and in the furrounding country, a greater variety of foffils is to be met with, than almoft in any {pace of the fame extent. | THE importance of a complete acquaintance with the pheno- mena which furround this city, cannot therefore, I think, be confidered of a trivial nature. Indeed, by the number of in- genious works already before the public, it may be thought that the fubject is exhaufted. But this is an error I am very defirous to combat, not only becaufe in my own experience I have found it to be one, but becaufe, as {cience advances, our ha- bits of inveftigation improve, phenomena become more fami- liar, we learn to trace and to feize not only the objects we are in purfuit of, but alfo to detect others, which our lefs practifed eye had originally paffed over unnoticed. We all think ourfelves perfectly acquainted with the rock, on which our Caftle ftands. But I fufpect there are many mem- bers of this Society, who will be furprifed to learn, that fand- ftone occurs near its fummit, and alfo at its bafe. Sa- lifbury 408 On the ROCKS in the lifbury Craig and Arthur’s Seat appear perfectly familiar to us ; there are phenomena belonging to both, however, of which, I _ have no doubt, many are yet ignorant. That any-circumftance of an interefting nature, fhould remain unobferved, can only be accounted for, by its being taken for granted, that thefe con{pi- cuous objects, having already undergone much critical examina- tion, nothing farther remains to be noticed. This is an opi- nion, which I fhall prove in the fequel, to be without founda- tion. Arthur’s Seat and Salisbury Craig, are naturally the objeéts, which firft attra@ the attention of the geological traveller, on his arrival in Edinburgh ; and to thefe places he is general- ly conducted by fome one of our amateurs, when the favourite theory is introduced, and each corroborative fact dwelt upon, with all the ufual keennefs of theoretic difcuffion. This was the ground which, in all probability, firft fuggefted the Theory of Hutton ; and it was perhaps here, that his comprehenfive mind originally laid the foundation, of the ftru@ture which he afterwards fo fuccefsfully reared. . But that theory, in itfelf fo beautiful, and in many points fo perfect, I am very far from embracing entirely. Iam very far, indeed, from follow- ing him through his formation and confolidation of ftrata, or the Eamerenion and arrangement of the materials, of which they are compofed. There are other circumftances alfo, which, ° though totally irreconcilable with any ether hypothefis, are yet but imperfe@ly explained by his. I particularly allude to the fin- _ gular contortions, exhibited in what are termed Transition ftra- ta, fo finely exemplified on the coaft of Berwickthire. [I wifh-to carry my inductions, juft as far as-facts will bear them out. It is therefore, only in the regions of unftratified rocks, er in their immediate vicinity, that I have as yet, been able to ‘difcover VEGINE TAG OR EDT ALR R GH. 409 difcover a language, which, if ftudied with due attention, can- not fail, I think, to become intelligible, and carry conviction to ‘ thofe, who choofe to reafon impartially on the fubjed. In the writings of Dr Hutton, we do not meet with deferip- tions of particular diftricts, his object being rather to eftablifh ageneral theory, by the particular facts which thefe diftrids aftorded. WE cannot, therefore, look to him for a mineralogical ac- count of the neighbourhood of Edinburgh; and we have to re- gret, that no other geologift has yet undertaken that tafk. ‘In a fhort notice, ih the Appendix of a work on another county, by Profeflor Jameson, this vicinity is mentioned as principally belonging, to what is termed the Coal Formation by WERNER, which, according to the fyftem of that celebrated na- turalift, forms part of the Fletz rocks. To render thefe terms intelligible to the general reader, it is neceflary to give fome explanation, as, without a confiderable knowledge of the fyftem to which they exclufively belong, they muft be totally incomprehenfible. WERNER is the only perfon, who has attempted a regular ar- rangement of rocks ; an arduous undertaking, which I have no doubt he has accomplifhed, with all the accuracy the fubject was fufceptible of, and fo far as the country he examined allow- ed *, But it appears very evident, that the facts he met with were fuch, that, in confequence of the hypothefis he had previoufly thought proper to adopt, it became neceflary to invent a theo- wOL. Vis be lh.” gk ry * Livxs from other quarters, having been fubfequently added to his forma- tion-fuites, by his pupils. 410 On the ROCKS in the ry capable of embracing all the phenomena, which the con- ftruction of his fyftematic arrangement led him to obferve. A -peculiar language was therefore indifpenfable; and as this language has been conftructed with fo much regard to his theo- ry, unlefs that is underftood and adopted, his terms become ufe- lefs. By a formation is meant, any feries or fuite of rocks which ufually occur together ; hence the Coal Formation is compo- fed of 1. Sandftone, - ©. Limeftone, 2. Coarfe Conglomerate, 7. Marl, 3. Slate-clay, 8. Clay-ironftone, 4. Bituminous Shale, = ——‘g..: Porphyritic Stone, 5- Indurated Clay, 10. Greenftone *, with which the Coal occurs in numerous beds, varying extreme- ly in thicknefs. Thefe, however, never all occur together, and it is no detriment to the Coal Formation fuite, even if Coal it- felf fhould not be found among them. AGAIN, the term Flatz is given to all the formations, contain- ed between the tranfition and alluvial rocks, and implies that they are diftinguifhed by their frequent occurrence in beds, which are much more nearly horizontal, than the primitive and tranfition * Greenftone is a literal tranflation from the German ; it is an extremely im- proper name; but as we have no other by which we can distinguish this variety of trap, we muft ufe it till a more appropriate is found, even at the expence of fuch language as red and blue greenftones, In the mean time, it muft be under- ftood merely as an arbitrary term. VICINITY of EDINBURGH. — ant tranfition rocks. If directly tranflated, the word fignifies flat, and may be correctly defcriptive of the diftricts originally exami- ned by WERNER; but as this conftruction will not apply uni- verfally to this clafs of rocks, and as it is particularly at vari- ance with thofe belonging to it in this country, it would be bet- ter to follow the example of Profeflor Davy, and ufe the term _ parallel rocks, which is much lefs liable to objection. Tue Hauttonian Theory has no language peculiar to itfelf, having nothing to defcribe, that cannot be done in the ufual phrafeology of any country. This, by the zealous admirers of that doctrine, may no doubt be lamented, as depriving it of an apparent fy{tematic arrangement, to which the oppofite theory is fo deeply indebted. tn forming a collection from the rocks in the neighbour-. hood of Edinburgh, the circumftances above narrated, indu-. ced me to begin with thofe of Salifbury Craig and its vicinity. The collection I have now the honour of prefenting to the So- ciety, I began fome years ago: it is only part of a {feries, which, as completed, I hope may be found worthy of a place in their cabinet. I confider it of very great importance, that every geological paper, fhould be accompanied with fpecimens, in order that if the former be found deferving of publication in your Transattions, thofe who perufe the defcription may: ‘know, that the fpecimens referred to, are to be feen in the re- - pofitories of this eftablifhment. 3F 2 SALISBURY. 412 On the ROCKS in the y | SALISBURY CRAIG is fituated on the north fide of Arthur’s Seat, againft which its fouthern extremity refts: from this it extends, in a northern direction, and rounds towards the eaft, fo as to form the fegment of a circle, about half a mile in length It is furmounted by a magnificent facade, which .is loweft at the extreme points; towards the mid- dle, the perpendicular rock may be from 80 to go feet high. From the bafe of this precipice, a floping bank, covered with debris, reaches to the valley below, altogether form- ing an elevation of nearly 4oo feet. From the upper edge of it, a regularly inclined plane, flopes gently, on the oppofite fide, at an angle of about 15°, in a north-eaft direction, and forms the left bank of the valley, called the Hunters Bog. On the right of this valley, the rocks again rife rapidly, affording indications of two or three feparate facades. Thefe are not characterized in the diftin@ manner of Salifbury Craig, but are furmounted by a furface, which, though a little rounded, pre- fents an inclination correfponding with that of the Craig, in a very ftriking manner. From the bafe of Salifbury Craig, or rather from the bafe of the debris by which it is encircled, towards the fouthern ex- tremity, the ground again rifes, and prefents an inclined plane, fimilar to its own, but of lefs magnitude. This is known by the name of St Leonard’s Hill. | -HENCE it appears, that there are three fimilarly inclined planes or terraces, of which Salifbury Craig forms the interme- diate one, each of them having a different elevation. From this ftructure we may eafily conceive the origin of the Swedifh word Trap, which has been employed as a generic term, for the VICINITY of EDINBURGH. 413 the clafs of rocks to which this appearance may generally be attributed *. Ir we imagine a vertical plane, to pafs from St Leonard’s Hill in an E.N.E. direction, which fhall cut Salifbury Craig, and continue through the right bank of the Hunters Bog, we fhall find the rocks difpofed in the following manner : St Leonard’s Hill. Sandftone. Porphyritic Greenftone. Sandftone. Salisbury Craig. Sanditone. Greenftone. Sandftone. Hunters’ Bog. Greenftone. Sandftone. Porphyritic Greenftone. Trap-Tuff. Bafalt. The * One of the greateft difficulties which geology as well as mineralogy has laboured under, is the multitude of fynonymous terms which have been applied to every individual foffil. Trap has fuffered from this difadvantage, perhaps more than any other variety of rocks; as above noticed, that name is derived from the fimilarity to the fteps of a ftair, which may generally be traced in the outline of a country, in which this rock abounds; and as it has been employed as a generic term by mineralogifts throughout Europe, I think it proper to use it, to the exclufion of whzn/tone, the name it bears in the writings of Dr Hutron ; a 414 On the ROCKS in the The two laft of thefe are not comprehended in the Coal For- mation fuite; they are confidered as members of another for- mation, denominated the Neweft Floetz-Trap. Tue upper fandftone of St Leonard’s Hill, and the lower fandfione of Salifbury Craig, are, fo far as we know, continu- ous; but as thefe, fuppofing the lines of the ftrata to be pro- jected, would form a bed of 450 feet thick, it is poflible alternations of greenftone may occur in it. Above, I have on-. ly mentioned fuch as are vifible. TuoseE on the right of the Hunters’ Bog, are not fo diftin@ly expofed as the reft ; but the foffils are all found in the order I have ftated. Occafionally fmall feams of reddifh-brown co- loured flaty clay, and clay-ironftone occur, principally inter-. mixed with the fand{tone; but they are fo thin, and fo uncon- nected, that they can fcarely be confidered.as ftrata. Tue feries of {pecimens I.am now about to defcribe, are thofe of St Leonard’s Hill and Salifbury Craig... ) No. 1. is a fpecimen of the Sandftone of St Leonard’s Hill ; it is of a reddifh-white colour, and extremely coarfe-grained. It was taken from the middle of the quarry, and preients a {pe- cies of conglomerate, the fragments of fandftone being agglu- tinated by a dark-red ferruginous patte.. . No. 2. from the fame quarry, is more compact, and prefents a ftreaked appearance, correfponding with the direction of the ftratum, There is a confiderable degree of irregularity to be obferved, in tracing the line of junction at St Leonard’s Hill. In fome places, two or three folds of the ftrata are cut off abruptly a name which, though perfeGtly underftood in this country, is not received abroad, and ought therefore to be relinquifhed.. VICINITY of EDINBURGH. 415 abruptly at each end by the greenftone; in another, that fub- ftance finks fuddenly as it weré into a gap in the ftrata, and be- ing loft in rubbifh, has fomewhat the appearance of a dike. Beyond this a double horizontal wedge of greenftone, with the ends turned downwards, appears among the ftrata; and a little farther, towards the north, a roundifh mafs of the fame fubftance _alfo occurs ; this has very much the appearance of an included fragment, but the decompofition of the fandftone has juft be- gun to expofe its connection with the rock above. . On the fandftone, Porphyritic Greenftone (No. 3.) refts. The colour of this is reddifh-brown; the texture is fine-grained ; and it contains fmall fpecks of flefh-coloured calcareous {par. It is traverfed in various places by veins of Hematitic Iron-ore (No. 4.) accompanied with fulphate of barytes.. Thefe two ' fpecimens have very much the character of fome varieties of porphyry-flate, and on breaking one ma({s, I obferved a tenden- cy to a flaty arrangement. In different places of this quar- ry, the greenftone affumes a variety of appearances (No. 5. and 6.), fome of which might be attributed to decompofition. I do not conceive, however, that any external caufe has ever had much effect upon this rock, although in fome places it has entirely loft its luftre, (No. 7.), and might be miftaken for — trap-tuff, were it not for the fhape of the crytftals. AsBoveE this, the rock graduates into a highly ory ftal- line Porphyritic ftone, (No. 8.) the pafte of which is of a brownifh-grey colour, very clofe-grained, with an uneven fplintery fracture, containing both cryftals of felfpar and horn- blende. In this quarry there are feveral inftances of slikensides, _ one of which is rather remarkable, it occurs in an inclined rent _ inthe fandftone: the traces of the flip, (No. 9.), are horizon- Saiki 22 4.16 3 On ihe ROCKS in the tal, and extremely well defined ; but immediately over it, in the greenftone, the appearance of the flip is not continued. Some indications of a flip appear a little to the right of it. In a part of the Greenftone which is confiderably decompo- fed, a vein, ftretching horizontally, of a dark-green fibrous fubftance occurs, (No. 10.); it is foft, and has a fhining fatiny luftre, like afbeftus. I have not anywhere in this vicinity met with any fimilar fubftance. WE now proceed to Salifbury Craig, where the circumftances I fhall principally notice, are, 1. The texture of the greenftone rock, with the foffils it con- tains. 2. The vein of greenftone by which the Craig is interfected. 3. The included mafs of fandftone which occurs in the green- {tone ; and, 4. The indurations and interruptions of the ftrata. No. 11. is a fpecimen of the greenftone taken from the lower edge of the bed, at the great quarry, where it touches the fand- ftone; the point of contact being marked by a {mall remaining fragment of the latter, at which the grain of the ftone is much finer than at the other extremity. The colour is iron-grey, with {mall fpecks of calcareous matter inter{perfed. Nos. 12, 13, & 14. are different gradations of texture, taken ina vertical line, from the edge towards the centre, where the ftone is always moft perfe@tly cryftallifed; from hence it again de- clines in grain towards the upper furface, where we find it in the fame earthy and uncryftallifed ftate (No. 15.) obferved at the bottom. In the laft fpecimen, there is a fmall detached fragment VICINITY of EDINBURGH. 4r7 fragment of the ftratified matter imbedded in the green- ftone, a circumftance connected with a very important clafs of facts. No. 16. This fpecimen of greenftone is remarkable, as exhi- biting a variety of colours; thefe are not blended, but diftin@- ly divided from each cther. The colours are iron-grey, light- grey, dark-red, and brick-red. No. 17. This fpecimen is a {trong example of the improprie- ty of the name which it bears; it is a greenftone, decidedly of ared colour. The fingular penetration of ferruginous matter, which is exhibited in various parts of this rock, is not eafily ac- counted for; but fuppofing it to have been once in a ftate of fu- fion, it may have obtained this fuperabundance of iron by ab- | forption, as the adjoining ftrata frequently abound in that mineral. In various parts of the Craig, veins of a peculiar nature may be obferved ; they are compofed precifely of the fame ingre- dients as the rock, and are diftinguifhable only by the red co- lour of the felfpar, (No. 18). Thefe are termed contemporancous veins, or veins of secretion; they are deeply waved, and gene- tally follow the direction of the bed. Some of them prefent a very bright brick-red colour, (No. 19.), mixed with fpecks of calcareous fpar. Nos. 20, 21. in thefe {pecimens, are fmall globules of a black earthy fubftance, which I am at a lofs to name. I fhould have confidered it Amphibole, but for the next fpecimen, : (No.2 2.), in which the fame fubftance appears: to occur in irregular fragments. No. 33. nalowne with cryftallifed Calcareous Spar. I be- fore noticed, that it was in the heart of the bed where the fabftance of the greenftone prefented the cryftalline texture in Vou. Vi. P. i. 3G the 418) On the ROCKS in the the higheft perfection. The occurrence of the analcime is ~ connected with the fame fact. I have never been able to find it on Salifbury Craig, excepting at one period, when an entire fection of the bed was quarried off, and about the middle of this the analcime occurred. No. 24. with Baphate of barytes, with calcareous {parry iron- ore.. No. 25. part of a very irregular vein. Its fides are formed of calcareous {parry iron-ore, which is followed by a coating of hematitic iron. Here the regular ftratification, as it is called, of the vein ends, and calcedony, firft femitranfparent, then opake, and common calcareous f{par, occupy the reft. No. 26. calcareous fparry iron-ore cryftallifed, with fome tran{parent cryftals of quartz. No. 27. large cryftals of calcareous {par, sith cry ttallifed. and radiated tufts of quartz. . No. 28. red axide of iron, with a vein of talearcons {parry - jron-ore. No. 29. green coloured quartz, with a coating of cryftallifed quartz. No. 30. cryftallifed quartz, with amethyft. Sucu are the minerals which occur on Salifbury Craig. Some of them are rare, and others to be found only when the rock is working in particular places. THE next circumftance I have to notice, is the vein of creenftone *. It occurs a little to the north of the fpot, to which. * The term dyke has been very generally applied to vezms of this defcription, and-I am not fatisfied that it is the leaft proper of the two; as there certainly is a marked diftin@tion between veins compofed of rocks, and what we general- ly _— VICINITY of EDINBURGH. 419 svhich the cart-road, along the bafe of the rock extends, a few feet beyond a gap, known by the name of the Cat’s Nick. I po not think this vein attracted the attention of geologifts an any particular manner, prior to 1805. It certainly was ob- ferved long before that period; but was not known to extend through the bed of greenftone, till Sir James Hatz and myfelf ‘noticed, that after cutting the fandftone, it continued its courfe uninterrupted to the top. This obfervation contributed very much to increafe our -curiofity, and a.man was employed to clear away the foil and rubbifh, which had accumulated on the furface. A confiderable portion of the rock was foon laid open, below the point from which it was at that time vi- fible. Nothing, however, of much intereft, was by this means difcovered. The dike, after bending a little to one fide, conti- nued its courfe downwards. ‘Tue fpace which this dike occupies, may be from fix to eight feet wide ; its width varies a little in fome parts, and thefe va- riations are apparently increafed, if the fection which is obfer- ved be not at right angles with the walls. That portion embraced by the ftrata, which we found principally co- vered with debris, was very much decompoied, prefenting on the furface a certain degree of nodular exfoliation, of a rufty- 3 Gre 7 brown ly underftand by mineral veins. The firit are formed of one uniform rock, compofed in all their parts. of the fame conftituents, and differing only in po- fition, from the beds thefe materials more ufually form; while the latter, though fometimes formed only of one fubftance, fuch as quartz or calcareous . {par, are generally compofed of a feries of foffils, arranged in lines parallel to the walls. No fuch appearance ever prevails in rock veins, or conftituting mountain mafles ; therefore, in ufing the term vezz, when applied to greenftone, granite, or the like, it muft be underftood as a generic term, of which these lat. cer, {pecify the variety. 420 ; -On the ROCKS in the brown colour. On breaking into the rock, it exhibited” (No. 31.) * the concentric lines fo common in decompofing greenf{tone; and beyond this, the ftone prefented a degree of frefhnefs, with a very coarfe grain of a peculiarly light afh-grey colour, and a very dull earthy texture, (No. 32.) Between this portion of the vein and that embraced by the greenftone, there is a very remarkable difference, the - latter being of the ufual iron-grey colour, and otherwife perfectly characteriftic. Before it leaves the fandftone ftrata, it feems to contain an unufually large proportion of calcareous matter. This may have aided the decompofition, together with the moifture retained by the debris, fo lately removed from its furface, and which has left it in a ftate eafily affected by the weather. Since I commenced writing this paper, I made an excurfion to the f{pot, and was greatly furprifed to obferve the devaftation of laft winter. Before the vein rifes above the level of the ftrata, a portion of it, ftill more decompofed than the reft, of a dark-purple co- lour, branches off, and embraces a wedge-fhaped mafs of the fandftone (No. 33. and 34.) indurated in a very high degree. Juft at the top of this indurated mafs, the whole dike changes its colour, and, I may alfo fay, its confiftence. It here prefents a light-greyifh afpect, deeply ftained, with red ferruginous marks, of a dull earthy texture, an even fracture, and a tolera- bly fine grain, (No. 35.) That portion correfponding with, and immediately over the included fandftone, I found much coarfer in the grain, (No. 36.), and in a more decompofed . {tate ; * Correfponding numbers will be found in the annexed engraving, which will explain more fully the relative pofition of the fpecimens, PLATE XI. Ss Vol IP. 420. Edie TransactionsRS bi fyhil? AAS we SS MS % i) Uy i yey) if Jil SSS RRS = /, ii = RSG V4 ARS " SS MY mana i AK LLL Ta J “ ALIEN i Senlps —— Azars —— )/p.3 VICTNIT Yof EDINBURG H. A2T: tate; while it differed from No..37., the ftone on the fides, which were perfectly fimilar to each other in compofi- tion.. Tracinc the friable purple-coloured portion upwards, I found it gradually became harder, and, of a fudden, chan ge to a. fine-grained blue-coloured greenftone ; and the part correfponde ing with the included mafs, alter to a hard coarfe-grained rock, (No. 38.) I foon obferved, that this coarfe-grained mafs, which. is about ten inches thick, continued. upwards, maintaining an uniform dimenfion and. pofition, in refpect to the walls of the. vein, as high as the eye could trace it in the rock, thus divid- ing it into two portions; that on the left fide being about eigh- teen inches wide, while the other is about five feet. On comparing the texture of the included ftripe, with that: of the walls on each fide, (No. 39. left fide; No. 40. included. ftripe; No. 41. right fide,) taken in a horizontal line, about fix. feet above the ftrata, I found as clofe a refemblance as it is poffible to conceive ; they are all coarfe-grained, and highly eryftallifed.. - This fimilarity is not more remarkable, than the difference be-- tween the fubftance of the vein and the included mafs.. Speci-- mens taken from the junction of thefe, mark this in a ftriking. manner. No. 42. is from the left fide of the right portion of the vein, to which the fine-grained part belongs. No..43, is. from the middle of this portion; and No. 44. from the fide next the right wall.. Thefe were alfo taken in a: horizontal line, and exhibit the fame gradation of grain noticed as exifting in the great bed. Even in the narrow portion of. eighteen inches, on the left fide, this circumftance is quite vi- fible; but the fpecimens taken from the other are highly illu-. ftrative of the fact. Ji 422 On the ROCKS in the I nave had an opportunity of examining many veins of greenftone; but I know of none more interefting im a geolo- gical point of view than this. [I THINK it can icarcely be doubted, that the fame effort which feparated the included portion of fandftone, cleft the correfponding ftripe of greenftone from the great bed. This, as well as the gradation of grain, everywhere obfervable in beds and veins of trap, are remarks, in my opinion, of ‘confiderable value to the Huttonian hypothefis. On a former’ occafion, when I had the honour of fubmitting fome remarks on the north of Ireland to the Society, [ took an opportunity of dwelling particularly on the laft.circumftance. Like the charring of coal, when that fubftance is found in contaé with whin, as has been ably remarked by Profeflor Pray- FAIR, “ few facts in the hiftory of foflils fo directly af-_ “ fmilates the operations of the mineral regions with thofe ‘“‘ which take place on the furface of the earth *.”’ This gra- dation of texture has a ftrong analogy to many accidental fads obfervable in furnaces, of glafshoufes and the like, and ftill more fo to thofe experiments made exprefsly for the purpofe of af- certaining the effe@s of flow cooling, by Sir James Hau and others. One additional argument for the igneous origin of thefe veins, has been added by the obfervations of Sir GEorcE Mac- KENZIE and his friends, in Iceland, in perfect correfpondence with the above fact. He there found many veins of this fub- ftance, coated on the fides with a glafly covering, exactly fimi- lar to melted greenftone, when rapidly cooled. I sHouLp expe the fame circumftance would be met with in veins of porphyry and granite; but I have not been able to ex- tend * Wuftrations of the Huttonian Theory, § 68. VICINITY of EDINBURGH. 423 tend my obfervations fo widely, as to embrace the facts refpect- ing thefe rocks. One remark I fhall, however, hazard in this place, refpecting an eflential difference between veins of granite and thofe of greenftone. The former feem to be of fimultaneous formation with the great body of that rock, to which they may generally be traced, and, fo far as I have hither- to obferved, are never found to cut it. Veins of greenftone, on- the other hand, I have never feen connected with the great beds of that fubftance; they traverfe thefe juft as they do every other kind of rock, and confequently are in all inftances of a pofterior formation. I am aware, that thefe ideas are ve- ry much at variance with certain received opinions. I there- fore wifh to. be underftood as fpeaking folely upon my own experience.. I HAVE now to mention the well-known included mafs of | fandftone. Along the edge of the ftrata, a number of inftances occur on Salifbury Craig, affording the moft unequivocal marks of difturbance; but it prefents only one example; of a mafs to- -tally enveloped in the fubftance of the green{ftone *. * Tus fpot has been the fcene of much controverfy, between contending geologifts. While the Huttonian confiders it as a moft incontrovertible proof of violence and of heat, the Werne- rian contends, that there is nothing in the leaft extraordinary in the appearance, and afferts, that the fuperficies of the apparently included mafs, is no.more than the fection of fome part of the ftratum, which, if traced, would be found to connect with the reft; that it had been enveloped in the fluid menftruum of the greenftone, when in this elevated pofition ; and that the rock be- ing * Since this paper was fent to prefs, others have been obferved in different parts of the rock. 424 On the. ROCKS in the ing cut in acertain direction, a fection having the appearance of an infulated mafs, would of courfe be expofed to view. There -is no doubt that fuch a circumfiance is perfectly poffible ; but, in the prefent inftance, this explanation will not be found at all applicable. In every othercafe, where the ftrata appear difplaced, they are not torn from the reft, nor has the greenftone infinuated itfelf, except as a wedge, fupporting the lifted maffes. The included mafs is of a light greenifh-grey colour, in fhape quadran- gular, and, when minutely examined, will be found fhivered in- to numerous diftiné fragments, with veins of greenftone run- ning through it in every dire@tion. It partly retains its ori- ginal ftratified texture (No. 45.) although indurated in a very high degree, and is fo firmly welded to the greenftone, that it is no difficult matter to obtain fpecimens (No. 46.) of the conjoined rocks; one {mall fpecimen (No. 47.) in the collec- ‘tion, is twice interfected by that fubftance. It, therefore, has no refemblance whatever to thofe pieces of ftrata, which are only in part detached, and which, if cut in a tranfverfe diredtion, would, in all probability, exhibit an infulated fection. ‘That fection, however, would not difplay the broken and diftorted . appearance defcribed above, at leaft if we may be allowed to judge by the integrity of the longitudinal fections, of which there are fo many examples in this vicinity. Befides, the colour of the included mafs is totally different from that of any of the {trata near it, which are here of a deep red (No. 48.), and at this particular fpot are remarkable for their apparent derange- ment. I therefore conclude, that there is every reafon to con- fider this, as a fragment detached from fome other part of the fandftone, and left fufpended in its prefent fituation, when the greenftone affumed a folid confiftence, as was originally con- jeGtured by Dr Hutron. é I VICINITY of EDINBURGH. 425° I Now come, as propofed, to that divifion of the fubje@ which relates to indurations. By zzduration is meant, a greater degree of compactnefs, obfervable in particular parts of ftratified rocks, than is ufual throughout their mafs. One part of a bed may be harder than another, confequently more indurated. But the in- duration here alluded to, is that which is fuppofed to have been _effected, by an alteration in the denfity of the ftone, in confe- quence of the adtion of heat. TueEsE phenomena are of a very ftriking nature, and were firft brought into notice by Dr Hurron ;, in them,. he found eyidence, to him perfeétly conclufive, of the igneous formation of whin, and, with that ingenuity and perfeverance which cha- racterife the whole of his works, he did not fail to generalife his obfervations, and to place the facts, firft noticed in this {pot, in fuch a light, as to render them effentially ufeful to his theory. Tue anxiety which the diciples of the Wernerian fchool have always evinced, to undervalue the merit of this obferva~ tion, is a fure mark of the eftimation in which they hold it; and it is, therefore, very properly confidered by the fupporters of the oppofite doctrine, as one of their ftrongeft holds. In the following lift, are comprehended moft of the varieties, which this indurated fandftone prefents on Salifbury Craig. No. 49. is a junétion fpecimen*, taken near the fouthern extremity, of the Craig; here the greenftone is of the deep red tinge noticed at No. 17. Mion. Vila bali te 3H. No. % By junéion /pecimen is meant, a fpecimen which exhibits the nee and the fandftone conjoined. 426 On the ROGKS in the No: 50. is another fpecimen of the fame kind ; the greenftone is here of the ufual colour, and the line of junction moft admi- rably defined. This was taken from the great quarry. The next, (No. s1.), is a fpecimen of the fandftone in its fuppofed unaltered ftate. Nos. 49, and 50. are beth from the lower junction. No. 52. is from the upper edge, taken about half- way between the higheft part of the Craig and Holyroodhoufe. Here the fandftone prefents a facetted appearance, an arrange- ~ ment which may be. owing to the fuperabundance of calcareous matter. No. 53. is highly indurated, of a deep red colour, with a conchoidal fra@ture, and a facetted texture. No. 54. has the fame facetted appearance. No. 55., extremely clofe-grained, is from one of the contor- ~ tions north of the dike. Nos. 56, & 57. Thefe are the varieties of the fandftone which have been called jasper. This is an improper name, as it con- founds two fubftances totally different. The moft compact con- tains a large proportion of lime, and in afpe¢t is very fimilar to fome of the limeftones of Gibraltar. Nos. 58, to 61. are varieties Re the fandftone, found near the © greenftone. No. 62. Although this fpecimen was taken very near the greenftone, ftill it does not exhibit the ufual induration. This exception occurs in different places on Salifbury Craig; and it even fometimes happens, that the {tone next the whin is lefs in- durated than the one below it. : No. 63. Containing a large proportion of ferruginous mat- ter. No. 64, to 66. Different fhades.and varieties of the fandftone, in an indurated flate. No. VICINITY of EDINBURGH. 427 No. 67. In this fpecimen there is fomething very like the ap- pearance of an agate; it, however, is not contained in the fub- ftance of the greenftone, but in the ftratified matter below it. No. 68. Another fpecimen of the fandftone, in its unaltered ftate, taken about thirty feet from the greenftone. Dr Hurron conceives, that the induration, fo very remark- able in the above fpecimens, was occafioned by the heat: of the whin, when it was injected between the ftrata of fand{ftone, cau- fing it to undergo a certain degree of fufion; and, to this idea, the facetted texture of fome of the fpecimens adds confiderable weight, fuch arrangements being very familiar in ftones which have undergone fufion. Tue Wernerian {chool, to account for the fate phenomenon, afferts, that as fandftone is generally porous, the fluid folution of the trap being introduced into the fiffure, naturally percola- ted to a greater or lefs extent *. Again, that it is owing to the intermixture of the matter of the vein, with the rock that forms its walls}; and, as a proof of this, it is added, that no indu- ration appears, where the traverfed rock is poffefled of a quart- zy bafe. ‘THESE arguments occur in different works, but they appear to me very little calculated to fupport the point in difpute, if not in fome refpects contradictory. On Salifbury Craig, and generally throughout the neighbourhood of Edinburgh, where- ever we find fandftone coming in contact with g¢reenftone, either in beds or veins, we are almoft certain, that an indura- tion will be exhibited along the edge of the ftrata. RET: Ir * Comparative View of the Huttonian and Neptunian Theory, p. 130.. +. System of Mineralogy, vol. iii. p, 365. 428 On the ROGKS in the Ir has already been obferved, that there are {pots on Salif- bury Craig, where this is not fo apparent as in others, and it very often happens, that {mall feams of clay occur, in a per- fedtly foft ftate. In Ineland, at Scrabo, in the county of Down, and at Fairhead in that.of Antrim, I found fandftone in the former, cut by veins, and in the latter, overlayed by a bed 300 feet thick, where no induration was to be feen. Now, it _ appears conclufive, that there could not have been a deficiency of induration in any fpeck of Salifbury Craig, far lefs a total abfence, as in the cafes quoted in Ireland, had it in any inftance been effected either by percolation, or by the intermixture of the matter of the vein. The fuperincumbent or included mat- ter, if in a fluid ftate, whatever its chemical powers were, would, to a certain extent, act mechanically, and be, in all cir- cumftances, poflefled of the fame power of communicating its moifture to the furrounding mafles. It is therefore impoffible to conceive, how it fhould have withheld it in one inftance, and parted with it fo amply in another, how it fhould have indu- rated the fandftone, and left the thin feams of clay in a foft and friable ftate. It is quite unimportant, of what bafe the fandftone may be formed; it is a fubftance, allowed as above to be generally porous, (and, in the cafes alluded to, it certain-, ly was fo); into that porofity, therefore, the fluid muft have percolated, whatever the bafe may have been. On the contrary, according to the Huttonian hypothefis, iz in- duration diftiné@ly depends, on the compofition of the ftrata ex- pofed to the influence of heat. Some ftrata may either whol- ly, or in part, be capable of refifting much higher temperatures than others. It is confequently to the ingredients of which they are formed, that we muft look either for the caufe of in- duration, or the abfence of it. This remark originated in ob- ferving, VICINITY of EDINBURGH. 429 ‘ferving, that all the indurated fandftones of this country, con- tained more or lefs calcareous matter, while the unindurated fpecimens from Ireland, did not afford ‘the flighteft indication of that fubftance, when fubjected to the fame feft. Beroxke I take leave of Salifbury Craig, I muff notice one more circumftance, which, fo far as I have hitherto {feen, is quite peculiar to the fpot. I mean the occurrence, in veins, of a fubftance in all refpects fimilar to the indurated fandftones, I have juft been defcribing. The firft of thefe I obferved, is about thirty paces north of the vein. The ground being cut’ away, in order to fee its connection with the ftrata, it branched out like the prongs of a fork, and had the interftice filled with a red decompofed fubftance (No. 69.), fimilar to that which occurred at the extremity of the included ftripe of greenftone in the vein. Where the prongs join, it is about three or four inches wide, and is there, partly compofed of indurated fandftone, and partly of hematitic iron-ore and calcareous fpar. _ (No. 70.) Higher up, where the vein is narrower, it is whol- ly compofed of fandftone, the fpecimen, No. 71., being the en- tire thicknefs of it. ~ Here the grain is finer than at firft, and, higher up, it becomes ftill more fo, (No. 72.) It ftill conti- ues to taper upwards, and even when reduced to lef{s than half an inch, the fubftance retains the ufual afpect ef indurated fand- ftone, (No. 73.) This vein rifes about twenty to thirty feet into the rock, always diminifhing, and about that height dif- appears. I have remarked other veins, alfo containing fub- ftances fimilar to indurated fandftone (No. 74.), one was of a much larger fize than that above defcribed (No. 75.), but the grain not near fo compact, (No. 76.) TueEsE veins all fet off from the lower furface, and fo long as they are of any confiderable thicknefs, the including rock is ftained ; 439 ‘On the ROCKS in the ftained with ferruginous matter. This fad feems connected . with the fingular appearances, which occur in the vein of green- ftone, at the level of the junction of the fandftone ftrata with the incumbent bed. WitHour offering any remarks on a fac as yet fo infulated, { content myfelf with merely mentioning it, in hopes that fimi- lar appearances may prefent themfelves to geologifts in other quarters, and perhaps throw fome light on a phenomenon, which by farther elucidation may prove interefting. BeErorE I clofe this paper, I fhall take the opportunity ef pre- fenting to the-Society, two fpecimens which were given to me by Sir Grorce Mackenzig, and which I efteem of con- fiderable value; one of them, a fragment of the rock of Salif-. bury Craig; the other, of the Calton Hill, marked in- the handwriting of the late Dr Kennepy, as the fubftances he ana- lyfed, and of which an account was given in the 5th volume of thefe Transactions. The great variety in the rock, both of Salifbury Craig and Calton Hill, makes it of importance to: afcertain with precifion the kind employed in’the refearch of that celebrated chemift; and asthe moft proper place for their reception, I depofite them in the cabinet of this Society, along. with my own collection, under the Nos. 77, and 78. roth VICINITY of EDINBURGH. 43% 19th March. Since I had the honour of reading the foregoing paper to the Society, a ftrong cafe in illuftration of what is mentioned in the commencement of it, has prefented itfelf; I mean, with refpect to the conftant occurrence of new and interefting mat- ter, even in the moft frequented ground. A Few days ago, Profeflor PLayrair mentioned to me, that . by the cutting down of a fection of the Craig, within thirty paces of the fouthern extremity, feveral maffes of fandftone had made their appearance, imbedded in the greenftone. I loft no time in vifiting the fpot, and was greatly pleafed to find, a confiderable addition to the mterefting facts, already exhibit- ed on Salifbury Craig. At this part of the rock, the greenftone becomes very thin, being no more than twenty to twenty-five feet thick; it has the appearance, however, of having once been of greater extent, the upper part being apparently cut away by fome operation of nature, of which we have now only to obferve the effects. It flopes rapidly towards the fouth, and is covered to a confider- able depth with foil and travelled ftones. It is at the upper furface of this, that the imbedded maffes occur; they appear to be portions of ftrata, which obferve the general inclination of the fandftone of Salifbury Craig, that is, dipping towards the north-eaft, while the expofed fetions are parallel to each other, and nearly horizontal ; confequently, being near the fur- face, they are cut off, or crop out, on the fouth fide. Their appearance, Rape On the ROCKS in the appearance, however, betpeaks their having been, at fome for- mer period, totally included in the greenftone. One mafs, in- deed, a little towards the north, is unequivocally fo; at leaft we: know with certainty, that a fhort time ago it was inclofed in the greenftone, and not to be feen ; and there is-at prefent, great apparent probability, that the next fection taken from the fame. part of the rock, will carry it away altogether. TILL now, we only knew of one included mafs in the green- {tone of Salifbury Craig; and with this, thefe now difcovered. have confiderable analogy ; they are of the fame colour, and al- though they appear to be only four or five diffin@ mafles, thefe maffes are all interfected vertically and diagonally, and © are fplit through the whole length of the horizontal line; fo that in examining a {ection of about ten feet perpendicular, no, lefs than nine difterent alternations of fandftone may be reckon- ed. Some of them are no doubt very minute ; but fill they were ail obfervable when I examined the rock. . From the moft northern mafs of included fandftone, I was enabled to procure a. few fpeciméns, which I have added: to the above collection. The rock rifes fo rapidly from the fouth, that although this mafs is nearly in the fame hori- zontal line with the others, all of which re out to the furface,. and although it is not diftant more than four or five yards, yet it appears to be fituated nearly about the middle, between the fandftone and the upper furface, from which it may natural- ly be inferred, that the maffles which now crop out, were like this, once entirely included in the fubftance of the greenftone. It is highly indurated, and at the extremities, is drawn out into minute veins. The thicknefs. of the principal mafs may be from ten to twelve inches, and in length from fix to eight feet.. Ehis body, as above noticed, is cut in all directions by the greenftone.. he ee Oe VEICTNITY Fi EDINBURG HZ. 433 greenftone. The fpecimen No. 79. fhews a portion of the f{andftone, with that fubftance traverfing its ftratified lines dia- gonally. No. 80. is a mafs of, the fandftone, containing a fmall portion of greenftone, much of the fame fhape as the double wedge of St Leonard’s Hill, and formed, as I conceive, exaétly in the fame manner. This wedge, on one fide of the fpecimen, is two inches long, but, on the oppofite, it is not one; and in the counter part of the fame fpecimen, (No. 81.) it is only to be feen on one furface ; it does not penetrate to the other fide, though fcarcely an inch thick. I am glad to find, that intereft has been made to prevent this valuable fet of facts from being foon deftroyed, as, in a few weeks, the rock in which thefe are contained would have been broken down, and carried off for the repair of the neigh- bouring roads. | Ir is on this account, that much activity is requifite to keep thefe perifhable phenomena from being loft, in the neighbour- hood of fuch a town as Edinburgh. Similar things are pre- fenting themfelves conftantly, but they are opened only for a day, and if not feized and recorded on the inftant, will be fhut up, and loft for ever. END OF THE SIXTH VOLUME. r _ Sag 18 ery alljoie Pe ol ey ai ae BS “Grad bhipwe by good os sing 4 Dd Zrth a > swodilgins of of gaisc e rt ylao ‘Ran2qo 518 yods, aud. jul ad five oe ada fio “pabrdo94 bits b Bieyie ode Pca) oh DIRECTIONS TO THE BINDER. The sheets marked a and b, containing the History of the Society and the Laws, . to be placed in front of the volume, immediately after the general title-page - and general table of Contents. The temporary title-pages and tables of: Contents for Parts I. and II. to be can- celled, as now unnecessary. The last leaf of signature 3 D; pp. 390, 400, and the first leaf of 3-E, pp. 401, . 402, are to be cancelled, and two new-leaves substituted.” Plate I. fronting p. 18. . II. and IIT. fronting p. 70. Sir James Hatu’s Plates, marked with small numerals, Pl. 1, 11, 11,-1¥, v. to front:: p- 186. Plates VI, and VII. fronting p. 244. —— VIII. - - 248. — IX. - - 344. — x. - - 376. — XI. > - 402. — XII. - - 420. *.* The Binder will take notice that the Plates must. fold out. ; * a \ ' ath ayeaeee rute nuh arate tatats. titphete ae! ‘i tet rt tories Videhetererede ane Crte en rs ce Hh +3) ‘ pyeplareeenaeetece avery inne eta Fe Fee he dieeee Vete latins