PROJECTILES

L. 195.

X 200. 15.

I. The EDINBURGH REVIEW, or CRITICAL JOURNAL, from its commencement, October 1802, to July 1809, Twenty-eight Numbers. 7l. 2s.; or done up in 1½ Vol. boards, 7l. 9s. Published Quarterly.

II. The FARMER'S MAGAZINE, a periodical work, published quarterly, exclusively devoted to Agriculture and Rural Affairs, for the Years 1800, 1801, 1802, 1803, 1804, 1805, 1806, 1807, 1808, 36 Numbers, (and Supplement to 1803). 4l. 3s.; or in 9 Volumes in boards, 4l. 7s. 6d.

*** The establishment of a Board for promoting agriculture and internal improvement brought husbandry into fashion, and directed public attention to an art which, before that period, had been undervalued and neglected. From this change of public sentiment, the Proprietors of the FARMER'S MAGAZINE were encouraged to bring forward a periodical work which both contained interesting discussions upon agricultural subjects, and furnished select and important information respecting the state of markets, produce of crops, rate of rents, and value of labour, in almost every district of the island. The design, at least the latter part of it, was new, therefore was not carried into execution till the assistance of numerous respectable agriculturists, both in Scotland and in England, was sought for and obtained; and to the active and steady exertions of these friends, may be attributed the uncommon and unprecedented success of the work since its commencement—a success far beyond that of any agricultural publication hitherto attempted in this, or in any other country.

In the volumes of the FARMER'S MAGAZINE already published, may be found regular essays or dissertations upon every agricultural subject which can be mentioned, together with an immense number of hints or observations, all calculated for the improvement of agriculture, and the benefit of those connected with it. What is of great importance to husbandmen, information is given in a plain and practical manner, neither clouded by theory, nor enveloped in technical terms. That eminent writer on husbandry, the Rev. Mr. Harte, in his Treatise on Agriculture, says, "The plain practical author pays his little contingent to the republic of knowledge with a bill of unstamped real bullion; whilst the vain-glorious man of science throws down an heap of glittering counters, which are gold to the eye, but lead to the touchstone."

III. The EDINBURGH MEDICAL and SURGICAL JOURNAL, for 1805, 1806, 1807, 1808, exhibiting a concise view of the latest and most important discoveries in Medicine, Surgery, and Pharmacy, (published quarterly), 16 Numbers, 2l. 8s.; or any single Number, 3s. Also Nos. 17, 18, and 19, 3s. each.

*** The object of the EDINBURGH MEDICAL and SURGICAL JOURNAL is the improvement of Medicine in all its branches, by the combined efforts of the Editors, and of their numerous and respectable correspondents. It affords to the profession at large an opportunity of recording, and of disseminating, very quickly and very widely, the knowledge of the remarkable facts which may occur in their practice, and of the theoretical opinions which may be the result of their observations; while the Editors endeavour, by a critical but candid analysis of the recent publications in the various departments of medicine, to make works of merit quickly known to their numerous readers, and to extract and condense whatever is most valuable in rare or expensive publications.

After the experience of four years, the Editors have the satisfaction of finding the patronage with which they have been encouraged, progressively increasing. The Fourth Volume is enriched by com-

munications from Doctors Alley, Bostock, Cheyne, Chisholm, Clarke, Corkindale, Dickson, Duncan, sen. and jun., Kellie, Paterson, Rutter, Tavarez, Thackeray, Vetch, Wilson, and Wright; and Messrs Arnoldi, Barlow, A. Burns, Cullen, Ellis, Fuller, Gibb, Hill, Jenkinson, Lawrence, Machel, Maunoir, Noble, Quarrier, Soden, and Wood.

IV. The SCOTS MAGAZINE, from its commencement in the year 1739 to 1808 inclusive, 70 Volumes, new and neatly bound; also odd Numbers and Volumes to complete sets.

The Scots Magazine was begun in January 1739, a few years after the first publication of the Gentleman's Magazine, and has ever since continued to be the standard work of the kind in Scotland. Few periodical publications have been held in higher estimation; and the present Editors have the satisfaction to reflect, that no pains have been spared by them to support and to raise its character, and that their efforts have been rewarded by the approbation of the Public. They have not rested satisfied with those voluntary communications, which have been liberally communicated to them, from persons often of the first distinction in Scottish literature. They have, besides, exerted themselves to the utmost in collecting from all quarters, whatever could contribute to the information and amusement of the Public.

The Biography of eminent persons deceased, has always formed an interesting department of periodical works. This the Editors have not overlooked; but have studied, to the utmost of their power, to let no eminent man depart, without some such tribute to his memory.

Antiquities are a subject replete with amusement, often with information, and peculiarly suited to the taste of the present age. Scotland, too, is rich in them; and the connexions and opportunities of the Editors have enabled them, and they trust will enable them, to present their readers with a constant succession of curious articles in this department.

Although, from the number of their communications, the Editors would find little difficulty in filling all their pages with original matter; yet this method, however easy for themselves, would, in many cases, be little to the advantage of their readers. It has appeared more eligible to insert only such as possessed superior merit in point of subject or manner; and, instead of the others, to introduce interesting extracts from rare or valuable works, which are not accessible to the generality of readers; and, particularly, translations from authors in foreign languages, which have not appeared in an English dress. From connexions which they have recently established, they hope to be able greatly to enlarge their command of foreign works for this latter purpose.

A few pages are appropriated to Poetry. The Scots Magazine has been the means of introducing to public notice several of the most popular Scottish poets, by whose communications it is still enriched.

In regard to Intelligence, both foreign and domestic, no periodical work has maintained a higher reputation than the Scots Magazine. During the long period through which it has been continued, it has always held the first rank as a work of reference. The present Editors have omitted nothing which could tend to support this reputation. A larger portion of their pages has been appropriated to this object than in any other magazine now published; and maps and plans are introduced, wherever they seem likely to be useful, in illustrating military operations, or important political events.

To every number is prefixed a view and description of some remarkable Scottish edifice, either such as is interesting from its antiquity, or an object of curiosity from its recent erection; and is published regularly on the first day of every month.

Works Published by
ARCHIBALD CONSTABLE & CO. EDINBURGH,
AND
CONSTABLE, HUNTER, PARK, & HUNTER, LONDON.

I. AN INQUIRY into the PRACTICAL MERITS of the SYSTEM for the GOVERNMENT of INDIA, under the Superintendence of the BOARD of CONTROUL. By the EARL of LAUDERDALE. 8vo. 7s. 6d. boards.

II. A SERIES OF DISCOURSES ON THE PRINCIPLES OF RELIGIOUS BELIEF, as connected with HUMAN HAPPINESS and IMPROVEMENT. By the Rev. R. MOREHEAD, A. M. of Balhol College, Oxford, Junior Minister of the Episcopal Chapel, Cowgate, Edinburgh. Second Edition. 9s. boards.

*** "It is the singular and unaffected benevolence of manner—the tone of genuine goodness and conciliating candour, so unlike the contemptuous arrogance of vulgar theologians, that forms the chief charm of the volume before us; and induces us to point it out to the attention of the public, as eminently calculated to fix the principles of the young and careless, and to improve the charity and mend the hearts of readers of every description."—Edinburgh Review, No. xxvii.

III. OBSERVATIONS on FUNGUS HÆMATODES, or SOFT CANCER, in several of the most important Organs of the Human Body; containing also a Comparative View of the Structure of Fungus Hæmatodes and Cancer, with Cases and Dissections. By JAMES WARDROP, F. R. S. E. Fellow of the Royal College of Surgeons, and one of the Surgeons of the Public Dispensary of Edinburgh. Illustrated by Eleven Plates. 12s. boards.

*** A few copies of this work have been printed in Royal 8vo, with proof impressions of the plates coloured. 1l. 1s. boards.

IV. ESSAYS on the MORBID ANATOMY of the HUMAN EYE, of which the various morbid appearances are illustrated by Coloured Engravings, by Meadows, Medland, Maddocks, Heath, &c. after Drawings by Mr Syme. In One Volume Royal 8vo. 1l. 1s. boards.

V. A DISSERTATION on the NUMBERS of MANKIND, in Antient and Modern times. By ROBERT WALLACE, D. D. late one of the Ministers of Edinburgh. Second Edition, revised and corrected by GEORGE WALLACE, Esq. Advocate. Octavo. 9s. boards.

*** The first edition of the above work was published in the year 1753, and is often referred to by Mr Malthus.

VI. NEW THEORY OF THE FORMATION OF VEINS; with its application to the Art of working Mines. By ABRAHAM GOTTLIEB WERNER, Counsellor of the Mines of Saxony, Professor of Mineralogy, and of the Art of Working Mines at Freyberg, &c. Translated from the German. To which is added, an Appendix, containing Notes illustrative of the subject. By CHARLES ANDERSON, M. D. Fellow of the Royal College of Surgeons, Member of the Chirurgical Society, of the Wernerman Natural History Society, &c. One Volume 8vo, 9s. boards.

VII. The VILLA GARDEN DIRECTORY, or MONTHLY INDEX of WORK to be done in Town and Villa Gardens, Parterres, &c.; with Hints on the Treatment of Plants and Flowers kept in the Green-Room, the Lobby, and the Drawing Room. By WALTER NICOL, Designer of Gardens, &c. and Author of "The Forcing, Fruit, and Kitchen-Gardener," "The Practical Planter," &c. Foolscap 8vo. 7s. 6d.

*** It is believed there is no book of gardening on the plan of this work. It is intended as an assistant to gentlemen whose business necessarily confines them to the chamber and to the counting-room, who seek health and recreation at their villas, and to those who take upon themselves the management of their own gardens and parterres.

VIII. A LETTER addressed to JOHN CARTWRIGHT, Esq. Chairman of the Committee at the Crown and Anchor, on the subject of Parliamentary Reform. By the EARL of SELKIRK. The Second Edition. 1s.

IX. AN ALPHABETICAL LIST of the NAMES of MINERALS at present most familiar in the English, French, and German languages, with Tables of Analyses. In one vol. 8vo. 5s. in boards.

X. THE PLOUGH-WRIGHT'S ASSISTANT; being a NEW PRACTICAL TREATISE on the PLOUGH, and on various other important IMPLEMENTS made use of in Agriculture. By ANDREW GRAY, Author of "The Experienced Mill-Wright." Royal Octavo, price 1l. 6s. boards. Containing Engravings of the Old Scots Plough—the Plough with a Convex twisted Mould-board—the Plough with a Concave twisted Mould-board—the Chain Plough—the Double or Two-Furrow Plough—of Harrows in general—the Break Harrow—the Common Harrow—a Machine whereby Land may be Harrowed in a Wet Season—the Roller—the Drain or Mole Plough—Plan, or Bird's-eye View of a Drill Machine—Plan, or Bird's-eye View of a Drill Machine with Indented Cylinders—the Cultivator, or Horse-hoe—a sowing Machine—of Wheels in general—of Cart-wheels—of Carts or Carriages in general—of Placing the Bushes into the Wheel—of the Method of finding the Dish of a Wheel—of the Axle-bed—of Contracting the Wheels before—of Wheel-carriages used in Husbandry—Plan, Elevation, and Section of a Thrashing Machine.

XI. The EXPERIENCED MILL-WRIGHT; or a Treatise on the Construction of some of the most useful Machines, with the latest Improvements; to which is prefixed, a short account of the general Principles of Mechanics, and of the Mechanical Powers. By ANDREW GRAY, Mill-Wright. Second Edition, 4to. with 44 Engravings. 2l. 2s. half-bound.

XII. ELEMENTS of MECHANICAL PHILOSOPHY, being the substance of a Course of Lectures on that Science. By JOHN ROBISON, LL. D. Professor of Natural Philosophy in the University of Edinburgh. Volume first, large 8vo. with 22 copperplates. 1l. 1s. boards.

XIII. A TRACT on MONASTIC ANTIQUITIES, with some account of a Recent Search for the Remains of the Kings Interred in the Abbey of Dunfermline. By JOHN GRAHAM DALYELL, Esq. With Specimens of the Chartulary of Dunfermline finely engraved. 8vo. 9s. boards.

XIV. The PASTORAL or LYRIC MUSE of SCOTLAND, a Poem, descriptive of the united influence of our national Poetry and Music, in softening the Passions, and civilizing the Manners of our feudal Ancestors on the Borders. By HECTOR MACNEILL, Esq. 4to. 7s. 6d. boards.

XV. MEMOIRS of CAPTAIN GEORGE CARLETON, an English Officer, who served in the Wars against France and Spain, containing an account of the EARL of PETERBOROUGH, and other General Officers, Admirals, &c. Octavo. 12s. boards.

*** While the eyes of the public are turned with hope and expectation towards the regeneration of the Spanish kingdom, all information respecting the character of the people and state of the country, particularly in a military point of view, must be highly acceptable. The Memoirs of Carleton were written during that memorable war, in which the Catalonian Insurgents, supported by an auxiliary British Force, drove the French from Madrid, and forced them to cross the Pyrenees; when it was, as now, the common cry, in the streets of the Spanish capital, "Paz con la Inglaterra, y con todo el mundo la guerra."

It is the work of an eyewitness and actor in the scenes he records; and was esteemed by the late Dr Johnson to contain the most authentic account of the campaigns of the gallant Earl of Peterborough.

XVI. The BATTLE of FLODDEN-FIELD, a Poem of the Sixteenth Century; with the various Readings of the different Copies, Historical Notes, a Glossary, and an Appendix, containing Ancient Poems and Historical Matter, relative to the Event. By HENRY WEBER, Esq. Embellished with Three Engravings. 8vo. Handsomely printed. 15s. boards. A very few in Royal Paper, 17. 7s. 6d. boards.

* * * The ancient Poem of Flodden-Field having become extremely scarce, is now, for the first time, published in an authentic form, the text being established by the collation of the different manuscripts and printed copies. Copious notes are subjoined, as also an Appendix, containing numerous ancient poems relating to the battle and its consequences, together with the minute accounts of the most creditable English historians. The engravings of the two standards carried by the Earls of Huntly and Marischall, and the sword and dagger of King James IV., are added as appropriate embellishments. The whole, it is hoped, will be found an interesting commentary to an event, which has latterly become so universally popular, by the publication of Mr Scott's Mormion.

XVII. The POOR MAN'S SABBATH, with other POEMS. By JOHN STRUTHERS. Third Edition. Foolscap 8vo. 5s. bds.

XVIII. A TREATISE on SCROFULA. By JAMES RUSSELL, Fellow of the Royal College of Surgeons, Professor of Clinical Surgery in the University of Edinburgh. 8vo. 5s. sewed.

XIX. HISTORY OF THE UNIVERSITY of EDINBURGH, from 1580 to 1646. By THOMAS CRAWFORD, A. M. Professor of Mathematics in the College of Edinburgh in the year 1646. To which is prefixed, the Charter granted to the College by King James, anno 1582. 8vo. 7s. 6d. sewed.

* * * Of this interesting tract only One Hundred Copies have been printed for sale.

XX. RESULT of an INQUIRY into the Nature and Causes of the Blight, the Rust, and the Mildew, which have particularly affected the Crops of Wheat on the Borders of England and Scotland; with some Observations on the Culture of Spring Wheat. By SIR JOHN SINCLAIR, Bart. M.P. &c. 8vo. 4s. sewed.

XXI. MEMOIRS of ROBERT CARY, Earl of Monmouth, written by himself. Published from an original MS. in the custody of the Earl of Corke and Orrery; to which is added, Fragmenta Regalia, being a History of Queen Elizabeth's Favourites, by SIR ROBERT NAUNTON; with explanatory Annotations. Handsomely printed in Octavo. 10s. 6d. boards. A few Copies on Royal Paper, price 17. 5s. boards.

* * * The Memoirs of Cary were first published from the original Manuscript by the Earl of Corke and Orrery. They contain an interesting account of some important transactions in Elizabeth's reign, and throw particular light upon the personal character of the Queen. To the present edition have been added additional explanatory Notes, particularly referring to Border Matters; and, as a suitable companion to these Memoirs, the Fragmenta Regalia of Sir Robert Naunton has also been reprinted.

XXII. The WHOLE WORKS of HENRY MACKENZIE, Esq. revised and corrected by the Author; with the addition of various Pieces never before published. Most beautifully printed in Eight Vol. Post 8vo, with a Portrait of the Author. 31. 8s. boards.

* * * This Edition, besides being published under the careful inspection and review of the Author, contains several hitherto unpublished works, particularly a Tragedy and a Comedy, in which it is believed the Public will be interested. Among the al-

ready published pieces, is included a Pamphlet published in the year 1790, the History of the Proceedings of the Parliament 1784, that Parliament in which Mr Pitt laid the foundation of all those measures to which the country has imputed the power which Great Britain exercised, under his auspices, of resisting the tyrannous encroachment of Buonaparte, so fatal to the rest of Europe. It adds a double interest to this publication, when we are informed, in a note by the Author, that it was anxiously revised and corrected by the hand of Mr Pitt himself.

XXIII. The ADVENTURES of ROBERT DRURY, during fifteen years captivity in the Island of Madagascar; containing a Description of that Island; an account of its Produce, Manufactures, and Commerce; with an account of the Manners and Customs, Wars, Religion, and Civil Policy of the Inhabitants; to which is added, a Vocabulary of the Madagascar Language. Written by himself, and now carefully revised and corrected from the original copy. With two Engravings. 8vo. 8s. boards.

* * * Among the numerous relations of voyages and travels, which combine so much instruction with delight, the adventures of Robert Drury will be found one of the most interesting and amusing; for the work is not that of one who passed rapidly through the scenes which he describes, but the experience of a long series of years spent in captivity. Indeed, the author was become so naturalized, that he found much difficulty in regaining his native language. The narration of the numerous destructive wars, and the surprising incidents and vicissitudes of fortune which his unfortunate condition induced, together with the descriptions of the customs, the religion, and the productions of the country, cannot fail highly to gratify every reader. The whole is delivered in an artist's style, which enhances the interest we feel for the author, as we are more immediately introduced to his situation of life, and more habituated to his own mode of thinking."

XXIV. A SERMON preached in the Episcopal Chapel, Cowgate, Edinburgh, November 16, 1806, the day after the Funeral of SIR WILLIAM FORBES, of Pittsigo, Bart. By ARCHIBALD ALISON, LL. B. F. R. S. Lond. and Edin., Prebendary of Sarum, &c. &c. &c. &c. and Senior Minister of that Chapel. 4to, 2s. 6d. 5. and 8vo, 1s. sewed.

XXV. ACCOUNT of the LIFE and WRITINGS of JAMES BRUCE of Kinnaid, Esq. F. R. S. Author of Travels to discover the Source of the Nile, in the Years 1768, 1769, 1770, 1771, 1772, and 1773. By A. MURRAY, F. A. S. E. and Secretary for Foreign Correspondence. Handsomely printed in Royal 4to, with 22 beautiful Engravings by HEATH. 27. 12s. 6d. boards.

* * * This work, being published in Quarto, forms a most appropriate Supplement to the First Edition of Bruce's Travels; for, besides the Life, it contains much important Correspondence between the Traveller and many of the first Literary Characters in Europe, as well as illustrations and testimonies in favour of the Authenticity of his Travels to Discover the Source of the Nile.

XXVI. QUEEN HOO-HALL, a Legendary Romance, interspersed with several original and beautiful Ballads; and ANCIENT TIMES, a Drama; both illustrative of the Domestic Manners and Amusements of the Fifteenth Century. By the late JOSPEH STRUTT, author of "Rural Sports and Pastimes of the People of England," &c. In Four neatly printed Volumes, small 8vo. Price 18s. in boards.

* * * "We have perused, with great pleasure, the interesting pages of this work, which contains a lively and well detailed picture of ancient customs, related with all the simplicity of native genius."—Miscell. Museum, July 1808.

Works in the Press, and preparing for Publication.

I. The GENEALOGY of the EARLS of SUTHERLAND, from the origin of that illustrious House to the year 1630, with the History of the Northern Parts of Scotland during that Period. By SIR ROBERT GORDON of Gordonstoun, Baronet, continued to the year 1651 by GILBERT GORDON of Sallagh. Published from the original manuscript in the possession of the Marchioness of STAFFORD. Handsomely printed in folio.

* * * The Public is here prefaced not only with an accurate genealogical history of the ancient House of Sutherland, but also with a minute detail of the principal transactions which occurred during a period of nearly 600 years, particularly in the counties of Sutherland and Caithness, and the Highlands of Scotland in general. The history of these parts, it is presumed, will receive more elucidation from this work than from any which the public is at present possessed of. The whole has been carefully transcribed by the kind permission of the Marchioness of Stafford, from the original manuscript preserved at Dunrobin Castle. An Appendix will be added, containing an inventory of writs of the Earldom, and the work will be illustrated by several engravings.

II. The WORKS of GAWIN DOUGLAS, Bishop of Dunkeld, with Historical and Critical Dissertations on his Life and Writings, Notes and a Glossary. By the Right Hon. SYLVESTER (DOUGLAS) LORD GLENBERVIE. 4 vol. 8vo. Elegantly printed.

* * * The whole works of Gawin Douglas, consisting of his Translation of Virgil's Æneid, the Palace of Honour, and King Hart, are now, for the first time, collected into one edition. Two Dissertations, the one on the Family of Douglas, the other on the Poet's Life and Writings, will be prefixed, and copious notes added. The text of Ruddiman's edition of the Æneid has been collated with the following five manuscripts; viz. two in the library of the University of Edinburgh, one in that of the Faculty of Advocates, a fourth in the possession of the Marquis of Bath at Longleat, and the fifth at Lambeth Palace. The excellent Glossary of Ruddiman is made the basis of that in the present work, but considerably enlarged, and extended to the other poems.

III. The DRAMATIC WORKS of JOHN FORD; with an Introduction and explanatory Notes. By HENRY WEBER, Esq. In Two Vol. 8vo.

WORKS IN THE PRESS, AND PREPARING FOR PUBLICATION.

IV. THE PEERAGE OF SCOTLAND; containing an 'Historical and Genealogical Account of the Nobility of that Kingdom, from their Origin to the present Generation.' Collected from the Public Records, and Ancient Chartularies of this Nation, the Charters, and other Writings of the Nobility, and the Works of our best Historians. By Sir ROBERT DOUGLAS of Glenbervie, Bart. Continued to the present time by J. P. WOOD, Esq. Handsomely printed in 2 vol. Folio, with the Arms of each Family beautifully engraven.

* * * A few Copies are printed on large Paper, forming two superb Volumes, with First Impressions of the Plates; and as the number printed is very limited, Noblemen and Gentlemen who wish to secure Copies, are respectfully requested to leave their names, either with ARCHIBALD CONSTABLE & Co. Edinburgh, or with CONSTABLE, HUNTER, PARK, & HUNTER, 10, Ludgate-Street, London, where Specimens of the work may now be seen.

V. LETTERS of the late ANNA SEWARD, written between the Years 1784 and 1807. 5 vol. post 8vo, with Portraits and other Plates.

VI. A SYSTEM OF SURGERY. By J. RUSSELL, F. R. S. E. Fellow of the Royal College of Surgeons, one of the Surgeons to the Royal Infirmary, and Professor of Clinical Surgery in the University of Edinburgh. 4 vol. 8vo.

VII. ILLUSTRATIONS of the HUTTONIAN THEORY. By JOHN PLAYFAIR, Professor of Natural Philosophy in the University of Edinburgh, F. R. S. London, and Secretary to the Royal Society, Edinburgh. Second Edition, with Additions. 1 vol. 4to, with Engravings.

VIII. CALEDONIA, or an Account, Historical and Topographical, of North Britain, from the most ancient to the present times. By GEORGE CHALMERS, Esq. F. R. S. Vol. II. 4to.

* * * The first volume of the above work, published last year, contains the Ancient History of North Britain. The second volume, which will appear in the course of 1809, will detail, after an Introductory Chapter of 26 Sections, the local History of its several Shires, in a correlative sequence; beginning with Roxburgh, the most Southern Shire, and proceeding successively, to Berwick, Haddington, Edinburgh, Linlithgow, Peebles, Selkirk, Dumfries, Kirkcudbright, Wigton, and perhaps Ayrshire: And the Local History of each Shire will be given in eight distinct sections;—1. Of its Name; 2. Of its Situation and Extent; 3. Of its Natural Objects; 4. Of its Antiquities; 5. Of its Establishment as a Shire; 6. Of its Civil History; 7. Of its Agriculture, Manufactures, Trade; 8. Of its Ecclesiastical History; the Account of each Shire concluding with a Supplemental State, which contains, in a Tabular form, the Names of the several Parishes, and the number of their Ministers; their Extent and Population in 1755, 1791, and 1801; with the Ministers' Stipends in 1755 and 1798, and their Patronage; forming, what Scotland does not now possess, a feet of Litter Regis.

This most interesting work will be completed by the publication of two other volumes. The third will contain the local history and description of the remaining counties, on the plan stated above. The fourth volume will consist of a Topographical Dictionary, containing whatever is interesting relative to all places and objects of any importance in this part of the United Kingdom. This volume will be preceded by an Historical View of the different Languages spoken in Scotland.

IX. THE ENGLISH ÆSOP, a Collection of Fables, Ancient and Modern, in Verse, translated, imitated, and original. By Sir BROOKE BOOTHBY, Bart. 2 vol. beautifully printed, Post 8vo.

X. THE HISTORY of SCOTLAND, by ROBERT LINDSAY of Pitcotrie. Edited from Ancient and Authentic Manuscripts, by JOHN GRAHAM DALYELL, Esq. 1 vol. 8vo, handsomely printed, with a Portrait of King James V. from an original Picture.

XI. THE GARDENER'S KALENDAR, or MONTHLY DIRECTORY of Operations in every Branch of HORTICULTURE. In 1 vol. 8vo. By WALTER NICOL, Author of 'The Villa Garden Directory,' 'The Forcing, Fruit, or Kitchen Gardener,' 'The Practical Planter,' &c.

* * * The Kalendar will be preceded by a Dissertation on the Situations proper for Gardens and Orchards; on Soils, and how to improve them; on Manures, and their application; and on the Rotation of Crops. It will exhibit the newest and most approved methods of cultivating all kinds of culinary Vegetables, Fruits, Shrubs, and Flowers; the Management of Hot-houses of every description, Hot-walls, Flued Pits, and Hot-beds; the Green-house, and the Conservatory; so as to form a complete assistant to the operative Gardener, and to the scientific Horticulturist.

D. WILLISON, PRINTER, EDINBURGH.

XII. METRICAL ROMANCES of the Thirteenth, Fourteenth, and Fifteenth Centuries. Published from Ancient Manuscripts, and illustrated by an Introduction, Notes, and a Glossary. By HENRY WEBER, Esq. In Three Volumes crown octavo.

'Of all manner of mischrafs,
And jiftours that tellen tales,
Both of sweeping and of gars,
And of all that bingeth into fars.' CHAUCER.

* * * The present publication is intended to comprehend the most valuable of those Romances, which have not yet been submitted to the public. The Life of Alexander, attributed by Warton to Adam Davis, and strongly recommended by him for publication, will form the first article; and will be followed by Richard Cœur de Lion, which, besides its very considerable poetical merit, must excite a strong national interest; and by others, selected either for the beauty of the tale, or some circumstances rendering them curious; among which a few Comical Romances will be found. To the Introduction, the Editor, at the request of several gentlemen most anxious for the publication, has subjoined a summary account of the German early Poetry and Romance; a subject of high interest, but as yet entirely unknown to this nation, and but little cultivated on the Continent. If the present publication should meet with the encouragement which the importance of this species of composition in the History of English Poetry deserves, a continuation, comprising those excluded from this selection, on account of its limited extent, will be published.

XIII. A TREATISE on the DISEASES and MANAGEMENT of SHEEP, with introductory Remarks on their Anatomical Structure; and an Appendix, containing Documents, exhibiting the Value of the Merino Breed, and their Progress in Scotland. By Sir GEORGE STEWART MACKENZIE of Coul, Bart. 1 vol. 8vo.

XIV. SHIPWRECKS and DISASTERS at SEA, according to the most Authentic Accounts, Ancient and Modern. 3 vol. 8vo.

XV. SWIFT'S WORKS, edited by WALTER SCOTT, Esq. with a Life of the Author, Notes Critical and Illustrative, &c. &c. Nineteen Volumes Octavo, handsomely printed, with a few copies on Royal Paper.

* * * The present edition of this incomparable English Classic is offered to the public on a plan different from that adopted by former editors. In the Life of the Author, it is proposed to collate and combine the various information which has been given by Mr Sheridan, Lord Orrery, Dr Delany, Mr Pilkington, Dean Swift, Dr Johnson, and others, into one distinct and comprehensive narrative; which, it is hoped, may prove neither a libel or apology for Swift, nor a collection from the pleadings of those who have written either; but a plain, impartial, and connected biographical narrative. By the favour of distinguished friends in Ireland, the Editor hopes to obtain considerable light upon some passages in the Dean's life, which have hitherto perplexed his biographers. In preparing the text and notes, no labour or expense has been spared to procure original information. The Tale of a Tub, for example, is illustrated with the marginal notes of the learned Bentley, transcribed from manuscript jettings on his own copy. Although neither long nor numerous, they offer some curious elucidations of the author, and afford a singular instance of the equanimity with which the satire even of Swift was borne by the venerable scholar against whom it was so unadvisedly levelled. Some preliminary critical observations are offered on the various literary productions of the Dean of St Patrick's; and historical explanations and anecdotes accompany his political treatises. All those pieces which, though hitherto admitted into Swift's works, are positively ascertained not to be of his composition, are placed in the Appendix, or altogether retrenched. On the other hand, the editor is encouraged to believe, that, by accurate research, some gleanings may yet be recovered, which have escaped even the laudable and undeniable industry of Swift's last editor. So that, upon the whole, he hopes the present edition will be fully more complete than those of late years. The work will appear in the course of 1810.

XVI. RESEARCHES into the ORIGIN and AFFINITY of the GREEK and TEUTONIC LANGUAGES. By A. MURRAY, F. A. S. E. and Secretary for Foreign Correspondence. One Volume Quarto.

* * * The immediate object of this work is, to illustrate the early state and connexion of these languages, on accurate and philological principles. The light which is thus thrown on the structure of the Greek tongue, gives a new and interesting form to the whole of classic philology; exhibits an extensive view of the process by which the mind invents and improves articulate speech; and leads to a development of the origin of the most ancient European nations. The notices ascertained in the course of investigation depend, not on conjecture, but on a comparison of almost every European language with those to which it is respectively allied. In the train of inquiry pursued in the researches above mentioned, particular regard has been paid to the Oriental tongues; those having been examined which bear no affinity to the Teutonic, as well as those which appear to be related to it. For a plan and outline of the whole work, reference may be had to page 505 of an 'Account of the Life and Writings of James Bruce of Kinnaid Esq., Author of Travels to discover the Source of the Nile, in the Years 1768—1773,' published last year (1808.)

be $\backslash frac\{1\}\{2a\}$. Moreover, the times in which the same velocity will be extinguished by different forces, acting uniformly, are inversely as the forces, and gravity would extinguish the velocity $v$ in the time $\backslash frac\{1\}\{g\}$, (in these measures) to $\backslash frac\{1\}\{u^2\}\; =\; \backslash frac\{2a\}\{u^2\}$. Therefore we have the following

proportion $\backslash frac\{1\}\{2a\}\; (=R)\; :\; \backslash frac\{u^2\}\{2a\}\; (=g)\; =\; \backslash frac\{2a\}\{u^2\}\; :\; 2a$, and $2a$ is equal to $E$, the time in which the velocity $v$ will be extinguished by the uniform action of the resistance competent to this velocity.

The velocity $v$ would in this case be extinguished after a motion uniformly retarded, in which the space described is one half of what would be uniformly described during the same time with the constant velocity $v$. Therefore the space thus described by a motion which begins with the velocity $v$, and is uniformly retarded by the resistance competent to this velocity, is equal to the height through which this body must fall in vacuo in order to acquire its terminal velocity in air.

All these circumstances may be conceived in a manner which, to some readers, will be more familiar and palpable. The terminal velocity is that where the resistance of the air balances and is equal to the weight of the body. The resistance of the air to any particular body is as the square of the velocity; therefore let $R$ be the whole resistance to the body moving with the velocity $v$, and $r$ the resistance to its motion with the terminal velocity $u$; we must have $r\; =\; R\; \backslash times\; u^2$, and this must be $=\; W$ the weight. Therefore, to obtain the terminal velocity, divide the weight by the resistance to the velocity $v$, and the quotient is the square of the terminal velocity, or $\backslash frac\{W\}\{R\}\; =\; u^2$: And this is a very expeditious method of determining it, if $R$ be previously known.

Then the common theorems give $a$, the fall necessary for producing this velocity in vacuo $=\; \backslash frac\{u^2\}\{2g\}$, and the time of the fall $=\; \backslash frac\{u\}\{g\}\; =\; e$, and $e\; u\; =\; 2a$, $=$ the space uniformly

described with the velocity $u$ during the time of the fall, or its equal, the time of the extinction by the uniform action of the resistance $r$; and, since $r$ extinguishes it in the time $e$, $R$, which is $u^2$ times smaller, will extinguish it in the time $u^2\; e$, and $R$ will extinguish the velocity $v$, which is $u$ times less than $u$, in the time $u\; e$, that is, in the time $2a$; and the body, moving uniformly during the time $2a$, $=\; E$, with the velocity $v$, will describe the space $2a$; and, if the body begin to move with the velocity $v$, and be uniformly opposed by the resistance $R$, it will be brought to rest when it has described the space $a$; and the space in which the resistance to the velocity $v$ will extinguish that velocity by its uniform action, is equal to the height through which that body must fall in vacuo in order to acquire its terminal velocity in air. And thus every thing is regulated by the time $E$ in which the velocity $v$ is extinguished by the uniform action of the corresponding resistance, or by $2a$, which is the space uniformly described during this time, with the velocity $v$. And $E$ and $2a$ must be expressed

VOL. XVII. Part II.

by the same number. It is a number of units, of time, or of length.

Having ascertained these leading circumstances for an The common unit of velocity, weight, and bulk, we proceed to de- parison made general. duce the similar circumstances for any other magnitude; and, to avoid unnecessary complications, we shall always suppose the bodies to be spheres, differing only in diameter and density.

First, then, let the velocity be increased in the ratio of $1$ to $v$.

The resistance will now be $\backslash frac\{v^2\}\{2a^2\}\; =\; r$.

The extinguishing time will be $\backslash frac\{E\}\{v\}\; =\; e\; =\; \backslash frac\{2a\}\{v\}$, and $e\; v\; =\; 2a$; so that the rule is general, that the space along which any velocity will be extinguished by the uniform action of the corresponding resistance, is equal to the height necessary for communicating the terminal velocity to that body by gravity. For $e\; v$ is twice the space through which the body moves while the velocity $v$ is extinguished by the uniform resistance.

In the 2d place, let the diameter increase in the proportion of $1$ to $d$. The aggregate of the resistance changes in the proportion of the surface similarly resisted, that is, in the proportion of $1$ to $d^2$. But the quantity of matter, or number of particles among which this resistance is to be distributed, changes in the proportion of $1$ to $d^3$. Therefore the retarding power of

the resistance changes in the proportion of $1$ to $\backslash frac\{1\}\{d\}$. When the diameter was $1$, the resistance to a velocity $v$ was $\backslash frac\{1\}\{2a\}$. It must now be $\backslash frac\{1\}\{2ad\}$. The time in which this diminished resistance will extinguish the velocity $v$ must increase in the proportion of the diminution of force, and must now be $E\; d$, or $2\; a\; d$, and the space uniformly described during this time with the initial velocity $v$ must be $2\; a\; d$; and this must fill be twice the height necessary for communicating the terminal velocity $w$ to this body. We must still have $g\; =\; \backslash frac\{w^2\}\{2\; a\; d\}$; and

therefore $w^2\; =\; 2\; g\; a\; d$, and $w\; =\; \backslash sqrt\{2\; g\; a\; d\}\; =\; \backslash sqrt\{2\; g\; a\}\; \backslash sqrt\{d\}$. But $u\; =\; \backslash sqrt\{2\; g\; a\}$. Therefore the terminal velocity $w$ for this body is $=\; u\; \backslash sqrt\{d\}$; and the height necessary for communicating it is $a\; d$. Therefore the terminal velocity varies in the subduplicate ratio of the diameter of the ball, and the fall necessary for producing it varies in the simple ratio of the diameter. The extinguishing time for the velocity $v$ must now be $\backslash frac\{E\; d\}\{v\}$.

If, in the 3d place, the density of the ball be increased in the proportion of $1$ to $m$, the number of particles among which the resistance is to be distributed is increased in the same proportion, and therefore the retarding force of the resistance is equally diminished; and if the density of the air is increased in the proportion of $1$ to $n$, the retarding force of the resistance increases in the same proportion: hence we easily deduce these general expressions.

The terminal velocity $=\; n\; \backslash sqrt\{\backslash frac\{d^2\; m\}\{n\}\}\; =\; \backslash sqrt\{2\; g\; a\; d\; \backslash frac\{m\}\{n\}\}$.

The producing fall in vacuo $=\; a\; d\; \backslash frac\{m\}\{n\}$.

The retarding power of resistance to any velocity =

The extinguishing time for any velocity $v\; =\; \backslash frac\{E\; d\; m\}\{v\; u\}$.

And thus we see that the chief circumstances are regulated by the terminal velocity, or are conveniently referred to it.

³⁶ Units necessary by which the quantities may be measured. To render the deductions from these premises perspicuous, and for communicating distinct notions or ideas, it will be proper to assume some convenient units, by which all these quantities may be measured; and, as this subject is chiefly interesting in the case of military projectiles, we shall adapt our units to this purpose. Therefore, let a second be the unit of time, a foot the unit of space and velocity, an inch the unit of diameter of a ball or shell, and a pound avoirdupois the unit of pressure, whether of weight or of resistance; therefore $g$ is 32 feet.

The great difficulty is to procure an absolute measure of $r$, or $u$, or $a$; any one of these will determine the others.

³⁷ Sir Isaac Newton's endeavours in this way. Sir Isaac Newton has attempted to determine $r$ by theory, and employs a great part of the second book of the Principia in demonstrating, that the resistance to a sphere moving with any velocity is to the force which would generate or destroy its whole motion in the time that it would uniformly move over $\backslash frac\{3\}\{4\}$ of its diameter with this velocity as the density of the air is to the density of the sphere. This is equivalent to demonstrating that the resistance of the air to a sphere moving through it with any velocity, is equal to half the weight of a column of air having a great circle of the sphere for its base, and for its altitude the height from which a body must fall in vacuo to acquire this velocity. This appears from Newton's demonstration; for, let the specific gravity of the air be to that of the ball as 1 to $m$; then, because the times in which the same velocity will be extinguished by the uniform action of different forces are inversely as the forces, the resistance to this velocity would extinguish it in the time of describing $\backslash frac\{3\}\{4\}\; m\; d$, $d$ being the diameter of the ball. Now 1 is to $m$ as the weight of the displaced air to the weight of the ball, or as $\backslash frac\{1\}\{3\}$ of the diameter of the ball to the length of a column of air of equal weight. Call this length $a$; $a$ is therefore equal to $\backslash frac\{3\}\{4\}\; m\; d$. Suppose the ball to fall from the height $a$ in the time $t$, and acquire the velocity $u$. If it moved uniformly with this velocity during this time, it would describe a space $=\; 2a$, or $\backslash frac\{3\}\{2\}\; m\; d$. Now its weight would extinguish this velocity, or destroy this motion, in the same time, that is, in the time of describing $\backslash frac\{3\}\{4\}\; m\; d$; but the resistance of the air would do this in the time of describing $\backslash frac\{3\}\{2\}\; m\; d$; that is, in twice the time. The resistance therefore is equal to half the weight of the ball, or to half the weight of the column of air whose height is the height producing the velocity. But the resistances to different velocities are as the squares of the velocities, and therefore, as their producing heights; and, in general, the resistance of the air to a sphere moving with any velocity, is equal to the half weight of a column of air of equal section, and whose altitude is the height producing the velocity. The result of this investigation has been acquiesced in by all Sir Isaac Newton's commentators. Many faults

have indeed been found with his reasoning, and even with his principles; and it must be acknowledged that although this investigation is by far the most ingenious of any in the Principia, and sets his acuteness and address in the most conspicuous light, his reasoning is liable to serious objections, which his most ingenious commentators have not completely removed. However, the conclusion has been acquiesced in, as we have already stated, but as if derived from other principles, or by more logical reasoning. We cannot, however, say that the reasonings or assumptions of these mathematicians are much better than Newton's: and we must add, that all the causes of deviation from the duplicate ratio of the velocities, and the causes of increased resistance, which the later authors have valued themselves for discovering and introducing into their investigations, were pointed out by Sir Isaac Newton, but purposely omitted by him, in order to facilitate the discussion in re difficillima. (See Schol. prop. 37. book ii.).

It is known that the weight of a cubic foot of water is 62$\backslash frac\{1\}\{2\}$ pounds, and that the medium density of the air is $\backslash frac\{1\}\{144\}$ of water; therefore, let $a$ be the height producing the velocity (in feet), and $d$ the diameter of the ball (in inches), and $\backslash pi$ the periphery of a circle whose diameter is 1; the resistance of the air will be $=\; \backslash frac\{62\backslash frac\{1\}\{2\}\}\{840\}\; \backslash times\; \backslash frac\{\backslash pi\}\{4\}\; \backslash times\; \backslash frac\{1\}\{144\}\; \backslash times\; \backslash frac\{a\}\{2\}\; \backslash times\; d^2\; =\; \backslash frac\{a\; d^2\}\{4928\backslash frac\{1\}\{2\}\}$ pounds, very nearly, $=\; \backslash frac\{v^3\}\{4928\backslash frac\{1\}\{2\}\; \backslash times\; 64\}\; d^2\; =\; \backslash frac\{v^3\; d^2\}\{315417\}$ pounds.

We may take an example. A ball of cast iron weighing 12 pounds, is 4$\backslash frac\{1\}\{2\}$ inches in diameter. Suppose this ball to move at the rate of 25$\backslash frac\{1\}\{2\}$ feet in a second (the reason of this choice will appear afterwards). The height which will produce this velocity in a falling body is 9$\backslash frac\{1\}\{2\}$ feet. The area of its great circle is 0.11044 feet, or $\backslash frac\{11044\}\{100000\}$ of one foot. Suppose water to be 840 times heavier than air, the weight of the air incumbent on this great circle, and 9$\backslash frac\{1\}\{2\}$ feet high, is 0.081151 pounds: half of this is 0.0405755 or $\backslash frac\{405755\}\{10000000\}$, or nearly $\backslash frac\{1\}\{25\}$ of a pound. This should be the resistance of the air to this motion of the ball.

In all matters of physical discussion, it is prudent to ³⁰ necessity confront every theoretical conclusion with experiment. This is particularly necessary in the present instance, because the theory on which this proposition is founded is extremely uncertain. Newton speaks of it with the most cautious diffidence, and secures the justness of the conclusions by the conditions which he assumes in his investigation. He describes with the greatest precision the state of the fluid in which the body must move, so as that the demonstrations may be strict, and leaves it to others to pronounce whether this is the real constitution of our atmosphere. It must be granted that it is not; and that many other suppositions have been introduced by his commentators and followers, in order to suit his investigation (for we must assert that little or nothing has been added to it) to the circumstances of the case.

Newton himself, therefore, attempted to compare his ⁴⁰ Newton's propositions with experiment. Some were made by dropping balls from the dome of St Paul's cathedral, and all these showed as great a coincidence with his theory as they did with each other; but the irregularities

ties were too great to allow him to say with precision what was the resistance. It appeared to follow the proportion of the squares of the velocities with sufficient exactness; and though he could not say that the resistance was equal to the weight of the column of air having the height necessary for communicating the velocity, it was always equal to a determinate part of it; and might be stated $=\; na$, $n$ being a number to be fixed by numerous experiments.

One great source of uncertainty in his experiments seems to have escaped his observation: the air in that dome is almost always in a state of motion. In the summer season there is a very sensible current of air downwards, and frequently in winter it is upwards: and this current bears a very great proportion to the velocity of the descents. Sir Isaac takes no notice of this.

He made another set of experiments with pendulums; and has pointed out some very curious and unexpected circumstances of their motions in a resisting medium. There is hardly any part of his noble work in which his address, his patience, and his astonishing penetration, appear in greater lustre. It requires the utmost intenseness of thought to follow him in these disquisitions; and we cannot enter on the subject at present: some notice will be taken of these experiments in the article RESISTANCE of Fluids. Their results were much more uniform, and confirmed his general theory; and, as we have said above, it has been acquiesced in by the first mathematicians of Europe.

⁴¹ Feasibility of the theory in practice. But the deductions from this theory were so inconsistent with the observed motions of military projectiles, when the velocities are prodigious, that no application could be made which could be of any service for determining the path and motion of cannon shot and bombs; and although Mr John Bernoulli gave, in 1718, a most elegant determination of the trajectory and motion of a body projected in a fluid which resists in the duplicate ratio of the velocities (a problem which even Newton did not attempt), it has remained a dead letter. Mr Benjamin Robins, equally eminent for physical science and mathematical genius, was the first who suspected the true cause of the imperfection of the usually received theories; and in 1737 he published a small tract, in which he showed clearly, that even the Newtonian theory of resistance must cause a cannon ball, discharged with a full allotment of powder, to deviate farther from the parabola, in which it would move in vacuo, than the parabola deviates from a straight line. But he farther asserted, on the authority of good reasoning, that in such great velocities the resistance must be much greater than this theory assigns; because, besides the resistance arising from the inertia of the air which is put in motion by the ball, there must be a resistance arising from a condensation of the air on the anterior surface of the ball, and a rarefaction behind it: and there must be a third resistance, arising from the statical pressure of the air on its anterior part, when the motion is so swift that there is a vacuum behind. Even these causes of disagreement with the theory had been foreseen and mentioned by Newton (see the Scholium to prop. 37. book ii. Princip.); but the subject seems to have been little attended to. The eminent mathematicians had few opportunities of making experiments; and the professional men, who were in the service of princes, and had their countenance and aid in

this matter, were generally too deficient in mathematical knowledge to make a proper use of their opportunities. The numerous and splendid volumes which these gentlemen have been enabled to publish by the patronage of sovereigns are little more than prolix extensions of the simple theory of Galileo. Some of them, however, such as St Remy, Antonini, and Le Blond, have given most valuable collections of experiments, ready for the use of the profound mathematician.

⁴³ Two or three years after this first publication, Mr Robins hit upon that ingenious method of measuring the great velocities of military projectiles, which has handed down his name to posterity with great honour, and relief. And having ascertained these velocities, he discovered also, the prodigious resistance of the air, by observing the diminution of velocity which it occasioned. This made him anxious to examine what was the real resistance to any velocity whatever, in order to ascertain what was the law of its variation; and he was equally fortunate in this attempt. His method of measuring the resistance has been fully described in the article GUNNERY, N^o 9, &c.

It appears (Robins's Math. Works, vol. i. page 205.) that a sphere of 4$\backslash frac\{1\}\{2\}$ inches in diameter, moving at the rate of 25$\backslash frac\{1\}\{2\}$ feet in a second, sustained a resistance of 0.04914 pounds, or $\backslash frac\{4914\}\{100000\}$ of a pound. This is a greater resistance than that of the Newtonian theory, which gave $\backslash frac\{4914\}\{100000\}$ in the proportion of 1000 to 1211, or very nearly in the proportion of five to six in small numbers. And we may adopt as a rule in all moderate velocities, that the resistance to a sphere is equal to $\backslash frac\{5\}\{100\}$ of the weight of a column of air having the great circle of the sphere for its base, and for its altitude the height through which a heavy body must fall in vacuo to acquire the velocity of projection.

This experiment is peculiarly valuable, because the ball is precisely the size of a 12 pound shot of cast iron; and its accuracy may be depended on. There is but one source of error. The whirling motion must have occasioned some whirl in the air, which would continue till the ball again passed through the same point of its revolution. The resistance observed is therefore probably somewhat less than the true resistance to the velocity of 25$\backslash frac\{1\}\{2\}$ feet, because it was exerted in a relative velocity which was less than this, and is, in fact, the resistance competent to this relative and smaller velocity.

—Accordingly, Mr Smeaton, a most sagacious naturalist, places great confidence in the observations of a Rouse and Mr Rouse of Leicestershire, who measured the resistance by the effect of the wind on a plane properly exposed to it. He does not tell us in what way the velocity of the wind was ascertained; but our deference for his great penetration and experience disposes us to believe that this point was well determined. The resistance observed by Mr Rouse exceeds that resulting from Mr Robins's experiments nearly in the proportion of 7 to 10. They differ widely in Chevalier de Borda made experiments similar to those of Mr Robins, and his results exceed those of Robins in the proportion of 5 to 6. These differences are so considerable, that we are at a loss what measure to abide by. It is much to be regretted, that in a subject so interesting both to the philosopher and the man of the world, experiments have not been multiplied. Nothing would tend so much to perfect the science

of gunnery; and indeed till this be done, all the labours of mathematicians are of no avail. Their investigations must remain an unintelligible cipher, till this key be supplied. It is to be hoped that Dr Charles Hutton of Woolwich, who has so ably extended Mr Robins's Examination of the Initial Velocities of Military Projectiles, will be encouraged to proceed to this part of the subject. We should wish to see, in the first place, a numerous set of experiments for ascertaining the resistances in moderate velocities; and, in order to avoid all error from the resistance and inertia of the machine, which is necessarily blended with the resistance of the ball, in Mr Robins's form of the experiment, and is separated with great uncertainty and risk of error, we would recommend a form of experiment somewhat different.

Let the axis and arm which carries the ball be connected with wheelwork, by which it can be put in motion, and gradually accelerated. Let the ball be so connected with a bent spring, that this shall gradually compress it as the resistance increases, and leave a mark of the degree of compression; and let all this part of the apparatus be screened from the air except the ball. The velocity will be determined precisely by the revolutions of the arm, and the resistance by the compression of the spring. The best method would be to let this part of the apparatus be made to slide along the revolving arm, so that the ball can be made to describe larger and larger circles. An intelligent mechanic will easily contrive an apparatus of this kind, held at any distance from the axis by a cord, which passes over a pulley in the axis itself, and is then brought along a perforation in the axis, and comes out at its extremity, where it is fitted with a swivel, to prevent it from snapping by being twisted. Now let the machine be put in motion. The centrifugal force of the ball and apparatus will cause it to fly out as far as it is allowed by the cord; and if the whole is put in motion by connecting it with some mill, the velocity may be most accurately ascertained. It may also be fitted with a bell and hammer like Gravefande's machine for measuring centrifugal forces. Now by gradually veering off more cord, the distance from the centre, and consequently the velocity and resistance increase, till the hammer is disengaged and strikes the bell.

Another great advantage of this form of the experiment is, that the resistance to very great velocities may be thus examined, which was impossible in Mr Robins's way. This is the great desideratum, that we may learn in what proportion of the velocities the resistances increase.

In the same manner, an apparatus, consisting of Dr Lynd's Anemometer, described in the article PNEUMATICS, No 311, &c. might be whirled round with prodigious rapidity, and the fluid on it might be made clammy, which would leave a mark at its greatest elevation, and thus discover the resistance of the air to rapid motions.

Nay, we are of opinion that the resistance to very rapid motions may be measured directly in the conduit pipe of some of the great cylinder bellows employed in blast furnaces: the velocity of the air in this pipe is ascertained by the capacity of the cylinder and the strokes of the piston. We think it our duty to point out

to such as have the opportunities of trying them methods which promise accurate results for ascertaining this most desirable point.

We are the more puzzled what measure to abide by, because Mr Robins himself, in his Practical Propositions, does not make use of the result of his own experiments, but takes a much lower measure. We must content ourselves, however, with this experimental measure, because it is as yet the only one of which any account can be given, or well-founded opinion formed. ⁴⁷ The result of Robins's experiments as yet most to be depended on.

Therefore, in order to apply our formulae, we must reduce this experiment, which was made on a ball of 4½ inches diameter, moving with the velocity of 252 feet per second, to what would be the resistance to a ball of one inch, having the velocity 1 foot. This will ⁴⁸ Applied to the formulae.

evidently give us $R\; =\; \backslash frac\{0.04914\}\{4.5^2\; \backslash times\; 25.2^2\}$, being diminished in the duplicate ratio of the diameter and velocity. This gives us $R\; =\; 0.0000381973$ pounds, or $\backslash frac\{3.81973\}\{100000\}$ of a pound. The logarithm is 4.58204. The resistance here determined is the same whatever substance the ball be of; but the retardation occasioned by it will depend on the proportion of the resistance to the vis inertia of the ball; that is, to its quantity of motion. This in similar velocities and diameters is as the density of the ball. The balls used in military service are of cast iron or of lead, whose specific gravities are 7.207 and 11.37 nearly, water being 1. There is considerable variety in cast iron, and this density is about the medium. These data will give us

These numbers are of frequent use in all questions on this subject.

Mr Robins gives an expeditious rule for readily finding a, which he calls F (see the article GUNNERY), by which it is made 900 feet for a cast iron ball of an inch diameter. But no theory of resistance which he professes to use will make this height necessary for producing the terminal velocity. His F therefore is an empirical quantity, analogous indeed to the producing height, but accommodated to his theory of the trajectory of cannon-shot, which he promised to publish, but did not live to execute. We need not be very anxious about this; for all our quantities change in the same proportion with R, and need only a correction by a multiplier or divisor, when R shall be accurately established.

We may illustrate the use of these formulae by an example or two.

1. Then, to find the resistance to a 24 pound ball moving with the velocity of 1670 feet in a second, which is nearly the velocity communicated by 16 lbs. of powder. The diameter is 5.603 inches. ⁴⁹ Examples of their use.

Log.

But it is found, by unequivocal experiments on the retardation of such a motion, that it is 504 lbs. This is owing to the causes often mentioned, the additional resistance to great velocities, arising from the condensation of the air, and from its pressure into the vacuum left by the ball.

2. Required the terminal velocity of this ball?

As the terminal velocity $u$, and its producing height $a$, enter into all computations of military projectiles, we have inserted the following Table for the usual sizes of cannon-shot, computed both by the Newtonian theory of resistance, and by the resistances observed in Robins's experiments.

Mr Muller, in his writings on this subject, gives a much smaller measure of resistance, and consequently a much greater terminal velocity: but his theory is a mistake from beginning to end (See his Supplement to his Treatise of Artillery art. 150, &c.) In art. 148. he assumes an algebraic expression for a principle of mechanical argument; and from its consequence draws erroneous conclusions. He makes the resistance of a cylinder one-third less than Newton supposes it; and his reason is false. Newton's measure is demonstrated by his commentators Le Seur and Jaquier to be even a little too small, upon his own principles, (Not. 277 Prop. 36. B. II.) Mr Muller then, without any seeming reason, introduces a new principle, which he makes the chief support of his theory, in opposition to the theories of other mathematicians. The principle is false, and even absurd, as we shall have occasion to show by and by. In consequence, however, of this principle, he is ena-

bled to compare the results with many experiments, and the agreement is very flattering. But we shall soon see that little dependence can be had on such comparisons. We notice these things here, because Mr Muller being head of the artillery school in Britain, his publications have become a sort of text-books. We are miserably deficient in works on this subject, and must have recourse to the foreign writers.

We now proceed to consider these motions through their whole course: and we shall first consider them as motions affected by the resistance only; then we shall consider the perpendicular ascents and descents of heavy bodies through the air; and, lastly, their motion in a curvilinear trajectory, when projected obliquely. This must be done by the help of the abstruser parts of fluxionary mathematics. To make it more perspicuous, we shall, by way of introduction, consider the simply resisted rectilinear motions geometrically, in the manner of Sir Isaac Newton. As we advance, we shall quit this track, and prosecute it algebraically, having by this time acquired distinct ideas of the algebraic quantities.

We must keep in mind the fundamental theorems of Preliminary varied motions.

1. The momentary variation of the velocity is proportional to the force and the moment of time jointly, and may therefore be represented by $\backslash pm\; \backslash dot\{v\}\; =\; f\; t$, where $v$ is the momentary increment or decrement of the velocity, $f$ the accelerating or retarding force, and $t$ the moment or increment of the time $t$.

2. The momentary variation of the square of the velocity is as the force, and as the increment or decrement of the space jointly; and may be represented by $\backslash pm\; \backslash dot\{v\}^2\; =\; f\; s$. The first proposition is familiarly known. The second is the 39th of Newton's Principia, B. I. It is demonstrated in the article ORRICES, and is the most extensively useful proposition in mechanics.

These things being premised, let the straight line AC (fig. 5.) represent the initial velocity $V$, and let only CO, perpendicular to AC, be the time in which this velocity would be extinguished by the uniform action of the resistance. Draw through the point A an equilateral hyperbola AEB, having OF, OCD for its asymptotes; then let the time of the resisted motion be represented by the line CB, C being the first instant of the motion. If there be drawn perpendicular ordinates $ae,\; gf,\; DB$, &c. to the hyperbola, they will be proportional to the velocities of the body at the instants $a,\; g,\; D$, &c. and the hyperbolic areas $ACae,\; ACgf,\; ACDB$, &c. will be proportional to the spaces described during the times $Cx,\; Cg,\; CB$, &c.

For, suppose the time divided into an indefinite number of small and equal moments, $Ce,\; Dd$, &c. draw the ordinates $ae,\; bd$, and the perpendiculars $b\backslash beta,\; aa$. Then, by the nature of the hyperbola, $AC\; :\; ae\; =\; OC\; :\; OC$; and $AC\; :\; ae\; =\; OC\; :\; OC$; that is, $AC\; :\; ae\; =\; OC\; :\; Cc$; and $AC\; :\; Cc\; =\; OC\; :\; OC$; = $AC\; :\; ae$; $AC\; :\; OC$; in like manner, $B\backslash beta\; :\; Dd\; =\; BD\; :\; BD$; $BD\; :\; OD$. Now $Dd\; =\; Ce$, because the moments of time were taken equal, and the rectangles $AC\; \backslash cdot\; CO$, $BD\; \backslash cdot\; DO$, are equal, by the nature of the hyperbola; therefore $AC\; :\; B\backslash beta\; =\; AC\; :\; ae$; $BD\; :\; bd$; but as the points $e,\; d$ continually approach, and ultimately coincide with $C,\; D$, the ultimate ratio of $AC\; \backslash cdot\; ae$ to $BD\; \backslash cdot\; bd$ is that of $AC^2$ to $BD^2$; therefore the momentary decrements of AC.

AC and BD are as $AC^2$ and $BD^2$. Now, because the resistance is measured by the momentary diminution of velocity, these diminutions are as the squares of the velocities; therefore the ordinates of the hyperbola and the velocities diminish by the same law; and the initial velocity was represented by AC; therefore the velocities at all the other instants $x,\; g,\; D$, are properly represented by the corresponding ordinates. Hence,

1. Since the abscissae of the hyperbola are as the times, and the ordinates are as the velocities, the areas will be as the spaces described, and $AC\; \backslash times\; e$ is to $Aegf$ as the space described in the time $Cx$ to the space described in the time $Cg$ (1st Theorem on varied motions).

2. The rectangle ACOF is to the area ACDB as the space formerly expressed by $2a$, or $E$ to the space described in the resisting medium during the time $CD$: for AC being the velocity $V$, and OC the extinguishing time $e$, this rectangle is $=\; eV$, or $E$, or $2a$, of our former disquisitions; and because all the rectangles, such as ACOF, BDOG, &c. are equal, this corresponds with our former observation, that the space uniformly described with any velocity during the time in which it would be uniformly extinguished by the corresponding resistance is a constant quantity, viz. that in which we always had $ev\; =\; E$, or $2a$.

3. Draw the tangent $Atx$; then, by the hyperbola $Cx\; =\; CO$: now $Cx$ is the time in which the resistance to the velocity AC would extinguish it; for the tangent coinciding with the elemental arc $Aa$ of the curve, the first impulse of the uniform action of the resistance is the same with the first impulse of its varied action. By this the velocity AC is reduced to $ae$. If this operated uniformly like gravity, the velocities would diminish uniformly, and the space described would be represented by the triangle $ACx$.

This triangle, therefore, represents the height through which a heavy body must fall in vacuo, in order to acquire the terminal velocity.

4. The motion of a body resisted in the duplicate ratio of the velocity will continue without end, and a space will be described which is greater than any assignable space, and the velocity will grow less than any that can be assigned; for the hyperbola approaches continually to the asymptote, but never coincides with it. There is no velocity BD so small, but a smaller ZP will be found beyond it; and the hyperbolic space may be continued till it exceeds any surface that can be assigned.

5. The initial velocity AC is to the final velocity BD as the sum of the extinguishing time and the time of the retarded motion, is to the extinguishing time alone: for $AC\; :\; BD\; =\; OD$ (or $OC\; +\; CD$) : $OC$; or $V\; :\; v\; =\; e\; :\; e\; +\; t$.

6. The extinguishing time is to the time of the retarded motion as the final velocity is to the velocity lost during the retarded motion: for the rectangles AFOC, BDOG are equal; and therefore AVGF and BVCD are equal, and $VC\; :\; VA\; =\; VG\; :\; VB$; therefore $t\; =\; e\; \backslash frac\{V-v\}\{v\}$, and $e\; =\; t\; \backslash frac\{v\}\{V-v\}$.

7. Any velocity is reduced in the proportion of $m$ to $n$ in the time $e\; \backslash frac\{m-n\}\{n\}$. For, let $AC\; :\; BD\; =\; m\; :\; n$;

then $DO\; :\; CO\; =\; m\; :\; n$, and $DC\; :\; CO\; =\; m\; -\; n\; :\; n$, and $DC\; =\; \backslash frac\{m-n\}\{n\}\; CO$, or $t\; =\; e\; \backslash frac\{m-n\}\{n\}$. Therefore any velocity is reduced to one half in the time in which the initial resistance would have extinguished it by its uniform action.

Thus may the chief circumstances of this motion be determined by means of the hyperbola, the ordinates and abscissae exhibiting the relations of the times and velocities, and the areas exhibiting the relations of both motion to the spaces described. But we may render the conception of these circumstances infinitely more easy and simple, by expressing them all by lines, instead of this combination of lines and surfaces. We shall accomplish this purpose by constructing another curve LKP, having the line ML, parallel to OD for its abscissa, and of such a nature, that if the ordinates to the hyperbola be produced till they cut this curve in and the abscissa in the ordinates may be proportional to the hyperbolic areas $eACx,\; fAeg,\; \backslash partial\; AK$. Let us examine what kind of curve this will be.

Make $OC\; :\; Ox\; =\; Ox\; :\; Og$; then Hamilton's Conicæ, IV. 14. Cor.), the areas $ACex,\; exgf$ are equal: therefore drawing $ps,\; nt$ perpendicular to $OM$, we shall have (by the assumed nature of the curve LKP), $Ms\; =\; st$; and if the abscissa $OD$ be divided into any number of small parts in geometrical progression (reckoning the commencement of them all from $O$), the axis $Vi$ of this curve will be divided by its ordinates into the same number of equal parts; and this curve will have its ordinates in geometrical progression, and its abscissae in arithmetical progression.

Also, let $KN,\; MV$ touch the curve in $K$ and $L$, and let $OC$ be supposed to be to $Oe$, as $OD$ to $Od$, and therefore $Ce$ to $Dd$ as $OC$ to $OD$; and let these lines $Ce,\; Dd$ be indefinitely small; then (by the nature of the curve) $Lo$ is equal to $Kr$: for the areas $aACe,\; bBDd$ are in this case equal. Also $ko$ is to $kr$, as $LM$ to $KI$, because $eC\; :\; dD\; =\; CO\; :\; DO$.

Therefore $IN\; :\; IK\; =\; rK\; :\; rk$
$IK\; :\; ML\; =\; rk\; :\; ol$
$ML\; :\; MV\; =\; ol\; :\; ol$
and $IN\; :\; MN\; =\; rK\; :\; oL$.

That is, the subtangent $IN$, or $MV$, is of the same magnitude, or is a constant quantity in every part of the curve.

Lastly, the subtangent $IN$, corresponding to the point $K$ of the curve, is to the ordinate $K\backslash partial$ as the rectangle $BDOG$ or $ACOF$ to the parabolic area $BDCA$.

For let $fg\; hn$ be an ordinate very near to $BD\; \backslash partial\; K$; and let $hn$ cut the curve in $n$, and the ordinate $KI$ in $q$; then we have

$Kq\; :\; qn\; =\; KI\; :\; IN$, or
$Dg\; :\; qn\; =\; DO\; :\; IN$;
but $BD\; :\; AC\; =\; CO\; :\; DO$;
therefore $BD\; :\; Dg\; :\; AC\; :\; qn\; =\; CO\; :\; IN$.

Therefore the sum of all the rectangles $BD.Dg$ is to the sum of all the rectangles $AC.qn$, as $CO$ to $IN$, but

but the sum of the rectangles $BD\; \backslash cdot\; Dg$ is the space $ACDB$; and, because $AC$ is given, the sum of the rectangles $AC\; \backslash cdot\; qn$ is the rectangle of $AC$ and the sum of all the lines $qn$; that is, the rectangle of $AC$ and $RL$: therefore the space $ACDB\; :\; AC\; \backslash cdot\; RL\; =\; CO\; :\; IN$, and $ACDB\; \backslash times\; IN\; =\; AC\; \backslash cdot\; CO\; \backslash cdot\; RL$; and therefore $IN\; :\; RL\; =\; AC\; \backslash cdot\; CO\; :\; ACDB$.

Hence it follows that $QL$ expresses the area $BVA$, and in general, that the part of the line parallel to $OM$, which lies between the tangent $KN$ and the curve $L\backslash rho\; K$, expresses the corresponding area of the hyperbola which lies without the rectangle $BDOG$.

And now, by the help of this curve, we have an easy way of convincing and computing the motion of a body through the air. For the subtangent of our curve now represents twice the height through which the ball must fall in vacuo, in order to acquire the terminal velocity; and therefore serves for a scale on which to measure all the other representatives of the motion.

56
The whole reduced to a simple arithmetical computation.

But it remains to make another observation on the curve $L\backslash rho\; K$, which will save us all the trouble of graphical operations, and reduce the whole to a very simple arithmetical computation. It is of such a nature, that when $MI$ is considered as the abscissa, and is divided into a number of equal parts, and ordinates are drawn from the points of division, the ordinates are a series of lines in geometrical progression, or are continual proportionals. Whatever is the ratio between the first and second ordinate, there is the same between the second and third, between the third and fourth, and so on; therefore the number of parts into which the abscissa is divided is the number of these equal ratios which is contained in the ratio of the first ordinate to the last: For this reason, this curve has got the name of the logistic or logarithmic curve; and it is of immense use in the modern mathematics, giving us the solution of many problems in the most simple and expeditious manner, on which the genius of the ancient mathematicians had been exercised in vain. Few of our readers are ignorant, that the numbers called logarithms are of equal utility in arithmetical operations, enabling us not only to solve common arithmetical problems with astonishing dispatch, but also to solve others which are quite inaccessible in any other way. Logarithms are nothing more than the numerical measures of the abscissa of this curve, corresponding to ordinates, which are measured on the same or any other scale by the natural numbers; that is, if $MI$ be divided into equal parts, and from the points of division lines be drawn parallel to $MI$, cutting the curve $L\backslash rho\; K$, and from the points of intersection ordinates be drawn to $MI$, these will divide $MI$ into portions, which are in the same proportion to the ordinates that the logarithms bear to their natural numbers.

In constructing this curve we were limited to no particular length of the line $LR$, which represented the space $ACDB$; and all that we had to take care of was, that when $OC,\; O\backslash pi,\; Oq$ were taken in geometrical progression, $Mx,\; Mt$ should be in arithmetical progression. The abscissæ having ordinates equal to might have been twice as long, as is shown in the dotted curve which is drawn through $L$. All the lines which serve to measure the hyperbolic spaces would then have been doubled. But $NI$ would also have been doubled, and

our proportions would have still held good; because this subtangent is the scale of measurement of our figure, as $E$ or $2a$ is the scale of measurement for the motions.

Since then we have tables of logarithms calculated for every number, we may make use of them instead of this geometrical figure, which still requires considerable trouble to suit it to every case. There are two sets of logarithmic tables in common use. One is called a table of hyperbolic or natural logarithms. It is suited to such a curve as is drawn in the figure, where the subtangent is equal to that ordinate $\backslash pi\; v$ which corresponds to the side $\backslash pi\; O$ of the square $\backslash pi\; O\; \backslash lambda\; O$ inserted between the hyperbola and its asymptotes. This square is the unit of surface, by which the hyperbolic areas are expressed; its side is the unit of length, by which the lines belonging to the hyperbola are expressed; $\backslash pi\; v$ is $=\; 1$, or the unit of numbers to which the logarithms are suited, and then $IN$ is also $1$. Now the square $\backslash pi\; O\; \backslash lambda\; O$ being unity, the area $BACD$ will be some number; $\backslash pi\; O$ being also unity, $OD$ is some number: Call it $x$. Then, by the nature of the hyperbola, $OB\; :\; O\backslash pi\; =\; \backslash pi\; O\; :\; DB$: That is, $x\; :\; 1\; =\; 1\; :\; \backslash frac\{1\}\{x\}$, so that $DB$ is $\backslash frac\{1\}\{x\}$.

Now calling $Dd\text{'}x$, the area $BD\; db$, which is the fluxion (ultimately) of the hyperbolic area, is $\backslash frac\{x\}\{x\}$. Now

in the curve $L\backslash rho\; K$, $MI$ has the same ratio to $NI$ that $BACD$ has to $\backslash pi\; O\; \backslash lambda\; O$: Therefore, if there be a scale of which $NI$ is the unit, the number on this scale corresponding to $MI$ has the same ratio to $1$ which the number measuring $BACD$ has to $1$; and $1$ is, which corresponds to $BD\; db$, is the fluxion (ultimately) of $MI$: Therefore, if $MI$ be called the logarithm of $x$, $\backslash frac\{x\}\{x\}$ is properly represented by the fluxion of $MI$. In

short, the line $MI$ is divided precisely as the line of numbers on a Gunter's scale, which is therefore a line of logarithms; and the numbers called logarithms are just the lengths of the different parts of this line measured on a scale of equal parts. Therefore, when we meet with such an expression as $\backslash frac\{x\}\{x\}$ viz. the fluxion

of a quantity divided by the quantity itself, we consider it as the fluxion of the logarithm of that quantity, because it is really so when the quantity is a number; and it is therefore strictly true that the fluent of $\backslash frac\{x\}\{x\}$ is the hyperbolic logarithm of $x$.

Certain reasons of convenience have given rise to another set of logarithms; these are suited to a logistic curve whose subtangent is only $\backslash frac\{1\}\{2\}\; \backslash pi\; v$ of the ordinate $\backslash pi\; v$, which is equal to the side of the hyperbolic square, and which is assumed for the unit of number. We shall suit our applications of the preceding investigation to both these, and shall first use the common logarithms whose subtangent is $0.43429$.

The whole subject will be best illustrated by taking an example of the different questions which may be proposed.

Recollect that the rectangle $ACOF$ is $=\; 2a$, or $\backslash frac\{u^2\}\{g\}$, or $F$.

$E$, for a ball of cast-iron one inch diameter, and if it has the diameter $d$, it is $\backslash frac\{u^2\; d\}\{g\}$, or $2ad$, or $Ed$.

I. It may be required to determine what will be the space described in a given time $t$ by a ball setting out with a given velocity $V$, and what will be its velocity $v$ at the end of that time.

Here we have $NI\; :\; MI\; =\; ACOF\; :\; BDCA$; now $NI$ is the subtangent of the logilic curve; $MI$ is the difference between the logarithms of $OD$ and $OC$; that is, the difference between the logarithms of $e+t$ and $e$; $ACOF$ is $2ad$, or $\backslash frac\{u^2\; d\}\{g\}$, or $Ed$.

Therefore by common logarithms $0.43429\; :\; \backslash log.\; \backslash frac\{e+t\}\{e\}\; =\; \backslash log.\; e\; =\; 2ad\; :\; S$, $S$ = space described,

or $0.43429\; :\; \backslash log.\; \backslash frac\{e+t\}\{e\}\; =\; 2ad\; :\; S$,

and $S\; =\; \backslash frac\{2ad\}\{0.43429\}\; \backslash times\; \backslash log.\; \backslash frac\{e+t\}\{e\}$,

by hyperbolic logarithms $S\; =\; 2ad\; \backslash times\; \backslash log.\; \backslash frac\{e+t\}\{e\}$.

Let the ball be a 12 pounder, and the initial velocity be 1600 feet, and the time 20 seconds. We must first find $e$, which is $\backslash frac\{2ad\}\{V\}$.

This must be considered as a common number by which we are to multiply $\backslash frac\{2ad\}\{0.43429\}$.

For the final velocity,

$OD\; :\; OC\; =\; AC\; :\; BD$, or $e+t\; :\; e\; =\; V\; :\; v$.

$23\text{'}\text{'}03\; :\; 3\text{'}\text{'}03\; =\; 1600\; :\; 210\backslash frac\{1\}\{2\}\; =\; v$.

The ball has therefore gone 3278 yards, and its velocity is reduced from 1600 to 210.

It may be agreeable to the reader to see the gradual progress of the ball during some seconds of its motion.

The first column is the time of the motion, the second is the space described, the third is the differences of the

spaces, showing the motion during each successive second; the fourth column is the velocity at the end of the time $t$; and the last column is the differences of velocity, showing its diminution in each successive second. We see that at the distance of 1000 yards the velocity is reduced to one half, and at the distance of less than a mile it is reduced to one-third.

II. It may be required to determine the distance at which the initial velocity $V$ is reduced to any other quantity $v$. This question is solved in the very same manner, by substituting the logarithms of $V$ and $v$ for those of $e+t$ and $e$; for $AC\; :\; BD\; =\; OD\; :\; OC$, and therefore $\backslash log.\; \backslash frac\{AC\}\{BD\}\; =\; \backslash log.\; \backslash frac\{OD\}\{OC\}$, or $\backslash log.\; \backslash frac\{V\}\{v\}\; =\; \backslash log.\; \backslash frac\{e+t\}\{e\}$.

Thus it is required to determine the distance in which the velocity 1780 of a 24 pound ball (which is the medium velocity of such a ball discharged with 16 pounds of powder) will be reduced to 1500.

Here $d$ is 5.68, and therefore the logarithm of $2ad$ is

This reduction will be produced in about $\backslash frac\{1\}\{2\}$ of a second.

III. Another question may be to determine the time which a ball, beginning to move with a certain velocity, employs in passing over a given space, and the diminution of velocity which it sustains from the resistance of the air.

We may proceed thus:

$2ad\; :\; S\; =\; 0.43429\; :\; \backslash log.\; \backslash frac\{e+t\}\{e\}\; =\; t$. Then to log. $\backslash frac\{e+t\}\{e\}$ add log. $e$, and we obtain log. $e+t$, and $e+t$; from which if we take $e$ we have $t$. Then to find $v$, say $e+t\; :\; e\; =\; V\; :\; v$.

We shall conclude these examples by applying this last rule to Mr Robins's experiment on a musket bullet of an experiment of $\backslash frac\{1\}\{4\}$ of an inch in diameter, which had its velocity reduced from 1670 to 1425 by passing through 100 feet of air. This we do in order to discover the resistance which it sustained, and compare it with the resistance to a velocity of 1 foot per second.

We must first ascertain the first term of our analogy. The ball was of lead, and therefore $2a$ must be multiplied by $d$ and by $m$, which expresses the ratio of the density of lead to that of cast-iron. $d$ is 0.75, and $m$ is $\backslash frac\{11.37\}\{7.21\}\; =\; 1.577$. Therefore $\backslash log.\; 2a$

and $2adm\; =\; 1274.2$.

Now $1274.2\; :\; 100\; =\; 0.43429\; :\; 0.03408\; =\; \backslash log.\; \backslash frac\{e+t\}\{e\}$.

But $e\; =\; \backslash frac\{2ad\}\{V\}\; =\; 0.763$, and its logarithm = 9.88252,

which, added to 0.03408, gives 9.91660, which is the log. of $e+t$, = 0.825, from which take $e$, and there remains

remains $t\; =\; 0\text{'}\text{'}.62$, or $\backslash frac\{62\}\{1000\}$ of a second, for the time of passage. Now, to find the remaining velocity, say $825\; :\; .763\; =\; 1670\; :\; 1544,\; =\; v$.

But in Mr Robins's experiment the remaining velocity was only 1425, the ball having lost 245; whereas by this computation it should have lost only 126. It appears, therefore, that the resistance is double of what it would have been if the resistance increased in the duplicate proportion of the velocity. Mr Robins says it is nearly triple. But he supposes the resistance to flow motions much smaller than his own experiment, so often mentioned, fully warrants.

The time $e$, in which the resistance of the air would extinguish the velocity is $0\text{'}\text{'}.763$. Gravity, or the weight of the bullet, would have done it in $\backslash frac\{1670\}\{32\}$ or $52\text{'}\text{'}$;

therefore the resistance is $\backslash frac\{52\}\{0.763\}$ times, or nearly 68 times its weight, by this theory, or 5.97 pounds. If we calculate from Mr Robins's experiment, we must say $\backslash log\; \backslash frac\{V\}\{v\}\; =\; 0.43429\; =\; 100\; :\; eV$, which will be 630.23, and

$e\; =\; \backslash frac\{630.23\}\{1670\}\; =\; 0\text{'}\text{'}.3774$, and $\backslash frac\{52\}\{0.3774\}$ gives 138 for the proportion of the resistance to the weight, and makes the resistance 12.07 pounds, fully double of the other.

It is to be observed, that with this velocity, which greatly exceeds that with which the air can rush into a void, there must be a statical pressure of the atmosphere equal to 64 pounds. This will make up the difference, and allows us to conclude that the resistance arising solely from the motion communicated to the air follows very nearly the duplicate proportion of the velocity.

The next experiment, with a velocity of 1690 feet, gives a resistance equal to 157 times the weight of the bullet, and this bears a much greater proportion to the former than $1690^2$ does to $1670^2$, which shows, that although these experiments clearly demonstrate a prodigious augmentation of resistance, yet they are by no means susceptible of the precision which is necessary for discovering the law of this augmentation, or for a good foundation of practical rules; and it is still greatly to be wished that a more accurate mode of investigation could be discovered.

Thus we have explained, in great detail, the principles and the process of calculation for the simple case of the motion of projectiles through the air. The learned reader will think that we have been unreasonably prolix, and that the whole might have been comprised in less room, by taking the algebraic method. We acknowledge that it might have been done even in a few lines. But we have observed, and our observation has been confirmed by persons well versed in such subjects, that in all cases where the fluxionary process introduces the fluxion of a logarithm, there is a great want of distinct ideas to accompany the hand and eye. The solution comes out by a sort of magic or legerdemain, we cannot tell either how or why. We therefore thought it our duty to furnish the reader with distinct conceptions of the things and quantities treated of. For this reason, after showing, in Sir Isaac Newton's manner, how the spaces described in the retarded motion of a projectile

followed the proportion of the hyperbolic areas, we shewed the nature of another curve, where lines could be found which increase in the very same manner as the path of the projectile increases; so that a point describing the abscissa MI of this curve moves precisely as the projectile does. Then, discovering that this line is the same with the line of logarithms on a Gunter's scale, we shewed how the logarithm of a number really represents the path or space described by the projectile.

Having thus, we hope, enabled the reader to conceive distinctly the quantities employed, we shall leave the geometrical method, and prosecute the rest of the subject in a more compendious manner.

We are, in the next place, to consider the perpendicular ascents and descents of heavy projectiles, where the resistance of the air is combined with the action of gravity: and we shall begin with the descents.

Let $u$, as before, be the terminal velocity, and $g$ the accelerating power of gravity: When the body moves with the velocity $u$, the resistance is equal to $g$; and in every other velocity $v$, we must have $u^2\; :\; v^2\; =\; g$:

$\backslash frac\{g\; v^2\}\{u^2\}\; =\; r$, for the resistance to that velocity. In the descent the body is urged by gravity $g$, and opposed by the resistance $\backslash frac\{g\; v^2\}\{u^2\}$: therefore the remaining ac-

celerating force, which we shall call $f$, is $g\; -\; \backslash frac\{g\; v^2\}\{u^2\}$, or $\backslash frac\{g\; u^2\; -\; g\; v^2\}\{u^2\}$, or $\backslash frac\{g(u^2\; -\; v^2)\}\{u^2\}\; =\; f$.

Now the fundamental theorem for varied motions is $f\; \backslash dot\{s\}\; =\; u\; \backslash dot\{v\}$, and $\backslash dot\{s\}\; =\; \backslash frac\{v\; \backslash dot\{v\}\}\{f\}\; =\; \backslash frac\{u^2\}\{g\}\; \backslash times\; \backslash frac\{v\; \backslash dot\{v\}\}\{u^2\; -\; v^2\}$, and $s\; =$

$\backslash frac\{u^2\}\{g\}\; \backslash times\; \backslash int\; \backslash frac\{v\; \backslash dot\{v\}\}\{u^2\; -\; v^2\}\; +\; C$. Now the fluent of $\backslash frac\{v\; \backslash dot\{v\}\}\{u^2\; -\; v^2\}$ is $=\; -\backslash text\{hyperb.\; log.\; of\; \}\; \backslash sqrt\{u^2\; -\; v^2\}$. For the fluxion of $\backslash sqrt\{u^2\; -\; v^2\}$ is $\backslash frac\{v\; \backslash dot\{v\}\}\{\backslash sqrt\{u^2\; -\; v^2\}\}$, and this divided by the quantity $\backslash sqrt\{u^2\; -\; v^2\}$, of which it is the fluxion, gives precisely $\backslash frac\{v\; \backslash dot\{v\}\}\{u^2\; -\; v^2\}$, which is therefore the fluxion of

its hyperbolic logarithm. Therefore $S\; =\; -\backslash frac\{u^2\}\{g\}\; \backslash times\; L\; \backslash sqrt\{u^2\; -\; v^2\}\; +\; C$. Where $L$ means the hyperbolic logarithm of the quantity annexed to it, and $\backslash lambda$ may be used to express its common logarithm. (See article FLUXIONS.)

The constant quantity $C$ for completing the fluent is determined from this consideration, that the space described is $s$, when the velocity is $0$: therefore $C\; =\; \backslash frac\{u^2\}\{g\}\; \backslash times\; L\; \backslash sqrt\{u^2\}\; =\; s$, and $C\; =\; \backslash frac\{u^2\}\{g\}\; \backslash times\; L\; \backslash sqrt\{u^2\}$, and the complete fluent $S\; =\; \backslash frac\{u^2\}\{g\}\; \backslash times\; L\; \backslash sqrt\{u^2\}\; -\; L\; \backslash sqrt\{u^2\; -\; v^2\}$, $=\; \backslash frac\{u^2\}\{g\}\; \backslash times\; L\; \backslash sqrt\{\backslash frac\{u^2\}\{u^2\; -\; v^2\}\}\; =\; \backslash frac\{u^2\}\{0.43429\; g\}\; \backslash times\; \backslash lambda\; \backslash sqrt\{\backslash frac\{u^2\}\{u^2\; -\; v^2\}\}$, or (putting $M$ for $0.43429$, the modulus or subtangent of the common logitistic curve) $=\; \backslash frac\{u^2\}\{M\; g\}\; \backslash times\; \backslash lambda\; \backslash sqrt\{\backslash frac\{u^2\}\{u^2\; -\; v^2\}\}$. This

This equation establishes the relation between the space fallen through, and the velocity acquired by the fall. We obtain by it $\backslash frac\{2gS\}\{u^2\}\; =\; L\; \backslash sqrt\{\backslash frac\{u^2\}\{u^2\; -\; v^2\}\}$, and $\backslash frac\{2gS\}\{u^2\}\; =\; L\; \backslash cdot\; \backslash frac\{u^2\}\{u^2\; -\; v^2\}$, or, which is still more convenient for us, $\backslash frac\{M\; \backslash times\; 2gS\}\{u^2\}\; =\; \backslash lambda\; \backslash frac\{u^2\}\{u^2\; -\; v^2\}$, that is, equal to the logarithm of a certain number: therefore having found the natural number corresponding to the fraction $\backslash frac\{M\; \backslash times\; 2gS\}\{u^2\}$, consider it as a logarithm, and take out the number corresponding to it: call this $n$. Then, since $n$ is equal to $\backslash frac\{u^2\}\{u^2\; -\; v^2\}$, we have $nu^2\; =\; u^2$, and $nv^2\; =\; u^2\; -\; u^2$, or $nv^2\; =\; u^2\; \backslash times\; n\; -\; u^2$, and $v^2\; =\; \backslash frac\{u^2\; \backslash times\; n\; -\; 1\}\{n\}$.

To expedite all the computations on this subject, it will be convenient to have multipliers ready computed for $M\; \backslash times\; 2g$, and its half,

viz. 27.794, whose log. is 1.44396
and 13.897 - - - - - 1.14293

But $v$ may be found much more expeditiously by observing that $\backslash sqrt\{\backslash frac\{u^2\}\{u^2\; -\; v^2\}\}$ is the secant of an arch of a circle whose radius is $u$, and whose fine is $v$, or whose radius is unity and fine $=\; \backslash frac\{v\}\{u\}$: therefore, considering the above fraction as a logarithmic secant, look for it in the tables, and then take the fine of the arc of which this is the secant, and multiply it by $u$; the product is the velocity required.

We shall take an example of a ball whose terminal velocity is 689$\backslash frac\{1\}\{2\}$ feet, and ascertain its velocity after a fall of 1848 feet. Here,

Log. of 0.10809 = log. $n$ - 9.03378
0.10809 is the logarithm of 1.2826 = $n$, and $n\; -\; 1\; =$
0.2826, and $\backslash frac\{u^2\; \backslash times\; n\; -\; 1\}\{n\}\; =\; 323.6^2\; =\; v^2$, and $v\; =$
323.6.

In like manner, 0.054045 (which is half of 0.10809) will be found to be the logarithmic secant of 28°, whose fine 0.46947 multiplied by 689$\backslash frac\{1\}\{2\}$ gives 324 for the velocity.

The process of this solution suggests a very perspicuous manner of conceiving the law of descent; and it may be thus expressed:

$M$ is to the logarithm of the secant of an arch whose fine is $\backslash frac\{v\}\{u\}$, and radius 1, as $2a$ is to the height through which the body must fall in order to acquire the velocity $v$. Thus, to take the same example.

1. Let the height $h$ be sought which will produce the velocity 323.62, the terminal velocity of the ball being 689.44. Here $2a$, or $\backslash frac\{u^2\}\{g\}$ is 14850, and $\backslash frac\{323.62\}\{689.44\}\; =$

0.46947, which is the fine of 28°. The logarithmic secant of this arch is 0.05407. Now $M$ or 0.43429: 0.05407 = 14850 : 1848, the height wanted.

2. Required the velocity acquired by the body by falling 1848 feet. Say 14850 : 1848 = 0.43429 : 0.05407. Look for this number among the logarithmic secants. It will be found at 28°, of which the logarithmic fine is - - - - - 9.67161

Add to this the log. of $u$ - - - - - 2.83844

The sum - - - - - 2.51005
is the logarithm of 323.62, the velocity required.

We may observe, from these solutions, that the acquired velocity continually approaches to, but never equals, the terminal velocity. For it is always expressed by the fine of an arch of which the terminal velocity is the radius. We cannot help taking notice here of a very strange assertion of Mr Muller, late professor of mathematics and director of the royal academy at Mr Muller's Woolwich. He maintains, in his Treatise on Gunnery, his Treatise of Fluxions, and in many of his numerous works, that a body cannot possibly move through the air with a greater velocity than this; and he makes this a fundamental principle, on which he establishes a theory of motion in a resisting medium, which he asserts with great confidence to be the only just theory; saying, that all the investigations of Bernoulli, Euler, Robins, Simpson, and others, are erroneous. We use this strong expression, because, in his criticisms on the works of those celebrated mathematicians, he lays aside good manners, and taxes them not only with ignorance, but with dishonesty; saying, for instance, that it required no small dexterity in Robins to confirm by his experiments a theory founded on false principles; and that Thomas Simpson, in attempting to conceal his obligations to him for some valuable propositions, by changing their form, had ignorantly fallen into gross errors.

Nothing can be more palpably absurd than this assertion of Mr Muller. A blown bladder will have but a small terminal velocity; and when moving with this velocity, or one very near it, there can be no doubt that it will be made to move much faster by a smart stroke. Were the assertion true, it would be impossible for a portion of air to be put into motion through the rest, for its terminal velocity is nothing. Yet this author makes this assertion a principle of argument, saying, that it is impossible that a ball can issue from the mouth of a cannon with a greater velocity than this; and that Robins and others are grossly mistaken, when they give them velocities three or four times greater, and resistances which are 10 or 20 times greater than is possible; and by thus compensating his small velocities by still smaller resistances, he confirms his theory by many experiments adduced in support of the others. No reason whatever can be given for the assertion. Newton, or perhaps Huygens, was the first who observed that there was a limit to the velocity which gravity could communicate to a body; and this limit was found by his commentators to be a term to which it was vastly convenient to refer all its other motions. It therefore became

became an object of attention; and Mr Muller, through inadvertency, or want of discernment, has fallen into this mistake, and with that arrogance and self-conceit which mark all his writings, has made this mistake a fundamental principle, because it led him to establish a novel set of doctrines on this subject. He was fretted at the superior knowledge and talents of Mr Simpson, his inferior in the academy, and was guilty of several mean attempts to hurt his reputation. But they were unsuccessful.

We might proceed to consider the motion of a body projected downwards. While the velocity of projection is less than the terminal velocity, the motion is determined by what we have already said: for we must compute the height necessary for acquiring this velocity in the air, and suppose the motion to have begun there. But if the velocity of projection be greater, this method fails. We pass it over (though not in the least more difficult than what has gone before), because it is of mere curiosity, and never occurs in any interesting case. We may just observe, that since the motion is swifter than the terminal velocity, the resistance must be greater than the weight, and the motion will be retarded. The very same process will give us for the space described

$$S\; =\; \backslash frac\{u^2\}\{g\}\; \backslash times\; L\; \backslash sqrt\{\backslash frac\{V^2\; -\; u^2\}\{V^2\; -\; u^2\}\},$$ V being the velocity of projection, greater than u. Now as this space evidently increases continually (because the body always falls), but does not become infinite in any finite time, the fraction $\backslash frac\{V^2\; -\; u^2\}\{V^2\; -\; u^2\}$ does not become infinite; that is, $V^2$ does not become equal to $u^2$: therefore although the velocity V is continually diminished, it never becomes so small as u. Therefore u is a limit of diminution as well as of augmentation.

We must now ascertain the relation between the time of the descent and the space described, or the velocity acquired. For this purpose we may use the other fundamental proposition of varied motions $f/i\; =\; v$, which, in the present case, becomes $\backslash frac\{f\; u^2\; -\; v^2\}\{u^2\}\; i\; =\; v$; therefore $i\; =$

$$\backslash frac\{u^2\}\{g\}\; \backslash times\; \backslash frac\{v\}\{u^2\; -\; v^2\},\; =\; \backslash frac\{u\}\{g\}\; \backslash times\; \backslash frac\{u\; v\}\{u^2\; -\; v^2\},\; \backslash text\{\; and\; \}\; t\; =\; \backslash frac\{u\}\{g\}\; \backslash times\; \backslash int\; \backslash frac\{u\; v\}\{u^2\; -\; v^2\}.$$ Now (art. FLUXIONS) $\backslash int\; \backslash frac\{u\; v\}\{u^2\; -\; v^2\}\; =\; L\; \backslash sqrt\{\backslash frac\{u+v\}\{u-v\}\}$. Therefore $t\; =\; \backslash frac\{u\}\{g\}\; \backslash times\; L\; \backslash sqrt\{\backslash frac\{u+v\}\{u-v\}\}\; =\; \backslash frac\{u\}\{Mg\}\; \backslash times\; \backslash lambda\; \backslash sqrt\{\backslash frac\{u+v\}\{u-v\}\}$. This fluent needs no constant quantity to complete it, or rather $C=0$; for $t$ must be $=0$ when $v=0$. This will evidently be the case: for then $L\; \backslash sqrt\{\backslash frac\{u+v\}\{u-v\}\}$ is $L\; \backslash sqrt\{\backslash frac\{u\}\{u\}\}$, $=\; L,\; =\; 0$.

But how does this quantity $\backslash frac\{u\}\{Mg\}\; \backslash times\; \backslash lambda\; \backslash sqrt\{\backslash frac\{u+v\}\{u-v\}\}$ signify a time? Observe, that in whatever numbers, or by whatever units of space and time, u and g are expressed, $\backslash frac\{u\}\{g\}$ expresses the number of units of time in which the velocity u is communicated or extinguished by gravity;

and $L\; \backslash sqrt\{\backslash frac\{u+v\}\{u-v\}\}$, or $\backslash frac\{\backslash lambda\}\{M\}\; \backslash sqrt\{\backslash frac\{u+v\}\{u-v\}\}$, is always an abstract number, multiplying this time.

We may illustrate this rule by the same example. In what time will the body acquire the velocity 323.62? Here $u+v\; =\; 1012.96$, $u-v\; =\; 365.72$; therefore $\backslash lambda\; \backslash sqrt\{\backslash frac\{u+v\}\{u-v\}\}\; =\; 0.22122$, and $\backslash frac\{u\}\{g\}$ (in feet and seconds) is 21", 542. Now, for greater perspicuity, convert the equation $t\; =\; \backslash frac\{u\}\{Mg\}\; \backslash times\; \backslash lambda\; \backslash sqrt\{\backslash frac\{u+v\}\{u-v\}\}$ into a proportion: thus $M\; :\; \backslash lambda\; \backslash sqrt\{\backslash frac\{u+v\}\{u-v\}\}\; =\; \backslash frac\{u\}\{g\}\; :\; t$, and we have 0.43429 : 0.22122 = 21", 542 : 10", 973, the time required.

This is by far the most distinct way of conceiving the subject; and we should always keep in mind that the numbers or symbols which we call logarithms are really parts of the line MI in the figure of the logistic curve, and that the motion of a point in this line is precisely similar to that of the body. The Marquis Poleni, in a dissertation published at Padua in 1725, has with great ingenuity constructed logarithms suited to all the cases which can occur. Herman, in his Phoronomia, has borrowed much of Poleni's methods, but has obscured them by an affectation of language geometrically precise, but involving the very obscure notion of abstract ratios.

It is easy to see that $\backslash sqrt\{\backslash frac\{u+v\}\{u-v\}\}$ is the cotangent of the $\backslash frac\{v\}\{u\}$ complement of an arch, whose radius is 1, and whose fine is $\backslash frac\{v\}\{u\}$: For let KC (fig. 6.) be $=u$, and

$BE\; =\; v$; then $KD\; =\; u+v$, and $DA\; =\; u-v$. Join KB and BA, and draw CG parallel to KB. Now GA is the tangent of $\backslash frac\{v\}\{u\}$ BA, $=\; \backslash frac\{v\}\{u\}$ complement of HB. Then, by similarity of triangles, $GA\; :\; AC\; =\; AB\; :\; BK\; =\; \backslash sqrt\{AD\}\; :\; \backslash sqrt\{DK\}\; =\; \backslash sqrt\{u-v\}\; :\; \backslash sqrt\{u+v\}$ and $\backslash frac\{AC\}\{GA\}\; (=\; \backslash cotan.$

$\backslash frac\{v\}\{u\}$ BA) $=\; \backslash sqrt\{\backslash frac\{u+v\}\{u-v\}\}$; therefore look for $\backslash frac\{v\}\{u\}$ among the natural fines, or for $\backslash frac\{v\}\{u\}$ among the logarithmic fines, and take the logarithmic cotangent of the half complement of the corresponding arch. This, considered as a common number, will be the second term of our proportion. This is a shorter process than the former.

By reversing this proportion we get the velocity corresponding to a given time.

To compare this descent of 1848 feet in the air with the fall of the body in vacuo during the same body in time, say $21",\; 542\; :\; 10",\; 973\; =\; 1848\; :\; 1926.6$, which makes a difference of 79 feet.

Cor. 1. The time in which the body acquires the velocity u by falling through the air, is to the time of acquiring the same velocity by falling in vacuo, as u.

$L\; \backslash sqrt\{\backslash frac\{u+v\}\{u-v\}\}$ to v: for it would acquire this velocity in

vacuo during the time $\backslash frac\{v\}\{g\}$, and it acquires it in the air in the time $\backslash frac\{u\}\{g\}\; \backslash sqrt\{\backslash frac\{u+v\}\{u-v\}\}$.

2. The velocity which the body acquires by falling through the air in the time $\backslash frac\{u\}\{g\}\; \backslash sqrt\{\backslash frac\{u+v\}\{u-v\}\}$, is to the velocity which it would acquire in vacuo during the same time, as $v$ to $u\; \backslash sqrt\{\backslash frac\{u+v\}\{u-v\}\}$: For the velocity which it would acquire in vacuo during the time $\backslash frac\{u\}\{g\}\; \backslash sqrt\{\backslash frac\{u+v\}\{u-v\}\}$ must be $u\; \backslash sqrt\{\backslash frac\{u+v\}\{u-v\}\}$ (because in any time $\backslash frac\{v\}\{g\}$ the velocity $v$ is acquired.)

In the next place, let a body, whose terminal velocity is $u$, be projected perpendicularly upwards, with any velocity $V$. It is required to determine the height to which it ascends, so as to have any remaining velocity $v$, and the time of its ascent; as also the height and time in which its whole motion will be extinguished.

We have now $\backslash frac\{g(u^2+v^2)\}\{u^2\}$ for the expression of $f$; for both gravity and resistance act now in the same direction, and retard the motion of the ascending body: therefore $\backslash frac\{g(u^2+v^2)\}\{u^2\}\; \backslash dot\{s\}\; =\; -v\; \backslash dot\{v\}$, and $\backslash dot\{s\}\; =\; -\backslash frac\{u^2\}\{g\}\; \backslash times\; \backslash frac\{v\; \backslash dot\{v\}\}\{u^2+v^2\}$, and $s\; =\; -\backslash frac\{u^2\}\{g\}\; \backslash times\; \backslash int\; \backslash frac\{v\; \backslash dot\{v\}\}\{u^2+v^2\}\; +\; C$, $=\; -\backslash frac\{u^2\}\{g\}\; \backslash times\; L\; \backslash sqrt\{u^2+v^2\}\; +\; C$ (see art. FLUXIONS). This must be $=0$ at the beginning of the motion, that is, when $v=V$, that is, $-\backslash frac\{u^2\}\{g\}\; \backslash times\; L\; \backslash sqrt\{u^2+V^2\}\; +\; C\; =\; 0$, or $C\; =\; \backslash frac\{u^2\}\{g\}\; \backslash times\; L\; \backslash sqrt\{u^2+V^2\}$, and the complete fluent will be $s\; =\; \backslash frac\{u^2\}\{g\}\; \backslash times\; L\; \backslash sqrt\{u^2+v^2\}\; -\; L\; \backslash sqrt\{u^2+V^2\}\; =\; \backslash frac\{u^2\}\{g\}\; \backslash times\; L\; \backslash left(\; \backslash sqrt\{\backslash frac\{u^2+v^2\}\{u^2+V^2\}\}\; -\; 1\; \backslash right)$.

Let $h$ be the greatest height to which the body will rise. Then $s\; =\; h$ when $v\; =\; 0$; and $h\; =\; \backslash frac\{u^2\}\{g\}\; \backslash times\; L\; \backslash sqrt\{\backslash frac\{u^2+V^2\}\{u^2\}\}\; =\; \backslash frac\{u^2\}\{g\}\; \backslash times\; L\; \backslash sqrt\{\backslash frac\{u^2+V^2\}\{u^2\}\}$. We have $\backslash lambda\; \backslash sqrt\{\backslash frac\{u^2+V^2\}\{u^2\}\}\; =\; s\; \backslash frac\{mg\}\{u^2\}$; therefore $\backslash lambda\; \backslash left(\; \backslash frac\{u^2+V^2\}\{u^2\}\; \backslash right)\; =\; \backslash frac\{2Mgs\}\{u^2\}$. Therefore let $n$ be the number whose common logarithm is $\backslash frac\{2Mgs\}\{u^2\}$; we shall have $n\; =\; \backslash frac\{u^2+V^2\}\{u^2+v^2\}$, and $v^2\; =\; \backslash frac\{u^2+V^2\}\{n\}\; -\; u^2$; and thus we obtain the relation of $s$ and $v$, as in the case of descents: but we obtain it still easier by observing that $\backslash sqrt\{u^2+V^2\}$ is the secant of an arch whose radius is $u$, and whose tangent is $V$, and that $\backslash sqrt\{u^2+v^2\}$ is the secant of another arch of the same circle, whose tangent is $v$.

Let the same ball be projected upwards with the velocity 411.05 feet per second. Required the whole height to which it will rise?

Here $\backslash frac\{V\}\{u\}$ will be found the tangent of 30.48°, the logarithmic secant of which is 0.06606. This, multiplied by $\backslash frac\{u^2\}\{Mg\}$, gives 22.59 feet for the height. It would have risen 26.40 feet in a void.

Suppose this body to fall down again. We can compare the velocity of projection with the velocity of projection with which it again reaches the ground. The ascent and descent are equal: therefore $\backslash sqrt\{\backslash frac\{u^2+V^2\}\{u^2\}\}$, which multiplies the constant factor in the ascent, is equal to

$\backslash sqrt\{\backslash frac\{u^2\}\{u^2-v^2\}\}$, the multiplier in the descent. The first is the secant of an arch whose tangent is $V$; the other is the secant of an arch whose sine is $v$. These secants are equal, or the arches are the same; therefore the velocity of projection is to the final returning velocity as the tangent to the sine, or as the radius to the cosine of the arch. Thus suppose the body projected with the terminal velocity, or $V=u$; then $v\; =\; \backslash frac\{u\}\{\backslash sqrt\{2\}\}$. If $V\; =\; 689,\; v\; =\; 487$.

We must in the last place ascertain the relation of the space and the time.

Here $\backslash frac\{g(u^2+v^2)\}\{u^2\}\; \backslash dot\{s\}\; =\; -\backslash dot\{v\}$, and $\backslash dot\{s\}\; =\; -\backslash frac\{u^2\}\{g\}\; \backslash times\; \backslash frac\{\backslash dot\{v\}\}\{u^2+v^2\}\; =\; -\backslash frac\{u\}\{g\}\; \backslash times\; \backslash frac\{u\; \backslash dot\{v\}\}\{u^2+v^2\}$ and $s\; =\; -\backslash frac\{u\}\{g\}\; \backslash times\; \backslash int\; \backslash frac\{u\; \backslash dot\{v\}\}\{u^2+v^2\}\; +\; C$. Now (art. FLUXIONS) $\backslash int\; \backslash frac\{u\; \backslash dot\{v\}\}\{u^2+v^2\}$ is an arch whose tangent $=\; \backslash frac\{v\}\{u\}$ and radius $1$; therefore $s\; =\; -\backslash frac\{u\}\{g\}\; \backslash times\; \backslash text\{arc.\; tan.\}\; \backslash frac\{v\}\{u\}\; +\; C$. This must be $=0$ when $v=V$, or $C\; =\; \backslash frac\{u\}\{g\}\; \backslash times\; \backslash text\{arc.\; tan.\}\; \backslash frac\{V\}\{u\}\; =\; 0$, and $C\; =\; \backslash frac\{u\}\{g\}\; \backslash times\; \backslash text\{arc.\; tan.\}\; \backslash frac\{V\}\{u\}$, and the complete fluent is $s\; =\; \backslash frac\{u\}\{g\}\; \backslash times\; \backslash left(\; \backslash text\{arc.\; tan.\}\; \backslash frac\{V\}\{u\}\; -\; \backslash text\{arc.\; tan.\}\; \backslash frac\{v\}\{u\}\; \backslash right)$. The quantities within the brackets express a portion of the arch of a circle whose radius is unity; and are therefore abstract numbers, multiplying $\backslash frac\{u\}\{g\}$, which we have shown to be the number of units of time in which a heavy body falls in vacuo from the height $a$, or in which it acquires the velocity $u$.

We learn from this expression of the time, that however great the velocity of projection, and the height to which this body will rise, may be, the time of its ascent is limited. It never can exceed the time of falling from the height $a$ in vacuo in a greater proportion than that of a quadrant arch to the radius, nearly the proportion of 8 to 5. A 24 pound iron ball cannot continue rising above 14 seconds, even if the resistance to quick motions did not increase faster than the square of the velocity. It probably will attain its greatest height in less than 12 seconds, let its velocity be ever so great.

In the preceding example of the whole ascent, $v=0$, and

and the time $t\; =\; \backslash frac\{u\}\{g\}\; \backslash times\; \backslash text\{arc.\; tan.\; \}\; \backslash frac\{V\}\{u\}$, or $\backslash frac\{u\}\{g\}\; \backslash text\{\; arc.\; \}\; 30^\{\backslash circ\}.48\text{'}$. Now $30^\{\backslash circ\}.48\text{'}\; =\; 1848\text{'}$, and the radius $r$ contains 3438; therefore the arch $=\; \backslash frac\{1848\}\{3438\}\; =\; 0.5376$; and $\backslash frac\{u\}\{g\}\; =\; 21\text{'}\text{'}\; .\; 54$. Therefore $t\; =\; 21\text{'}\text{'}\; .\; 54\; \backslash times\; 0.5376\; =\; 11\text{'}\text{'}\; .\; 58$, or nearly $11\backslash frac\{1\}\{2\}$ seconds. The body would have risen to the same height in a void in $10\backslash frac\{1\}\{2\}$ seconds.

68
This time compared in bodies projected in air and in vacuo.

Cor. 1. The time in which a body, projected in the air with any velocity $V$, will attain its greatest height, is to that in which it would attain its greatest height in vacuo, as the arch whose tangent expresses the velocity is to the tangent; for the time of the ascent in the air is $\backslash frac\{u\}\{g\}\; \backslash times\; \backslash text\{arch\}$; the time of the ascent in vacuo is $\backslash frac\{V\}\{g\}$. Now $\backslash frac\{V\}\{g\}$ is $=\; \backslash tan.$ and $V\; =\; u\; \backslash times\; \backslash tan.$ and $\backslash frac\{V\}\{g\}\; =\; \backslash frac\{u\}\{g\}\; \backslash times\; \backslash tan.$

Fig. 6. It is evident, by inspecting fig. 6. that the arch $AI$ is to the tangent $AG$ as the sector $ICA$ to the triangle $GCA$; therefore the time of attaining the greatest height in the air is to that of attaining the greatest height in vacuo (the velocities of projection being the same), as the circular sector to the corresponding triangle.

If therefore a body be projected upwards with the terminal velocity, the time of its ascent will be to the time of acquiring this velocity in vacuo as the area of a circle to the area of the circumscribed square.

2. The height $H$ to which a body will rise in a void, is to the height $h$ to which it would rise through the air when projected with the same velocity $V$ as $M\; \backslash cdot\; V^2$ to

$u^2\; \backslash times\; \backslash lambda\; \backslash frac\{u^2\; +\; V^2\}\{u^2\}$; for the height to which it will rise in vacuo is $\backslash frac\{V^2\}\{2g\}$, and the height to which it rises in the air is

$\backslash frac\{u^2\}\{Mg\}\; \backslash lambda\; \backslash sqrt\{\backslash frac\{u^2\; +\; V^2\}\{u^2\}\}$; therefore $H\; :\; h\; =\; \backslash frac\{V^2\}\{2g\}$;

$\backslash frac\{u^2\}\{Mg\}\; \backslash lambda\; \backslash sqrt\{\backslash frac\{u^2\; +\; V^2\}\{u^2\}\}\; =\; V^2\; :\; \backslash frac\{u^2\}\{M\}\; \backslash times\; \backslash lambda\; \backslash sqrt\{\backslash frac\{u^2\; +\; V^2\}\{u^2\}\}\; =\; V^2\; :$

$\backslash frac\{u^2\}\{M\}\; \backslash times\; \backslash lambda\; \backslash frac\{u^2\; +\; V^2\}\{u^2\}\; =\; M\; \backslash cdot\; V^2\; :\; u^2\; \backslash times\; \backslash lambda\; \backslash frac\{u^2\; +\; V^2\}\{u^2\}.$

Therefore if the body be projected with its terminal velocity, so that $V\; =\; u$, the height to which it will rise in the air is $\backslash frac\{30103\}\{43429\}$ of the height to which it will rise

in vacuo, or $\backslash frac\{5\}\{7\}$ in round numbers.

We have been thus particular in treating of the perpendicular ascents and descents of heavy bodies through the air, in order that the reader may conceive distinctly the quantities which he is thus combining in his algebraic operations, and may see their connection in nature with each other. We shall also find that, in the present state of our mathematical knowledge, this simple state of the case contains almost all that we can determine with any confidence. On this account it were to be wished that the professional gentlemen would make many experiments on these motions. There is no way that promises so much for assisting us in forming accurate no-

tions of the air's resistance. Mr Robins's method with the pendulum is impracticable with great shot; and the experiments which have been generally resorted to for this purpose, viz. the ranges of shot and shells on a horizontal plane, are so complicated in themselves, that the utmost mathematical skill is necessary for making any inferences from them; and they are subject to such irregularities, that they may be brought to support almost any theory whatever on this subject. But the perpendicular flights are affected by nothing but the initial velocity and the resistance of the air; and a considerable deviation from their intended direction does not cause any sensible error in the consequences which we may draw from them for our purpose.

But we must now proceed to the general problem, of 7^o to determine the motion of a body projected in any direction, and with any velocity. Our readers will be jectio-n-lieve beforehand that this must be a difficult subject, when they see the simplest cases of rectilinear motion abundantly abstruse: it is indeed so difficult, that Sir Isaac Newton has not given a solution of it, and has thought himself well employed in making several approximations, in which the fertility of his genius appears in great lustre. In the tenth and subsequent propositions of the second book of the Principia, he shows what state of density in the air will comport with the motion of a body in any curve whatever: and then, by applying this discovery to several curves which have some similarity to the path of a projectile, he finds one which is not very different from what we may suppose to obtain in our atmosphere. But even this approximation was involved in such intricate calculations, that it seemed impossible to make any use of it. In the second edition of the Principia, published in 1713, Newton corrects some mistakes which he had committed in the first, and carries his approximations much farther, but still does not attempt a direct investigation of the path which a body will describe in our atmosphere. This is somewhat surprising. In prop. 14. &c. he shows how a body, actuated by a centripetal force, in a medium of a density varying according to certain laws, will describe an eccentric spiral, of which he assigns the properties, and the law of description. Had he supposed the density constant, and the difference between the greatest and least distances from the centre of centripetal force exceedingly small in comparison with the distances themselves, his spiral would have coincided with the path of a projectile in the air of uniform density, and the steps of his investigation would have led him immediately to the complete solution of the problem. For this is the real state of the case. A heavy body is not acted on by equal and parallel gravity, but by a gravity inversely proportional to the square of the distance from the centre of the earth, and in lines tending to that centre nearly; and it was with the view of simplifying the investigation, that mathematicians have adopted the other hypothesis.

Soon after the publication of this second edition of 7^o the Principia, the dispute about the invention of the among fluxionary calculus became very violent, and the great British and foreign promoters of that calculus upon the continent were in the habit of proposing difficult problems to exercise the talents of the mathematician. Challenges of this kind frequently passed between the British and foreigners.

Dr Keill of Oxford had keenly espoused the claim of Sir Isaac Newton to this invention, and had engaged in a very acrimonious altercation with the celebrated John Bernoulli of Basse. Bernoulli had published in the Acta Eruditorum Lipsiae an investigation of the law of forces, by which a body moving in a resisting medium might describe any proposed curve, reducing the whole to the simplest geometry. This is perhaps the most elegant specimen which he has given of his great talents. Dr Keill proposed to him the particular problem of the trajectory and motion of a body moving through the air, as one of the most difficult. Bernoulli very soon solved the problem in a way much more general than it had been proposed, viz. without any limitation either of the law of resistance, the law of the centripetal force, or the law of density, provided only that they were regular, and capable of being expressed algebraically. Dr Brook Taylor, the celebrated author of the Method of Increments, solved it at the same time, in the limited form in which it was proposed. Other authors since that time have given other solutions. But they are all (as indeed they must be) the same in substance with Bernoulli's. Indeed they are all (Bernoulli's not excepted) the same with Newton's first approximations, modified by the steps introduced into the investigation of the spiral motions mentioned above; and we still think it most strange that Sir Isaac did not perceive that the variation of curvature, which he introduced in that investigation, made the whole difference between his approximations and the complete solution. This we shall point out as we go along. And we now proceed to the problem itself, of which we shall give Bernoulli's solution, restricted to the case of uniform density and a resistance proportional to the square of the velocity. This solution is more simple and perspicuous than any that has since appeared.

PROBLEM. To determine the trajectory, and all the circumstances of the motion of a body projected through the air from A (fig. 7.) in the direction AB, and resisted in the duplicate ratio of the velocity.

Let the arch AM be put $=\backslash infty$, the time of describing it $t$, the abscissa $AP=\backslash infty$, the ordinate $PM=y$. Let the velocity in the point $M=v$, and let $MN=\backslash dot\{x\}$, be described in the moment $t$; let $r$ be the resistance of the air, $g$ the force of gravity, measured by the velocity which it will generate in a second; and let $a$ be the height through which a heavy body must fall in vacuo to acquire the velocity which would render the resistance of the air equal to its gravity: so that we have $r\; =\; \backslash frac\{v^2\}\{2a\}$; because, for any velocity $u$, and producing height $h$, we have $g\; =\; \backslash frac\{u^2\}\{2h\}$.

Let $Mm$ touch the curve in $M$; draw the ordinate $pN$, and draw $Mo$, $Nn$ perpendicular to $Np$ and $Mm$. Then we have $MN=\backslash dot\{x\}$, and $Mo=\backslash infty$, also $mo$ is ultimately $=y$ and $Mm$ is ultimately $=MN$ or $\backslash dot\{x\}$. Lastly, let us suppose $\backslash dot\{x\}$ to be a constant quantity, the elementary ordinates being supposed equidistant.

The action of gravity during the time $t$ may be measured by $mN$, which is half the space which it

would cause the body to describe uniformly in the time $t$ with the velocity which it generates in that time. Let this be resolved into $mN$, by which it deflects the body into a curvilinear path, and $mn$, by which it retards the ascent and accelerates the descent of the body along the tangent. The resistance of the air acts solely in retarding the motion, both in ascending and descending, and has no deflective tendency. The whole action of gravity then is to its accelerating or retarding tendency as $mN$ to $mn$, or (by similarity of triangles) as $mM$ to

$mo$. Or $\backslash dot\{x\}\; :\; \backslash dot\{y\}\; =\; g\; :\; \backslash frac\{r\backslash dot\{y\}\}\{\backslash dot\{x\}\}$, and the whole retardation in the ascent will be $r\; +\; \backslash frac\{g\backslash dot\{y\}\}\{\backslash dot\{x\}\}$. The same fluxionary symbol

will express the retardation during the descent, because in the descent the ordinates decrease, and $\backslash dot\{y\}$ is a negative quantity.

The diminution of velocity is $-\backslash dot\{v\}$. This is proportional to the retarding force and to the time of its action

jointly, and therefore $-\backslash dot\{v\}\; =\; r\; +\; \backslash frac\{g\backslash dot\{y\}\}\{\backslash dot\{x\}\}\; \backslash times\; t$; but the time

$t$ is as the space $\backslash dot\{x\}$ divided by the velocity $v$; therefore $-\backslash dot\{v\}\; =\; r\; +\; \backslash frac\{g\backslash dot\{y\}\}\{\backslash dot\{x\}\}\; \backslash times\; \backslash frac\{\backslash dot\{x\}\}\{v\}\; =\; -\backslash frac\{r\backslash dot\{y\}\; +\; g\backslash dot\{y\}\}\{v\}$, and $-\backslash dot\{v\}\; =\; -\backslash frac\{r\backslash dot\{y\}\; +\; g\backslash dot\{y\}\}\{v\}$.

$r\backslash dot\{x\}\; -\; g\backslash dot\{y\}\; =\; \backslash frac\{v^2\; \backslash dot\{x\}\}\{2a\}\; -\; g\backslash dot\{y\}$. Because $mN$ is the deflection by gravity, it is as the force $g$ and the square of the time $t$ jointly (the momentary action being held as uniform). We have therefore $mN$, or $-\backslash dot\{y\}\; =\; g\backslash dot\{t\}^2$. (Observe that $mN$ is in fact only the half of $-\backslash dot\{y\}$; but $g$ being twice the fall of a heavy body in a second, we have $-\backslash dot\{y\}$ strictly equal to $g\backslash dot\{t\}^2$.) But $\backslash dot\{t\}^2\; =\; \backslash frac\{\backslash dot\{x\}^2\}\{v^2\}$; therefore $-\backslash dot\{y\}\; =\; \backslash frac\{g\backslash dot\{x\}^2\}\{v^2\}$.

and $v^2\; =\; \backslash frac\{g\backslash dot\{x\}^2\}\{-\backslash dot\{y\}\}$, and $-\backslash dot\{v\}\; \backslash dot\{y\}\; =\; g\backslash dot\{x\}^2$. The fluxion of this equation is $-\backslash dot\{v\}\; \backslash dot\{y\}\; -\; 2v\backslash dot\{y\}\backslash dot\{v\}\; =\; 2g\backslash dot\{x\}\backslash dot\{x\}$; but, because $\backslash dot\{x\}\; :\; \backslash dot\{y\}\; =\; mM\; :\; mo\; =\; mN\; :\; mn\; =\; y\; :\; \backslash dot\{x\}$, we have $\backslash dot\{x\}\backslash dot\{x\}\; =\; \backslash dot\{y\}\backslash dot\{y\}$. Therefore $2g\backslash dot\{y\}\backslash dot\{y\}\; =\; 2g\backslash dot\{x\}\backslash dot\{x\}\; =\; -v^2\backslash dot\{y\}\; -\; 2v\backslash dot\{y\}\backslash dot\{v\}$, and $-2v\backslash dot\{y\}\backslash dot\{v\}\; =\; v^2\backslash dot\{y\}$, $-2g\backslash dot\{y\}\backslash dot\{y\}$, and $v\backslash dot\{v\}\; =\; \backslash frac\{v^2\backslash dot\{y\}\}\{2y\}\; -\; g\backslash dot\{y\}$. But we have already $-\backslash dot\{v\}\; =\; -\backslash frac\{r\backslash dot\{y\}\; +\; g\backslash dot\{y\}\}\{v\}$.

$\backslash frac\{v^2\backslash dot\{x\}\}\{2a\}\; -\; g\backslash dot\{y\}$; therefore $\backslash frac\{v^2\backslash dot\{y\}\}\{y\}\; =\; \backslash frac\{v^2\backslash dot\{x\}\}\{a\}$, and finally $\backslash frac\{\backslash dot\{y\}\}\{y\}\; =\; \backslash frac\{\backslash dot\{x\}\}\{a\}$, or $a\backslash dot\{y\}\; =\; \backslash dot\{x\}\backslash dot\{y\}$, for the fluxionary equation of the curve.

If we put this into the form of a proportion, we have $a\; :\; \backslash dot\{x\}\; =\; \backslash dot\{y\}\; :\; \backslash dot\{y\}$. Now this evidently establishes a relation between the length of the curve and its variation of the curvature; and between the curve itself and its evolution, which are the very circumstances introduced by Newton curvature.

75
Relation
between
the length
of the curve
and its varia-
tion of
Newton curvature.

Newton into his investigation of the spiral motions. And the equation $\backslash frac\{\backslash dot\{x\}\}\{a\}\; =\; \backslash frac\{\backslash dot\{y\}\}\{y\}$ is evidently an equation connected with the logarithmic curve and the logarithmic spiral. But we must endeavour to reduce it to a lower order of fluxions, before we can establish a relation between $x$, $x$, and $y$.

Let $\backslash rho$ express the ratio of $\backslash dot\{y\}$ to $\backslash dot\{x\}$, that is, let $\backslash rho$ be $\backslash frac\{\backslash dot\{y\}\}\{\backslash dot\{x\}\}$, or $\backslash rho\; \backslash dot\{x\}\; =\; \backslash dot\{y\}$. It is evident that this expresses the inclination of the tangent at $M$ to the horizon, and that $\backslash rho$ is the tangent of this inclination, radius being unity. Or it may be considered merely as a number, multiplying $\backslash dot\{x\}$, so as to make it $\backslash dot\{y\}$. We now have $y^2\; =\; \backslash rho^2\; x^2$, and since $\backslash dot\{x\}^2\; =\; \backslash dot\{x\}^2\; +\; \backslash dot\{y\}^2$, we have $\backslash dot\{x\}^2\; =\; \backslash dot\{x\}^2\; +\; \backslash rho^2\; \backslash dot\{x\}^2$, $=\; (1\; +\; \backslash rho^2)\; \backslash dot\{x\}^2$, and $\backslash dot\{x\}\; =\; x\; \backslash sqrt\{1\; +\; \backslash rho^2\}$.

Moreover, because we have supposed the abscissa $x$ to increase uniformly, and therefore $\backslash dot\{x\}$ to be constant, we have $\backslash ddot\{x\}\; =\; \backslash dot\{x\}\; \backslash dot\{\backslash rho\}$, and $\backslash dot\{y\}\; =\; \backslash dot\{x\}\; \backslash dot\{\backslash rho\}$. Now let $q$ express the ratio of $\backslash dot\{\backslash rho\}$ to $\backslash dot\{x\}$, that is, make $\backslash frac\{\backslash dot\{\backslash rho\}\}\{\backslash dot\{x\}\}\; =\; q$, or $q\; \backslash dot\{x\}\; =\; \backslash dot\{\backslash rho\}$. This gives us $\backslash dot\{x\}\; \backslash dot\{q\}\; =\; \backslash dot\{\backslash rho\}$, and $\backslash dot\{x\}^2\; \backslash dot\{q\}\; =\; \backslash dot\{x\}\; \backslash dot\{\backslash rho\}\; =\; \backslash dot\{y\}$.

By these substitutions our former equation $a\; \backslash dot\{y\}\; =\; \backslash ddot\{x\}\; \backslash dot\{y\}$ changes to $a\; \backslash dot\{x\}^2\; \backslash dot\{q\}\; =\; \backslash dot\{x\}\; \backslash sqrt\{1\; +\; \backslash rho^2\}\; \backslash dot\{x\}\; \backslash dot\{\backslash rho\}$, or $a\; \backslash dot\{y\}\; =\; \backslash dot\{p\}\; \backslash sqrt\{1\; +\; \backslash rho^2\}$, and, taking the fluent on both sides, we have $a\; q\; =\; \backslash int\; \backslash dot\{p\}\; \backslash sqrt\{1\; +\; \backslash rho^2\}\; +\; C$, $C$ being the constant quantity required for completing the fluent according to the limiting conditions of the case. Now $\backslash dot\{x\}\; =\; \backslash frac\{\backslash rho\}\{q\}$, and $\backslash frac\{\backslash dot\{x\}\}\{q\}\; =$

Also, since $\backslash dot\{y\}\; =\; \backslash rho\; \backslash dot\{x\}$, $=\; \backslash frac\{\backslash rho\; \backslash dot\{p\}\}\{q\}$, we have $y\; =$

Also $\backslash ddot\{x\}\; =\; \backslash dot\{x\}\; \backslash sqrt\{1\; +\; \backslash rho^2\}\; =\; \backslash frac\{a\; \backslash dot\{p\}\; \backslash sqrt\{1\; +\; \backslash rho^2\}\}\{\backslash dot\{p\}\; \backslash sqrt\{1\; +\; \backslash rho^2\}\; +\; C\}$

The values of $\backslash dot\{x\}$, $\backslash dot\{y\}$, $\backslash ddot\{x\}$, give us

The process therefore of describing the trajectory is, $1/\backslash rho$. To find $q$ in terms of $\backslash rho$ by the area of the curve whose abscissa is $\backslash rho$ and the ordinate is $\backslash sqrt\{1\; +\; \backslash rho^2\}$.

2d, We get $x$ by the area of another curve whose abscissa is $\backslash rho$, and the ordinate is $\backslash frac\{1\}\{q\}$.

3d, We get $y$ by the area of a third curve whose abscissa is $\backslash rho$, and the ordinate is $\backslash frac\{\backslash rho\}\{q\}$.

The problem of the trajectory is therefore completely solved, because we have determined the ordinate, abscissa, and arch of the curve for any given position of its tangent. It now only remains to compute the magnitudes of these ordinates and abscissae, or to draw them by a geometrical construction. But in this consists the difficulty. The areas of these curves, which express the lengths of $x$ and $y$, can neither be computed nor exhibited geometrically, by any accurate method yet discovered, and we must content ourselves with approximations. These render the description of the trajectory exceedingly difficult and tedious, so that little advantage has as yet been derived from the knowledge we have got of its properties. It will however greatly assist our conception of the subject to proceed some length in this construction; for it must be acknowledged that very few distinct notions accompany a mere algebraic operation, especially if in any degree complicated, which we confess is the case in the present question.

Let $BmNR$ (fig. 8.) be an equilateral hyperbola, of which $B$ is the vertex, $BA$ the semitransverse axis, which we shall assume for the unity of length. Let $AV$ be the semiconjugate axis $=\; BA$, $=$ unity, and $AS$ the asymptote, bisecting the right angle $BAV$. Let $PN$, $\backslash rho\; n$ be two ordinates to the conjugate axis, exceedingly near to each other. Join $BP$, $AN$, and draw $B\backslash beta$, $N\backslash beta$ perpendicular to the asymptote, and $BC$ parallel to $AP$. It is well known that $BP$ is equal to $NP$. Therefore $PN^2\; =\; BA^2\; +\; AP^2$. Now since $BA\; =\; 1$, if we make $AP\; =\; \backslash rho$ of our formulae, $PN$ is $\backslash sqrt\{1\; +\; \backslash rho^2\}$, and $P\backslash rho$ is $\backslash dot\{\backslash rho\}$, and the area $BAPNB\; =\; \backslash int\; \backslash dot\{p\}\; \backslash sqrt\{1\; +\; \backslash rho^2\}$: That is to say, the number $\backslash int\; \backslash dot\{p\}\; \backslash sqrt\{1\; +\; \backslash rho^2\}$ (for it is a number) has the same proportion to unity of number that the area $BAPNB$ has to $BCVA$, the unit of surface. This area consists of two parts, the triangle $APN$, and the hyperbolic sector $ABN$. $APN\; =\; \backslash frac\{1\}\{2\}\; AP\; \backslash times\; PN\; =\; \backslash frac\{1\}\{2\}\; \backslash rho\; \backslash sqrt\{1\; +\; \backslash rho^2\}$, and the hyperbolic sector $ABN\; =\; BN\; \backslash beta$, which is equivalent to the hyperbolic logarithm of the number represented by $A\backslash beta$ when $A\backslash beta$ is unity. Therefore it is equal to $\backslash frac\{1\}\{2\}$ the logarithm of $\backslash rho\; +\; \backslash sqrt\{1\; +\; \backslash rho^2\}$. Hence we see by the bye that $\backslash int\; \backslash dot\{p\}\; \backslash sqrt\{1\; +\; \backslash rho^2\}\; =\; \backslash frac\{1\}\{2\}\; \backslash rho\; \backslash sqrt\{1\; +\; \backslash rho^2\}\; +\; \backslash frac\{1\}\{2\}$ hyperbolic logarithm $\backslash rho\; +\; \backslash sqrt\{1\; +\; \backslash rho^2\}$.

Now let $AMD$ be another curve, such that its ordinates $Vm$, $PD$, &c. may be proportional to the areas $ABmV$, $ABNP$, and may have the same proportion to $AB$, the unity of length, which these areas have to $ABCV$, the unity of surface. Then $VM\; :\; VC\; =\; Vm\; BA\; :\; VCBA$, and $PD\; :\; P\backslash beta\; =\; PNBA\; :\; VCBA$, &c. These ordinates will now represent $\backslash int\; \backslash dot\{p\}\; \backslash sqrt\{1\; +\; \backslash rho^2\}$ with reference to a linear unit, as the areas to the hyperbola represented it in reference to a superficial unit.

Again,

Again, in every ordinate make $PD\; :\; P\backslash bar\{z\}\; =\; P\backslash bar\{z\}\; :\; PO$, and thus we obtain a reciprocal to $PD$, or to $\backslash int\; \backslash dot\{p\}\; \backslash sqrt\{1+p^2\}$, or equivalent to $\backslash int\; \backslash frac\{1\}\{\backslash dot\{p\}\; \backslash sqrt\{1+p^2\}\}$. This

will evidently be $\backslash frac\{\backslash dot\{x\}\}\{a\backslash dot\{p\}\}$, and $PO$ or $p$ will be $\backslash frac\{\backslash dot\{x\}\}\{a\}$, and the area contained between the lines $AF$, $AW$, and the curve $GEOH$, and cut off by the ordinate $PO$, will represent $\backslash frac\{\backslash dot\{x\}\}\{a\}$.

Lastly, make $PO\; :\; PQ\; =\; AV\; :\; AP\; =\; 1\; :\; p$; and then $PQ$ or $p$ will represent $\backslash frac\{\backslash dot\{y\}\}\{a\}$, and the area $ALEQP$ will represent $\backslash frac\{\backslash dot\{y\}\}\{a\}$.

But we must here observe, that the fluents expressed by these different areas require what is called the correction to accommodate them to the circumstances of the case. It is not indifferent from what ordinate we begin to reckon the areas. This depends on the initial direction of the projectile, and that point of the abscissa $AP$ must be taken for the commencement of all the areas which gives a value of $p$ suited to the initial direction. Thus, if the projection has been made from $A$ (fig. 7.) at an elevation of $45^\backslash circ$, the ratio of the fluxions $\backslash dot\{x\}$ and $\backslash dot\{y\}$ is that of equality; and therefore the point $E$ of fig. 8. where the two curves intersect and have a common ordinate, evidently corresponds to this condition. The ordinate $EV$ passes through $V$, so that $AV$ or $p\; =\; AB\; =\; 1$, = tangent $45^\backslash circ$, as the case requires. The values of $x$ and of $y$ corresponding to any other point of the trajectory, such as that which has $AP$ for the tangent of the angle which it makes with the horizon, are now to be had by computing the areas $VEOP$, $VEQP$.

Another curve might have been added, of which the ordinates would exhibit the fluxions of the arch of the trajectory $\backslash dot\{z\}\; =\; \backslash frac\{a\backslash dot\{p\}\; \backslash sqrt\{1+p^2\}\}\{\backslash int\; \backslash dot\{p\}\; \backslash sqrt\{1+p^2\}\; +\; C\}$ and of which the area

would exhibit the arch itself. And this would have been very easy, for it is $\backslash dot\{z\}\; =\; a\; \backslash frac\{\backslash dot\{p\}\; \backslash sqrt\{1+p^2\}\}\{\backslash int\; \backslash dot\{p\}\; \backslash sqrt\{1+p^2\}\; +\; C\}$,

which is evidently the fluxion of the hyperbolic logarithm of $\backslash int\; \backslash dot\{p\}\; \backslash sqrt\{1+p^2\}$. But it is needless, since $\backslash dot\{z\}\; =\; \backslash dot\{x\}\; \backslash sqrt\{1+p^2\}$, and we have already got $\backslash dot\{x\}$. It is only increasing $PO$ in the ratio of $BA$ to $BP$.

And thus we have brought the investigation of this problem a considerable length, having ascertained the form of the trajectory. This is surely done when the ratio of the arch, absciss, and ordinate, and the position of its tangent, is determined in every point. But it is still very far from a solution, and much remains to be done before we can make any practical application of it. The only general consequence that we can deduce from the premises is, that in every case where the resistance in any point bears the same proportion to the force of gravity, the trajectory will be similar. Therefore, two balls, of the same density, projected in the same direction, will

describe similar trajectories if the velocities are in the subduplicate ratio of the diameters. This we shall find to be of considerable practical importance. But let us now proceed to determine the velocity in the different points of the trajectory, and the time of describing its several portions.

Recollect, therefore, that $v^2\; =\; \backslash frac\{-g\; \backslash dot\{x\}^2\}\{\backslash dot\{y\}\}$, and that $\backslash dot\{x\}^2\; =\; \backslash dot\{x\}^2(1+p^2)$ and $\backslash dot\{y\}\; =\; \backslash dot\{x\}\; \backslash dot\{p\}$. This gives $v^2\; =\; \backslash frac\{-g\; \backslash dot\{x\}^2(1+p^2)\}\{\backslash dot\{p\}\}$.

But $\backslash dot\{p\}\; =\; q\; \backslash dot\{x\}$. Therefore $v^2\; =\; \backslash frac\{-g\; \backslash dot\{x\}^2(1+p^2)\}\{q\}$, = $\backslash frac\{-ag\; \backslash sqrt\{1+p^2\}\}\{\backslash int\; \backslash dot\{p\}\; \backslash sqrt\{1+p^2\}\; +\; C\}$, and $v\; =\; \backslash sqrt\{\backslash frac\{-g\; \backslash sqrt\{1+p^2\}\}\{q\}\}$, =

Also $i$ was found = $\backslash frac\{\backslash dot\{z\}\}\{v\}\; =\; \backslash frac\{\backslash dot\{x\}\; \backslash sqrt\{1+p^2\}\}\{v\}\; =$

$\backslash frac\{\backslash dot\{p\}\; \backslash sqrt\{1+p^2\}\}\{q\; v\}$. If we now substitute for $v$ its value

just found, we obtain $i\; =\; \backslash frac\{\backslash dot\{p\}\}\{\backslash sqrt\{-gq\}\}$, and $t\; =\; \backslash int\; \backslash frac\{\backslash dot\{p\}\}\{\backslash sqrt\{-gq\}\}$,

The greatest difficulty still remains, viz. the accommodation of these formulae, which appear abundantly simple, to the particular cases. It would seem at first sight, that all trajectories are similar; since the ratio of the fluxions of the ordinate and abscissa corresponding to any particular angle of inclination to the horizon seems the same in them all: but a due attention to what has been hitherto said on the subject will show us that we have as yet only been able to ascertain the velocity in the point of the trajectory, which has a certain inclination to the horizon, indicated by the quantity $p$, and the time (reckoned from some assigned beginning) when the projectile is in that point.

To obtain absolute measures of these quantities, the term of commencement must be fixed upon. This will be expressed by the constant quantity $C$, which is assumed for completing the fluent of $\backslash dot\{p\}\; \backslash sqrt\{1+p^2\}$, which is the basis of the whole construction. We there found $q\; =$

$\backslash int\; \backslash dot\{p\}\; \backslash sqrt\{1+p^2\}$. This fluent is in general $q\; =$

and the constant quantity $C$ is to be accommodated to some circumstances of the case. Different authors have selected different circumstances.

Euler,

80 Euler, in his Commentary on Robins, and in a dissertation in the Memoirs of the Academy of Berlin published in 1753, takes the vertex of the curve for the beginning of his abscissa and ordinate. This is the simplest method of any, for C must then be so chosen that the whole fluent may vanish when $\backslash rho\; =\; 0$, which is the case in the vertex of the curve, where the tangent is parallel to the horizon. We shall adopt this method.

Fig. 9. Therefore, let $AP$ (fig. 9.) $=\; x$, $PM\; =\; y$, $AM\; =\; z$. Put the quantity C which is introduced into the fluent equal to $\backslash frac\{n\}\{a\}$. It is plain that $n$ must be a number; for it must be homologous with $\backslash dot\{\backslash rho\}\; \backslash sqrt\{1\; +\; \backslash rho^2\}$, which is a number. For brevity's sake let us express the fluent of $\backslash dot\{\backslash rho\}\; \backslash sqrt\{1\; +\; \backslash rho^2\}$ by the single letter $P$; and thus we shall have $x\; =\; a\; \backslash times\; \backslash int\; \backslash frac\{\backslash dot\{\backslash rho\}\}\{n\; +\; P\}$, $y\; =\; a\; \backslash times\; \backslash int\; \backslash frac\{\backslash dot\{\backslash rho\}\; \backslash rho\}\{n\; +\; P\}$, $z\; =\; a\; \backslash times\; \backslash int\; \backslash frac\{\backslash dot\{\backslash rho\}\; \backslash sqrt\{1\; +\; \backslash rho^2\}\}\{n\; +\; P\}$. And $v^2\; =\; \backslash frac\{-ag(1\; +\; \backslash rho^2)\}\{n\; +\; P\}$. Now the height $h$ necessary for communicating any velocity $v$ is $\backslash frac\{v^2\}\{2g\}\; =\; \backslash frac\{-ag(1\; +\; \backslash rho^2)\}\{2g(n\; +\; P)\}\; =\; \backslash frac\{-\backslash frac\{1\}\{2\}a(1\; +\; \backslash rho^2)\}\{n\; +\; P\}$. And lastly, $z\; =\; \backslash frac\{\backslash sqrt\{a\}\}\{\backslash sqrt\{g\}\}\; \backslash int\; \backslash frac\{\backslash dot\{\backslash rho\}\}\{\backslash sqrt\{n\; +\; P\}\}$.

These fluents, being all taken so as to vanish at the vertex, where the computation commences, and where $\backslash rho$ is $=\; 0$ (the tangent being parallel to the horizon), we obtain in this case $h\; =\; \backslash frac\{\backslash frac\{1\}\{2\}a\}\{n\}$, $=\; \backslash frac\{a\}\{2n\}$, and $n\; =\; \backslash frac\{a\}\{2h\}$.

Hence we see that the circumstance which modifies all the curves, distinguishing them from each other, is the velocity (or rather its square) in the highest point of the curve. For $h$ being determined for any body whose terminal velocity is $u$, $n$ is also determined; and this is the modifying circumstance. Considering it geometrically, it is the area which must be cut off from the area $DMAP$ of fig. 8. in order to determine the ordinates of the other curves.

We must farther remark, that the values now given relate only to that part of the area where the body is descending from the vertex. This is evident; for, in order that $y$ may increase as we recede from the vertex, its fluxion must be taken in the opposite sense to what it was in our investigation. There we supposed $y$ to increase as the body ascended, and then to diminish during the descent; and therefore the fluxion of $y$ was first positive and then negative.

The same equations, however, will serve for the ascending branch $CNA$ of the curve, only changing the sign of $P$; for if we consider $y$ as decreasing during

the ascent, we must consider $q$ as expressing $\backslash frac\{\backslash dot\{\backslash rho\}\}\{x\}$, and therefore $P$, or $\backslash int\; \backslash dot\{\backslash rho\}\; \backslash sqrt\{1\; +\; \backslash rho^2\}$, which is $=\; \backslash frac\{q\}\{a\}$, must be taken negatively. Therefore, in the ascending branch, we have $AQ$ or $x$ (increasing as we recede from $A$) $=$

$a\; \backslash times\; \backslash int\; \backslash frac\{\backslash dot\{\backslash rho\}\}\{n\; -\; P\}$, $QN$ or $y\; =\; a\; \backslash times\; \backslash int\; \backslash frac\{\backslash dot\{\backslash rho\}\; \backslash rho\}\{n\; -\; P\}$, $AN$ or $z\; =$

Vol. XVII. Part II.

$a\; \backslash times\; \backslash int\; \backslash frac\{\backslash dot\{\backslash rho\}\; \backslash sqrt\{1\; +\; \backslash rho^2\}\}\{n\; -\; P\}$, $z\; =\; \backslash frac\{\backslash sqrt\{a\}\}\{\backslash sqrt\{g\}\}\; \backslash times\; \backslash int\; \backslash frac\{\backslash dot\{\backslash rho\}\}\{\backslash sqrt\{n\; -\; P\}\}$, and the height producing the velocity at $N\; =\; \backslash frac\{\backslash frac\{1\}\{2\}a(1\; +\; \backslash rho^2)\}\{n\; -\; P\}$.

Hence we learn by the bye, that in no part of the Remark-ascending branch can the inclination of the tangent be so much such that $P$ shall be greater than $n$; and that if we suppose the pole $P$ equal to $n$ in any point of the curve, the velocity in that point will be infinite. That is to say, there is a certain assignable elevation of the tangent which cannot be exceeded in a curve which has this velocity in the vertex. The best way for forming a conception of this circumstance in the nature of the curve, is to invert the motion, and suppose an accelerating force, equal and opposite to the resistance, to act on the body in conjunction with gravity. It must describe the same curve, and this branch $ANC$ must have an asymptote $LO$, which has this limiting position of the tangent. For, as the body descends in this curve, its velocity increases to infinity by the joint action of gravity and this accelerating force, and yet the tangent never approaches so near the perpendicular position as to make $P\; =\; n$. This remarkable property of the curve was known to Newton, as appears by his approximations, which all lead him to curves of a hyperbolic form, having one asymptote inclined to the horizon. Indeed it is pretty obvious: For the resistance increasing faster than the velocity, there is no velocity of projection so great but that the curve will come to deviate so from the tangent, that in a finite time it will become parallel to the horizon. Were the resistance proportional to the velocity, then an infinite velocity would produce a rectilinear motion, or rather a deflection from it less than any that can be assigned.

We now see that the particular form and magnitude of what of this trajectory depends on two circumstances, $a$ and its form and $n$. $a$ affects chiefly the magnitude. Another circumstance might indeed be taken in, viz. the diminution of the accelerating force of gravity by the statical effect of the air's gravity. But, as we have already observed, this is too trifling to be attended to in military projectiles.

$\backslash frac\{y\}\{x\}$ was made equal to $\backslash dot\{\backslash rho\}$. Therefore the radius of curvature, determined by the ordinary methods, is $\backslash frac\{x(1\; +\; \backslash rho^2)(\backslash sqrt\{1\; +\; \backslash rho^2\})\}\{\backslash dot\{\backslash rho\}\}$, and, because $\backslash frac\{x\}\{\backslash dot\{\backslash rho\}\}$ is simpson's Fluxion, $\backslash frac\{1\}\{65\}$, &c. $=\; \backslash frac\{a\}\{n\; +\; P\}$ for the descending branch of the curve, the

radius of curvature at $M$ is $\backslash frac\{a\; \backslash times\; 1\; +\; \backslash rho^2\; \backslash times\; \backslash sqrt\{1\; +\; \backslash rho^2\}\}\{n\; +\; P\}$, and, in the ascending branch at $N$, it is $\backslash frac\{a\; \backslash times\; 1\; +\; \backslash rho^2\; \backslash times\; \backslash sqrt\{1\; +\; \backslash rho^2\}\}\{n\; -\; P\}$.

On both sides therefore, when the velocity is infinitely great, and $P$ by this means supposed to equal or exceed $n$, the radius of curvature is also infinitely great. We also see that the two branches are unlike each other, and that when $\backslash rho$ is the same in both, that is, when the tangent is equally inclined to the horizon, the radius of curvature, the ordinate, the absciss, and the arch, are all greater in the ascending branch. This is pretty obvious.

vious. For as the resistance acts entirely in diminishing the velocity, and does not affect the deflection occasioned by gravity, it must allow gravity to incurvate the path so much the more (with the same inclination of its line of action) as the velocity is more diminished. The curvature, therefore, in those points which have the same inclination of the tangent, is greatest in the descending branch, and the motion is swiftest in the ascending branch. It is otherwise in a void, where both sides are alike. Here $u$ becomes infinite, or there is no terminal velocity; and $n$ also becomes infinite, being $\backslash frac\{a\}\{2h\}$.

It is therefore in the quantity $P$, or $f\; \backslash dot\{p\}\; \backslash sqrt\{1+p^2\}$,

that the difference between the trajectory in a void and in a resisting medium consists; it is this quantity which expresses the accumulated change of the ratio of the increments of the ordinate and abscissa. In vacuo the second increment of the ordinate is constant when the first increment of the abscissa is so, and the whole increment of the ordinate is as $1+p$. And this difference is so much the greater as $P$ is greater in respect of $n$. $P$ is nothing at the vertex, and increases along with the angle $MTP$; and when this is a right angle, $P$ is infinite. The trajectory in a resisting medium will come therefore to deviate infinitely from a parabola, and may even deviate farther from it than the parabola deviates from a straight line. That is, the distance of the body in a given moment from that point of its parabolic path where it would have been in a void, is greater than the distance between that point of the parabola from the point of the straight line where it would have been, independent of the action of gravity. This must happen whenever the resistance is greater than the weight of the body, which is generally the case in the beginning of the trajectory in military projectiles; and this (were it now necessary) is enough to show the inutility of the parabolic theory.

Although we have no method of describing this trajectory, which would be received by the ancient geometers, we may ascertain several properties of it, which will assist us in the solution of the problem. In particular, we can align the absolute length of any part of it by means of the logistic curve. For because $P$

$=\; f\; \backslash dot\{p\}\; \backslash sqrt\{1+p^2\}$, we have $\backslash dot\{p\}\; \backslash sqrt\{1+p^2\}\; =\; \backslash dot\{P\}$, and there-

fore $\backslash pi$, which was $=\; a\; \backslash times\; f\; \backslash frac\{\backslash dot\{p\}\; \backslash sqrt\{1+p^2\}\}\{f\; \backslash dot\{p\}\; \backslash sqrt\{1+p^2\}\}\; +\; C$, or $=\; a\; \backslash times$

$f\; \backslash frac\{\backslash dot\{P\}\}\{n+P\}$, may be expressed by logarithms; or $\backslash pi\; =\; a$

$\backslash times\; \backslash text\{hyp.\; log.\; of\; \}\; \backslash frac\{n+P\}\{n\}$, since at the vertex $A$, where $\backslash pi$ must be $=\; 0$, $P$ also $=\; 0$.

Being able, in this way, to ascertain the length $AM$ of the curve (counted from the vertex), corresponding to any inclination $\backslash rho$ of the tangent at its extremity $M$, we can ascertain the length of any portion of it, such as $Mm$, by first finding the length of the part $Am$, and then of the part $AM$. This we do more expeditiously thus: Let $\backslash rho$ express the position of the tangent in $M$, and

$q$ its position at $m$; then $AM\; =\; a\; \backslash times\; \backslash log.\; \backslash frac\{n+P\}\{n\}$ and $Am$

$=\; a\; \backslash times\; \backslash log.\; \backslash frac\{n+Q\}\{n\}$, and therefore $Mm\; =\; a\; \backslash times\; \backslash log.\; \backslash frac\{n+Q\}\{n+P\}$.

Thus we can find the values of a great number of small portions, and the inclination of the tangents at their extremities. Then to each of these portions we can assign its proportion of the abscissa and ordinate, without having recourse to the values of $x$ and $y$.

For the portion of abscissa corresponding to the arch $Mm$, whose middle point is inclined to the horizon in the angle $b$, will be $Mm\; \backslash times\; \backslash cosine\; b$, and the corresponding portion of the ordinate will be $Mm\; \backslash times\; \backslash sin\; b$. Then we obtain the velocity in each part of the curve by the equation $h\; =\; \backslash frac\{a\; \backslash times\; \backslash sqrt\{1+p^2\}\}\{n+P\}$; or, more directly the velocity

$v$ at $M$ will be $=\; \backslash sqrt\{ag\}\; \backslash frac\{\backslash sqrt\{1+p^2\}\}\{\backslash sqrt\{n+P\}\}$. Lastly, divide the

length of the little arch by this, and the quotient will be the time of describing $Mm$ very nearly. Add all these together, and we obtain the whole time of describing the arch $AM$, but a little too great, because the motion in the small arch is not perfectly uniform. The error, however, may be as small as we please, because we may make the arch as small as we please; and for greater accuracy, it will be proper to take the $\backslash rho$ by which we compute the velocity, a medium between the $\backslash rho$ for the beginning and that for the end of the arch.

This is the method followed by Euler, who was one of the most expert analysts, if not the very first, in Euler's method preferred. It is not the most elegant, and the methods of some other authors, who approximate directly to the areas of the curves which determine the values of $x$ and $y$, have a more scientific appearance; but they are not ultimately very different: For, in some methods, these areas are taken piecemeal, as Euler takes the arch; and by the methods of others, who give the value of the areas by Newton's method of describing a curve of the parabolic kind through any number of given points, the ordinates of these curves, which express $x$ and $y$, must be taken singly, which amounts to the same thing, with the great disadvantage of a much more complicated calculus, as any one may see by comparing the expressions of $x$ and $y$ with the expression of $\backslash pi$. As to those methods which approximate directly to the areas or values of $x$ and $y$ by an infinite series, they all, without exception, involve us in most complicated expressions, with coefficients of sines and tangents, and ambiguous signs, and engage us in a calculation almost endless. And we know of no series which converges fast enough to give us tolerable accuracy, without such a number of terms as is sufficient to deter any person from the attempt. The calculation of the arches is very moderate, so that a person tolerably versant in arithmetical operations may compute an arch with its velocity and time in about five minutes. We have therefore no hesitation in preferring this method of Euler's to all that we have seen, and therefore proceed to determine some other circumstances which render its application more general.

85
Its applica-
tion made
more gen-
eral.

If there were no resistance, the smallest velocity would be at the vertex of the curve, and it would immediately increase by the action of gravity conferring (in however small degree) with the motion of the body. But in a resisting medium, the velocity at the vertex is diminished by a quantity to which the acceleration of gravity in that point bears no assignable proportion. It is therefore diminished, upon the whole, and the point of smallest velocity is a little way beyond the vertex. For the same reasons, the greatest curvature is a little way beyond the vertex. It is not very material for our present purpose to ascertain the exact positions of those points.

The velocity in the descending branch augments continually: but it cannot exceed a certain limit, if the velocity at the vertex has been less than the terminal velocity; for when the curve is infinite, $\backslash rho$ is also infinite, and $h\; =\; \backslash frac\{a\backslash rho^2\}\{P\}$, because $n$ in this case is nothing in respect of $P$, which is infinite; and because $\backslash rho$ is infinite, the number $\backslash text\{hyp.\; log.\; \}\; \backslash rho\; \backslash times\; \backslash sqrt\{1+\backslash rho^2\}$, though infinite, vanishes in comparison with $\backslash rho\; +\; \backslash sqrt\{1+\backslash rho^2\}$; so that in this case $P\; =\; \backslash frac\{1\}\{2\}\backslash rho^2$, and $h\; =\; a$, and $v\; =\; \backslash text\{terminal\; velocity\}$.

If, on the other hand, the velocity at the vertex has been greater than the terminal velocity, it will diminish continually, and when the curve has become infinite, $v$ will be equal to the terminal velocity.

In either case we see that the curve on this side will have a perpendicular asymptote. It would require a long and pretty intricate analysis to determine the place of this asymptote, and it is not material for our present purpose. The place and position of the other asymptote LO is of the greatest moment. It evidently distinguishes the kind of trajectory from any other. Its position depends on this circumstance, that if $\backslash rho$ marks the position of the tangent, $n\; =\; P$, which is the denominator of the fraction expressing the square of the velocity, must be equal to nothing, because the velocity is infinite: therefore, in this place, $P\; =\; n$, or $n\; =\; \backslash frac\{1\}\{2\}\backslash rho\; \backslash sqrt\{1+\backslash rho^2\}\; +\; \backslash frac\{1\}\{2\}\; \backslash log\; \backslash frac\{\backslash rho\; +\; \backslash sqrt\{1+\backslash rho^2\}\}\{\backslash rho\; -\; \backslash sqrt\{1+\backslash rho^2\}\}$. In order, therefore, to find the point L, where the asymptote LO cuts the horizontal line AL, put $P\; =\; n$, then will $AL\; =\; x\; =\; \backslash frac\{2x\}\{y\}\; =\; a\; \backslash times\; \backslash left(\; \backslash int\; \backslash frac\{\backslash rho\}\{n-P\}\; -\; \backslash frac\{1\}\{\backslash rho\}\; \backslash int\; \backslash frac\{\backslash rho\; \backslash rho\}\{n-P\}\; \backslash right)$.

Through the whole of this article $f$ means $\backslash log$.

It is evident that the logarithms used in these expressions are the natural or hyperbolic. But the operations may be performed by the common tables, by making the value of the arch $Mm$ of the curve $=\; \backslash frac\{a\}\{M\}\; \backslash times\; \backslash log$.

$\backslash frac\{n+Q\}\{n+P\}$ &c. where $M$ means the sub-tangent of the common logarithms, or 0.43429; also the time of describing this arch will be expeditiously had by taking a medium $\backslash mu$ between the values of $\backslash frac\{\backslash sqrt\{1+\backslash rho^2\}\}\{\backslash sqrt\{n+P\}\}$ and $\backslash frac\{\backslash sqrt\{1+q^2\}\}\{\backslash sqrt\{n+Q\}\}$ and making the time $=\; \backslash frac\{\backslash sqrt\{a\}\}\{M\backslash mu\backslash sqrt\{g\}\}\; \backslash times\; \backslash log\; \backslash frac\{n+Q\}\{n+P\}$.

86
Mode of applying this process in practice.

Such then is the process by which the form and magnitude of the trajectory, and the motion in it, may be determined. But it does not yet appear how this is to be applied to any question in practical artillery. In this

process we have only learned how to compute the motion from the vertex in the descending branch till the ball has acquired a particular direction, and the motion to the vertex from a point of the ascending branch where the ball has another direction, and all this depending on the greatest velocity which the body can acquire by falling, and the velocity which it has in the vertex of the curve. But the usual question is, "What will be the motion of the ball projected in a certain direction with a certain velocity?"

The mode of application is this: Suppose a trajectory computed for a particular terminal velocity, produced by the fall $a$, and for a particular velocity at the vertex, which will be characterized by $n$, and that the velocity at that point of the ascending branch where the inclination of the tangent is $30^\backslash circ$ is 920 feet per second. Then, we are certain, that if a ball, whose terminal velocity is that produced by the fall $a$, be projected with the velocity of 920 feet per second, and an elevation of $30^\backslash circ$, it will describe this very trajectory, and the velocity and time corresponding to every point will be such as is here determined.

Now this trajectory will, in respect to form, answer an infinity of cases: for its characteristic is the proportion of the velocity in the vertex to the terminal velocity. When this proportion is the same, the number $n$ will be the same. If, therefore, we compute the trajectories for a sufficient variety of these proportions, we shall find a trajectory that will nearly correspond to any case that can be proposed; and an approximation sufficiently exact will be had by taking a proportional medium between the two trajectories which come nearest to the case proposed.

87
Accordingly, a set of tables or trajectories have been computed by the English translator of Euler's Commentary on Robins's Gunnery. They are in number 18, distinguished by the position of the asymptote of the ascending branch. This is given for $3^\backslash circ$, $10^\backslash circ$, $15^\backslash circ$, &c. to $85^\backslash circ$, and the whole trajectory is computed as far as it can ever be supposed to extend in practice. The following table gives the value of the number $n$ corresponding to each position of the asymptote.

Since the path of a projectile is much less incurved, and more rapid in the ascending than in the descending branch, and the difference is so much the more remarkable in great velocities; it must follow, that the range on a horizontal or inclined plane depends most on the ascending branch: therefore the greatest range will not be made with that elevation which bisects the angle of position, but with a lower elevation; and the deviation from the bisecting elevation will be greater as the initial

velocities are greater. It is very difficult to frame an exact rule for determining the elevation which gives the greatest range. We have subjoined a little table which gives the proper elevations (nearly) corresponding to the different initial velocities.

It was computed by the following approximation, which will be found the same with the series used by Newton in his Approximation.

Let $e$ be the angle of elevation, $a$ the height producing the terminal velocity, $h$ the height producing the initial velocity, and $e$ the number whose hyperbolic logarithm is 1 (i. e. the number 2.718). Then,

&c. Make $y\; =\; v$, and take the maximum by varying $e$, we obtain $\backslash sin.\; e\; +\; \backslash frac\{a\; \backslash cdot\; \backslash sin.\; e\}\{2h\}\; =\; \backslash text\{hyperbol.\; log.\}$

The numbers in the first column, multiplied by the terminal velocity of the projectile, give us the initial velocity; and the numbers in the last column, being multiplied by the height producing the terminal velocity, and by 2.3026, give us the greatest range. The middle column contains the elevation. The table is not computed with scrupulous exactness, the question not requiring it. It may, however, be depended on within one part of 20:20.

To make use of this table, divide the initial velocity by the terminal velocity $u$, and look for the quotient in the first column. Opposite to this will be found the elevation giving the greatest range; and the number in the last column being multiplied by 2.3026 $\backslash times\; u$ (the height producing the terminal velocity) will give the range.

TABLE of Elevations giving the greatest Range.

Such is the solution which the present state of our mathematical knowledge enables us to give of this celebrated problem. It is exact in its principle, and the application of it is by no means difficult, or even onerous.

But let us see what advantage we are likely to derive from it.

In the first place, it is very limited in its application. There are few circumstances of general coincidence, and almost every case requires an appropriated calculus. Perhaps the only general rules are the two following:

1. Balls of equal density, projected with the same elevation, and with velocities which are as the square-roots of their diameters, will describe similar curves.—This is evident, because, in this case, the resistance will be in the ratio of their quantities of motion. Therefore all the homologous lines of the motion will be in the proportion of the diameters.

2. If the initial velocities of balls projected with the same elevation are in the inverse subduplicate ratio of the whole resistances, the ranges, and all the homologous lines of their track, will be inversely as those resistances.

These theorems are of considerable use: for by means of a proper series of experiments on one ball projected with different elevations and velocities, tables may be constructed which will ascertain the motions of an infinity of others.

But when we take a retrospective view of what we have done, and consider the conditions which were assumed in the solution of the problem, we shall find that much yet remains before it can be rendered of great practical use, or even satisfy the curiosity of the man of science. The resistance is all along supposed to be in the duplicate ratio of the velocity; but even theory points out many causes of deviation from this law, such as the pressure and condensation of the air, in the case of very swift motions; and Mr Robins's experiments are sufficient to show us that the deviations must be exceedingly great in such cases. Mr Euler and all subsequent writers have allowed that it may be three times greater, even in cases which frequently occur; and Euler gives a rule for ascertaining with tolerable accuracy what this increase and the whole resistance may amount to. Let $H$ be the height of a column of air whose weight is equivalent to the resistance taken in the duplicate ratio of the velocity. The whole resistance will be expressed by $H\; +\; \backslash frac\{H^2\}\{28843\}$. This number 28843 is the

height in feet of a column of air whose weight balances its elasticity. We shall not at present call in question his reasons for assigning this precise addition. They are rather reasons of arithmetical convenience than of physical import. It is enough to observe, that if this measure of the resistance is introduced into the process of investigation, it is totally changed; and it is not too much to say, that with this complication it requires the knowledge and address of a Euler to make even a partial and very limited approximation to a solution.—Any law of the resistance, therefore, which is more complicated than what Bernoulli has assumed, namely, that of a simple power of the velocity, is abandoned by all the mathematicians, as exceeding their abilities; and they have attempted to avoid the error arising from the assumption of the duplicate ratio of the velocity, either by supposing the resistance throughout the whole trajectory to be greater than what it is in general, or they have divided the trajectory into different portions, and assigned different resistances to each, which vary,

vary, through the whole of that portion, in the duplicate ratio of the velocities. By this kind of patchwork they make up a trajectory and motion which corresponds, in some tolerable degree, with what? With an accurate theory? No; but with a series of experiments. For, in the fifth place, every theoretical computation that we make, proceeds on a supposed initial velocity; and this cannot be ascertained with any thing approaching to precision, by any theory of the action of gunpowder that we are yet possessed of. In the next place, our theories of the resisting power of the air are entirely established on the experiments on the flights of shot and shells, and are corrected and amended till they tally with the most approved experiments we can find. We do not learn the ranges of a gun by theory, but the theory by the range of the gun. Now the variety and irregularity of all the experiments which are appealed to are so great, and the acknowledged difference between the resistance to slow and swift motions is also so great, that there is hardly any supposition which can be made concerning the resistance, that will not agree in its results with many of those experiments. It appears from the experiments of Dr Hutton of Woolwich, in 1784, 1785, and 1786, that the shots frequently deviated to the right or left of their intended track 200, 300, and sometimes 400 yards. This deviation was quite accidental and anomalous, and there can be no doubt but that the shot deviated from its intended and supposed elevation as much as it deviated from the intended vertical plane, and this without any opportunity of measuring or discovering the deviation. Now, when we have the whole range from one to three to choose among for our measure of resistance, it is evident that the confirmations which have been drawn from the ranges of shot are but feeble arguments for the truth of any opinion. Mr Robins finds his measures fully confirmed by the experiments at Metz and at Minorca. Mr Muller finds the same. Yet Mr Robins's measure both of the initial velocity and of the resistance are at least treble of Mr Muller's; but by compensation they give the same results. The Chevalier Borda, a very expert mathematician, has adduced the very same experiments in support of his theory, in which he abides by the Newtonian measure of the resistance, which is about $\backslash frac\{1\}\{2\}$ of Mr Robins's, and about $\backslash frac\{1\}\{3\}$ of Muller's.

⁹⁰ Cause of its insubst. What are we to conclude from all this? Simply this, that we have hardly any knowledge of the air's resistance, and that even the solution given of this problem has not as yet greatly increased it. Our knowledge consists only in those experiments, and mathematicians are attempting to patch up some notion of the motion of a body in a resisting medium, which shall tally with them.

There is another essential defect in the conditions assumed in the solution. The density of the air is supposed uniform; whereas we are certain that it is less by one-fifth or one-sixth towards the vertex of the curve, in many cases which frequently occur, than it is at the beginning and end of the flight. This is another latitude given to authors in their assumptions of the air's resistance. The Chevalier de Borda has, with considerable ingenuity, accommodated his investigation to this circumstance, by dividing the trajectory into portions, and, without much trouble, has made one equation answer them all. We are disposed to think that his solution of the problem (in the Memoirs of the

Academy of Paris for 1769) corresponds better with the physical circumstances of the case than any other. But this process is there delivered in too concise a manner to be intelligible to a person not perfectly familiar with all the resources of modern analysis. We therefore prefered John Bernoulli's, because it is elementary and rigorous.

After all, the practical artillerist must rely chiefly on the records of experiments contained in the books of practice at the academies, or those made in a more public manner. Even a perfect theory of the air's resistance can do him little service, unless the force of gunpowder were uniform. This is far from being the case even in the same powder. A few hours of a damp day will make a greater difference than occurs in any theory; and, in service, it is only by trial that every thing is performed. If the first shell fall very much short of the mark, a little more powder is added; and, in cannonading, the correction is made by varying the elevation.

We hope to be forgiven by the eminent mathematicians for these observations on their theories. They by no means proceed from any disrespect for their labours. We are not ignorant of the almost insuperable difficulty of the task, and we admire the ingenuity with which some of them have contrived to introduce into their analysis reasonable substitutions for those terms which would render the equations intractable. But we must still say, upon their own authority, that these are but ingenious guesses, and that experiment is the touchstone by which they mould these substitutions; and when they have found a coincidence, they have no motive to make any alteration. Now, when we have such a latitude for our measure of the air's resistance, that we may take it of any value, from one to three, it is no wonder that compensations of errors should produce a coincidence; but where is the coincidence? The theorist supposes the ball to set out with a certain velocity, and his theory gives a certain range; and this range agrees with observation—but how? Who knows the velocity of the ball in the experiment? This is concluded from a theory incomparably more uncertain than that of the motion in a resisting medium.

The experiments of Mr Robins and Dr Hutton show, in the most incontrovertible manner, that the resistance to a motion exceeding 1100 feet in a second, is almost three times greater than in the duplicate ratio to the resistance to moderate velocities. Euler's translator, in his companion of the author's trajectories with experiment supposes it to be so greater. Yet the coincidence is very great. The same may be said of the Chevalier de Borda's. Nay, the same may be said of Mr Robins's own practical rules: for he makes his $F$, which corresponds to our $a$, almost double of what these authors do, and yet his rules are confirmed by practice. Our observations are therefore well founded.

But it must not be inferred from all this, that the physical theory is of no use to the practical artillerist. It plainly shows him the impropriety of giving the projectile an enormous velocity. This velocity is of no use in fact after 200 or 300 yards at farthest, because it is so rapidly reduced by the prodigious resistance of the air. Mr Robins has deduced several practical maxims of the greatest importance from what we already know of this subject, and which could hardly have been even conjectured without this knowledge. See GUNNERY.

And it must still be acknowledged, that this branch of physical science is highly interesting to the philosopher; nor should we despair of carrying it to greater perfection. The defects arise almost entirely from our ignorance of the law of variation of the air's resistance. Experiments may be contrived much more conducive to our information here than those commonly resorted to. The oblique flights of projectiles are, as we have seen, of very complicated investigation, and ill fitted for instructing us; but numerous and well contrived experiments on the perpendicular ascents are of great simplicity, being affected by nothing but the air's resistance. To make them instructive, we think that the following plan might be pursued. Let a set of experiments be premised for ascertaining the initial velocities. Then let shells be discharged perpendicularly with great varieties of density and velocity, and let nothing be attended to but the height and the time; even a considerable deviation from the perpendicular will not affect either of these circumstances, and the effect of this circumstance can easily be computed. The height can be ascertained with sufficient precision for very valuable information by their light or smoke. It is evident that these experiments will give direct information of the air's retarding force; and every experiment gives us two measures, viz. the ascent and descent: and the comparison of the times of ascent and descent, combined with the observed height in one experiment made with a great initial velocity, will give us more information concerning the air's resistance than 50 ranges. If we should suppose the resistance as the square of the velocity, this comparison will give in each experiment an exact determination of the initial and final velocities, which no other method can give us. These, with experiments on the time of horizontal flights, with known initial velocities, will give us more instruction on this head than any thing that has yet been done; and till something of this kind is carefully done, we presume to say that the motion of bodies in a resisting medium will remain in the hands of the mathematicians as a matter of curious speculation. In the mean time, the rules which Mr Robins has delivered in his Gunnery are very simple and easy in their use, and seem to come as near the truth as any we have met with. He has not informed us upon what principles they are founded, and we are disposed to think that they are rather empirical than scientific. But we profess great deference for his abilities and penetration, and doubt not but that he had framed them by means of as scientific a discussion as his knowledge of this new and difficult subject enabled him to give it.

We shall conclude this article, by giving two or three tables, computed from the principles established above, and which serve to bring into one point of view the chief circumstances of the motion in a resisting medium. Although the result of much calculation, as any person who considers the subject will readily see, they must not be considered as offering any very accurate results; or that, in comparison with one or two experiments, the differences shall not be considerable. Let any person peruse the published registers of experiments which have been made with every attention, and he will see such enormous irregularities, that all expectations of perfect agreement with them must cease. In the experiments at Woolwich in 1735, which were continued for several days, not only do

the experiments of one day differ among themselves, but the mean of all the experiments of one day differs from the mean of all the experiments of another no less than one fourth of the whole. The experiments in which the greatest regularity may be expected, are those made with great elevations. When the elevation is small, the range is more affected by a change of velocity, and still more by any deviation from the supposed or intended direction of the shot.

The first table shows the distance in yards to which a ball projected with the velocity 1600 will go, while its velocity is reduced one-tenth, and the distance at which it drops 16 feet from the line of its direction. This table is calculated by the resistance observed in Mr Robins's experiments. The first column is the weight of the ball in pounds. The second column remains the same whatever be the initial velocity; but the third column depends on the velocity. It is here given for the velocity which is very usual in military service, and its use is to assist us in directing the gun to the mark. If the mark at which a ball of 24 pounds is directed is 474 yards distant, the axis of the piece must be pointed 16 feet higher than the mark. These defections from the line of direction are nearly as the squares of the distances.

The next table contains the ranges in yards of a 2 pound shot, projected at an elevation of 45°, with the different velocities in feet per second, expressed in the first column. The second column contains the distances to which the ball would go in vacuo in a horizontal plane; and the third contains the distances to which it will go through the air. The fourth column is added, to show the height to which it rises in the air; and the fifth shows the ranges corrected for the diminution of the air's density as the bullet ascends, and may therefore be called the corrected range.

PROJECTILES.

Plate CCCCXLI.

Fig. 2.

Fig. 3.

Fig. 1.

Fig. 4.

Fig. 5.

Fig. 6.

Fig. 7.

Fig. 8 is a geometric diagram illustrating projectile trajectories. It features several horizontal lines and a series of intersecting curves. Key points are labeled: B, A, L, G, P, D, N, M, E, F, Q, R, S, T, U, V, W, X, Y, Z, and WHK. Angles are marked with values such as 20°, 40°, 60°, and 80°. The diagram shows how different launch angles result in different parabolic paths.

Fig. 9 is a geometric diagram showing a projectile path. It includes a horizontal line with points B, Q, I, A, T, P, R, and Z. A curve starts at O, passes through G, N, and M, and ends at H. Vertical lines are drawn from B, Q, I, A, T, P, R, and Z down to the curve. Points D and F are also indicated. The diagram demonstrates the geometric construction of a projectile's path based on its launch point and angle.

Fig. 10 is a complex geometric diagram showing multiple projectile trajectories. It consists of two main parts. On the left, a series of lines radiate from a common point A to various points on a horizontal line, with numerical labels 4, 9, 16, 21, 26, 32, 39, 46, 52, and 60. On the right, a series of curves originate from point A and intersect with a horizontal line at points labeled 4, 9, 16, 21, 26, 32, 39, 46, 52, and 60. Points M, N, L, B, A, P, Q, R, and G are also marked. The diagram illustrates the relationship between the launch angle and the resulting trajectory of a projectile.

The initial velocities can never be pushed as far as we have calculated for in this table; but we mean it for a table of more extensive use than appears at first sight. Recollect, that while the proportion of the velocity at the vertex to the terminal velocity remains the same, the curves will be similar: therefore, if the initial velocities are as the square-roots of the diameters of the balls, they will describe similar curves, and the ranges will be as the diameters of the balls.

Therefore, to have the range of a 12 pound shot, if projected at an elevation of 45, with the velocity 1500; suppose the diameter of the 12 pounder to be $d$, and that of the 24 pounder $D$; and let the velocities be $v$ and $V$: Then say, $\backslash sqrt\{d\}\; :\; \backslash sqrt\{D\}\; =\; 1500$, to a fourth proportional $V$. If the 24 pounder be projected with the velocity $V$, it will describe a curve similar to that described by the 12 pounder, having the initial velocity 1500. Therefore find (by interpolation) the range of the 24 pounder, having the initial velocity $V$. Call this $R$. Then $D\; :\; d\; =\; R\; :\; r$, the range of the 12 pounder which was wanted, and which is nearly 3380 yards.

We see by this table the immense difference between the motions through the air and in a void. We see that the ranges through the air, instead of increasing in the duplicate ratio of the initial velocities, really increase slower than those velocities in all cases of military service; and in the most usual cases, viz. from 800 to 1600, they increase nearly as the square-roots of the velocities.

A set of similar tables, made for different elevations, would almost complete what can be done by theory, and would be much more expeditious in their use than Mr Euler's Trajectories, computed with great labour by his English translator.

The same table may also serve for computing the ranges of bomb-shells. We have only to find what must be the initial velocity of the 24 pound shot which corresponds to the proposed velocity of the shell. This must be deduced from the diameter and weight of the shell, by making the velocity of the 24 pounder such, that the ratio of its weight to the resistance may be the same as in the shell.

That the reader may see with one glance the relation of those different quantities, we have given this table, expressed in a figure (fig. 10). The abscissa, or axis $DA$, is the scale of the initial velocities in feet per second, measured on a scale of 400 equal parts in an inch. The ordinates to the curve $ACG$ express the yards of the range on a scale containing 800 yards in an inch. The ordinates to the curve $Axy$ express (by the same scale) the height to which the ball rises in the air.

The ordinate $BC$ (drawn through the point of the abscissa which corresponds to the initial velocity 2000) is divided in the points 4, 9, 12, 18, 24, 32, 42, in the ratio of the diameters of cannon-shot of different weights; and the same ordinate is produced on the other side of the axis, till $BO$ be equal to $BA$; and then $BO$ is divided in the subduplicate ratio of the same diameters. Lines are drawn from the point $A$, and from any point $D$ of the abscissa, to these divisions.

We see distinctly by this figure how the effect of the initial velocity gradually diminishes, and that in very great velocities the range is very little increased by its augmentation. The dotted curve $APQR$, shows what the ranges in vacuo would be.

By this figure may the problems be solved. Thus, to find the range of the 12 pounder, with the initial velocity 1500. Set off 1500 from $B$ to $F$; draw $FH$ parallel to the axis, meeting the line $12A$ in $H$; draw the ordinate $HK$; draw $KL$ parallel to the axis, meeting 24 $B$ in $L$; draw the ordinate $LM$, cutting 12 $B$ in $N$. $MN$ is the range required.

If curves, such as $ACG$, were laid down in the same manner for other elevations, all the problems might be solved with great dispatch, and with much more accuracy than the theory by which the curves are drawn can pretend to.

Note, that fig. 10. as given on Plate CCCCXLII. is one-half less than the scale according to which it is described; but the practical mathematician will find no difficulty in drawing the figure on the enlarged scale to correspond to the description.

PROJECTION OF THE SPHERE.

THE PROJECTION OF THE SPHERE is a perspective representation of the circles on the surface of the sphere; and is variously denominated according to the different positions of the eye and plane of projection.

There are three principal kinds of projection; the stereographic, the orthographic, and gnomic. In the stereographic projection the eye is supposed to be placed on the surface of the sphere; in the orthographic it is supposed to be at an infinite distance; and in the gnomic projection the eye is placed at the centre of the sphere. Other kinds of projection are, the globular, Merentor's, scenographic, &c. for which see the articles GEOGRAPHY, NAVIGATION, PERSPECTIVE, &c.

DEFINITIONS.

1. The plane upon which the circles of the sphere are described, is called the plane of projection, or the

primitive circle. The pole of this circle is the pole Stereographic Projection of the Sphere.

2. The line of measures of any circle of the sphere is that diameter of the primitive, produced indefinitely, which passes through the centre of the projected circle.

AXIOM.

The projection, or representation of any point, is where the straight line drawn from it to the projecting point intersects the plane of projection.

SECTION I.

Of the Stereographic Projection of the Sphere.

In the stereographic projection of the sphere, the eye.

eye is placed on the surface of the sphere in the pole of the great circle upon which the sphere is to be projected. The projection of the hemisphere opposite to the eye falls within the primitive, to which the projection is generally limited; it, however, may be extended to the other hemisphere, or that wherein the eye is placed, the projection of which falls without the primitive.

As all circles in this projection are projected either into circles or straight lines, which are easily described, it is therefore more generally understood, and by many preferred to the other projections.

PROPOSITION I. THEOREM I.

Every great circle which passes through the projecting point is projected into a straight line passing through the centre of the primitive; and every arch of it, reckoned from the other pole of the primitive, is projected into its semitangent.

Let ABCD (fig. 1.) be a great circle passing through A, C, the poles of the primitive, and intersecting it in the line of common section BED, E being the centre of the sphere. From A, the projecting point, let there be drawn straight lines AP, AM, AN, AQ, to any number of points P, M, N, Q, in the circle ABCD: these lines will intersect BED, which is in the same plane with them. Let them meet it in the points p, m, n, q; then p, m, n, q, are the projections of P, M, N, Q: hence the whole circle ABCD is projected into the straight line BED, passing through the centre of the primitive.

Again, because the pole C is projected into E, and the point M into m; therefore the arch CM is projected into the straight line E m, which is the semitangent of the arch CM to the radius AE. In like manner, the arch CP is projected into its semitangent, E p, &c.

COROLLARIES.

1. Each of the quadrants contiguous to the projecting point is projected into an indefinite straight line, and each of those that are remote into a radius of the primitive.

2. Every small circle which passes through the projecting point is projected into that straight line which is its common section with the primitive.

3. Every straight line in the plane of the primitive, and produced indefinitely, is the projection of some circle on the sphere passing through the projecting point.

4. The projection of any point in the surface of the sphere, is distant from the centre of the primitive, by the semitangent of the distance of that point from the pole opposite to the projecting point.

PROPOSITION II. THEOREM II.

Every circle on the sphere which does not pass through the projecting point is projected into a circle.

If the given circle be parallel to the primitive, then a straight line drawn from the projecting point to any point in the circumference, and made to revolve about the circle, will describe the surface of a cone; which being cut by the plane of projection parallel to the base, the section will be a circle. See CONIC SECTIONS.

But if the circle MN (fig. 2.) be not parallel to the primitive circle BD, let the great circle ABCD, passing through the projecting point, cut it at right angles in the diameter MN, and the primitive in the diameter BD. Through M, in the plane of the great circle, let MF be drawn parallel to BD; let AM, AN be joined, and meet BD in m, n. Then, because AB, AD are quadrants, and BD, MF parallel, the arch AM is equal to AF, and the angle AMF or A m n is equal to ANM. Hence the conic surface described by the revolution of AM about the circle MN is cut by the primitive in a subcontrary position; therefore the section is in this case likewise a circle.

COROLLARIES.

1. The centres and poles of all circles parallel to the primitive have their projection in its centre.

2. The centre and poles of every circle inclined to the primitive have their projections in the line of measures.

3. All projected great circles cut the primitive in two points diametrically opposite; and every circle in the plane of projection, which passes through the extremities of a diameter of the primitive, or through the projections of two points that are diametrically opposite on the sphere, is the projection of some great circle.

4. A tangent to any circle of the sphere, which does not pass through the projecting point, is projected into a tangent to that circle's projection; also, the circular projections of tangent circles touch one another.

5. The extremities of the diameter, on the line of measures of any projected circle, are distant from the centre of the primitive by the semitangents of the least and greatest distances of the circle on the sphere, from the pole opposite to the projecting point.

6. The extremities of the diameter, on the line of measures of any projected great circle, are distant from the centre of the primitive by the tangent and cotangent of half the great circle's inclination to the primitive.

7. The radius of any projected circle is equal to half the sum, or half the difference of the semitangents of the least and greatest distances of the circle from the pole opposite to the projecting point, according as that pole is within or without the given circle.

PROPOSITION III. THEOREM III.

An angle formed by two tangents at the same point in the surface of the sphere, is equal to the angle formed by their projections.

Let FGI and GH (fig. 3.) be the two tangents, and A the projecting point; let the plane AGF cut the sphere in the circle AGL, and the primitive in the line BML. Also, let MN be the line of common section of the plane AGH with the primitive: then the angle FGH = LMN. If the plane FGH be parallel to the primitive BLD, the proposition is manifest. If not, through any point K in AG produced, let the plane FKH, parallel to the primitive, be extended to meet FGH in the line FH. Then, because the plane AGF meets the two parallel planes BLD, FKH, the lines of common section LM, FK are parallel; there-
fore

Stereographic projection of the Sphere.
fore the angle $AML\; =\; AKF$. But since A is the pole of BLD, the chords, and consequently the arches AB AL, are equal, and the arch ABG is the sum of the arches AL, BG; hence the angle AML is equal to an angle at the circumference standing upon AG, and therefore equal to AGI or FGK; consequently the angle FGK = FKG, and the side FG = FK. In like manner HG = HK: hence the triangles GHF, KHF are equal, and the angle FGH = FKH = LMN.

1. An angle contained by any two circles of the sphere is equal to the angle formed by their projections. For the tangents to these circles on the sphere are projected into straight lines, which either coincide with, or are tangents to, their projections on the primitive.

2. An angle contained by any two circles of the sphere is equal to the angle formed by the radii of their projections at the point of intersection.

The centre of a projected great circle is distant from the centre of the primitive; the tangent of the inclination of the great circle to the primitive, and its radius, is the secant of its inclination.

Let MNG (fig. 4.) be the projection of a great circle, meeting the primitive in the extremities of the diameter MN, and let the diameter BD, perpendicular to MN, meet the projection in F, G. Biseft FG in H, and join NH. Then, because any angle contained by two circles of the sphere is equal to the angle formed by the radii of their projections at the point of intersection; therefore the angle contained by the proposed great circle and the primitive is equal to the angle ENH, of which EH is the tangent, and NH the secant, to the radius of the primitive.

1. All circles which pass through the points M, N are the projections of great circles, and have their centres in the line BG; and all circles which pass through the points F, G, are the projections of great circles, and have their centres in the line HI, perpendicular to BG.

2. If NF, NH be continued to meet the primitive in L, F; then BL is the measure of the great circle's inclination to the primitive; and $MT\; =\; 2BL$.

The centre of projection of a less circle perpendicular to the primitive, is distant from the centre of the primitive, the secant of the distance of the less circle from its nearest pole; and the radius of projection is the tangent of that distance.

Let MN (fig. 5.) be the given less circle perpendicular to the primitive, and A the projecting point. Draw AM, AN to meet the diameter BD produced in G and H; then GH is the projected diameter of the less circle: biseft GH in C, and C will be its centre; join NE, NC. Then because AE, NI are parallel, the angle $INE\; =\; NEA$; but $NEA\; =\; 2NMA$.

Stereographic projection of the Sphere.
$=\; 2NHG\; =\; NCG$: hence $ENC\; =\; INE\; +\; INC\; =\; NCG$. $+\; INC\; =$ a right angle; and therefore NC is a tangent to the primitive at N; but the arch ND is the distance of the less circle from its nearest pole D; hence NC is the tangent, and EC the secant of the distance of the less circle from its pole to the radius of the primitive.

The projection of the poles of any circle, inclined to the primitive, are, in the line of measures, distant from the centre of the primitive, the tangent, and cotangent, of half its inclination.

Let MN (fig. 6.) be a great circle perpendicular to the primitive ABCD, and A the projecting point; then P, $\backslash rho$ are the poles of MN, and of all its parallels $\backslash pi$, $\backslash rho$, &c. Let AP, A $\backslash rho$ meet the diameter BD in F, f, which will therefore be the projected poles of MN and its parallels. The angle BEM is the inclination of the circle MEN, and its parallels, to the primitive; and because BC and MP are quadrants, and MC common to both; therefore $PC\; =\; BM$; and hence PEC is also the inclination of MN and its parallels. Now EF is the tangent of EAF, or of half the angle PEC the inclination; and Ef is the tangent of the angle EAf; but EAf is the complement of EAF, hence Ef is the cotangent of half the inclination.

1. The projection of that pole which is nearest to the projecting point is without the primitive, and the projection of the other within.

2. The projected centre of any circle is always between the projection of its nearest pole and the centre of the primitive; and the projected centres of all circles are contained between their projected poles.

Equal arches of any two great circles of the sphere will be intercepted between two other circles drawn on the sphere through the remote poles of those great circles.

Let AGB, CFD (fig. 7.) be two great circles of the sphere, whose remote poles are E, P; through which draw the great circle PBEC, and less circle PGE, intersecting the great circles AGB, CFD, in the points B, G, and D, F; then the arch BG is equal to the arch DF.

Because E is the pole of the circle AGB, and P the pole of CFD, therefore the arches EB, PD are equal; and since BD is common to both, hence the arch ED is equal to the arch PB. For the same reason, the arches EF, PG are equal; but the angle DEF is equal to the angle BPG; hence these triangles are equal, and therefore the arch DF is equal to the arch BG.

If from either pole of a projected great circle, two straight lines be drawn to meet the primitive and the projection, they will intercept similar arches of these circles.

On the plane of projection $AGB$ (fig. 7.) let the great circle $CFD$ be projected into $cfd$, and its pole $P$ into $p$; through $p$ draw the straight lines $pd,\; pf$, then are the arches $GB,\; fd$ similar.

Since $pd$ lies both in the plane $AGB$ and $APBE$, it is in their common section, and the point $B$ is also in their common section; therefore $pd$ passes through the point $B$. In like manner it may be shown that the line $pf$ passes through $G$. Now the points $D,\; F$ are projected into $d,\; f$; hence the arches $FD,\; fd$ are similar; but $GB$ is equal to $FD$, therefore the intercepted arch of the primitive $GB$ is similar to the projected arch $fd$.

Hence, if from the angular point of a projected spherical angle two straight lines be drawn through the projected poles of the containing sides, the intercepted arch of the primitive will be the measure of the spherical angle.

To describe the projection of a great circle through two given points in the plane of the primitive.

Let $P$ and $B$ be given points, and $C$ the centre of the primitive.

Fig. 8. 1. When one point $P$ (fig. 8.) is the centre of the primitive, a diameter drawn through the given points will be the great circle required.

Fig. 9. 2. When one point $P$ (fig. 9.) is in the circumference of the primitive. Through $P$ draw the diameter $PD$; and an oblique circle described through the three points $P,\; B,\; D$, will be the projection of the required great circle.

Fig. 10. 3. When the given points are neither in the centre nor circumference of the primitive. Through either of the given points $P$ (fig. 10.) draw the diameter $ED$, and at right angles thereto draw the diameter $FG$. From $F$ through $P$ draw the straight line $FPH$, meeting the circumference in $H$; draw the diameter $HI$, and draw the straight line $FIK$, meeting $ED$ produced in $D$; then an arch, terminated by the circumference, being described through the three points, $P,\; B,\; K$, will be the great circle.

To describe the representation of a great circle about any given point as a pole.

Let $P$ be the given pole, and $C$ the centre of the primitive.

Fig. 11. 1. When $P$ (fig. 11.) is in the centre of the primitive, then the primitive will be the great circle required.

Fig. 12. 2. When the pole $P$ (fig. 12.) is in the circumference of the primitive. Through $P$ draw the diameter $PE$, and the diameter $AB$ drawn at right angles to $PE$ will be the projected great circle required.

Plate ccccxliv. 3. When the given pole is neither in the centre nor circumference of the primitive. Though the pole $P$ (fig. 12.) draw the diameter $AB$, and draw the diameter $DE$ perpendicular to $AB$; through $E$ and $P$ draw the straight line $EPF$, meeting the circumference in $F$. Make $FG$ equal to $ED$; through $E$ and $G$ draw the

straight line $EGH$, meeting the diameter $AB$ produced if necessary in $H$; then from the centre $H$, with the radius $HE$, describe the oblique circle $DIE$, and it will be the projection of the great circle required.

Or, make $DK$ equal to $FA$; join $EK$, which intersects the diameter $AB$ in $I$; then through the three points, $D,\; I,\; E$, describe the oblique circle $DIE$.

To find the poles of a great circle.

1. When the given great circle is the primitive, its centre is the pole.

2. To find the pole of the right circle $ACB$ (fig. 13.) Draw the diameter $PE$ perpendicular to the given circle $AB$; and its extremities $P,\; E$ are the poles of the circle $ACB$.

3. To find the pole of the oblique circle $DEF$ (fig. Fig. 13.) Join $DF$, and perpendicular thereto draw the diameter $AB$, cutting the given oblique circle $DEF$ in $E$. Draw the straight line $FEG$, meeting the circumference in $G$. Make $GI,\; GH$, each equal to $AD$; then $FI$ being joined, cuts the diameter $AB$ in $P$, the lower pole; through $F$ and $H$ draw the straight line $FH$, meeting the diameter $AB$ produced in $\backslash rho$, which will be the opposite or exterior pole.

To describe a less circle about any given point as a pole, and at any given distance from that pole.

1. When the pole of the less circle is in the centre of the primitive; then from the centre of the primitive, with the semitangent of the distance of the given circle from its pole, describe a circle, and it will be the projection of the less circle required.

2. If the given pole is in the circumference of the primitive, from $C$ (fig. 14.) the centre of the primitive, set off $CE$ the secant of the distance of the less circle from its pole $P$; then from the centre $E$, with the tangent of the given distance, describe a circle, and it will be the less circle required. Or, make $PG,\; PF$ each equal to the chord of the distance of the less circle from its pole. Through $B,\; G$, draw the straight line $BGD$ meeting $CP$ produced in $D$; bisect $GD$ in $H$, and draw $HE$ perpendicular to $GD$; and meeting $PD$ in $E$, then $E$ is the centre of the less circle.

3. When the given pole is neither in the centre nor circumference of the primitive. Through $P$ (fig. 15.) the given pole, and $C$ the centre of the primitive, draw the diameter $AB$, and draw the diameter $DE$ perpendicular to $AB$; join $EP$, and produce it to meet the primitive in $\backslash rho$; make $\backslash rho\; F,\; \backslash rho\; G$, each equal to the chord of the distance of the less circle from its pole; join $EF$ which intersects the diameter $AB$ in $H$; from $E$ through $G$ draw the straight line $EGI$, meeting the diameter $AB$ produced in $I$; bisect $HI$ in $K$. Then a circle described from the centre $K$, at the distance $KH$ or $KI$, will be the projection of the less circle.

To find the poles of a given less circle.

The poles of a less circle are also those of its parallel great

Stereographic Projection of the Sphere.
great circle. If therefore the parallel great circle be given, then its poles being found by Prob. III. will be those of the less circle. But if the parallel great circle be not given, let $HMIN$ (fig. 15.) be the given less circle. Through its centre, and $C$ the centre of the primitive, draw the line of measures $IAHB$; and draw the diameter $DE$ perpendicular to it, also draw the straight line $EHF$ meeting the primitive in $F$; make $F\backslash rho$ equal to the chord of the distance of the less circle from its pole; join $E\backslash rho$, and its intersection $P$ with the diameter $AB$ is the interior pole. Draw the diameter $\backslash rho\; CI$ through $E$ and $I$, draw $ELq$ meeting the diameter $AB$ produced in $q$; then $q$ is the external pole. Or thus: Join $EI$ intersecting the primitive in $G$; join also $EH$, and produce it to meet the primitive in $F$; bisect the arch $GF$ in $\backslash rho$; from $E$ to $\backslash rho$ draw the straight line $EP\backslash rho$, and $P$ is the pole of the given less circle.

1. Arches of the primitive are measured on the line of chords.

2. Right circles are measured on the line of semitangents, beginning at the centre of the primitive. Thus, the measure of the portion $AC$ (fig. 16.) of the right circle $DE$, is found by applying it to the line of semitangents. The measure of the arch $DB$ is found by subtracting that of $BC$ from $90^\backslash circ$: the measure of the arch $AF$, lying partly on each side of the centre, is obtained by adding the measures of $AC$ and $CF$. Lastly, To measure the part $AB$, which is neither terminated at the centre or circumference of the primitive, apply $CA$ to the line of semitangents; then $CB$, and the difference between the measures of these arches, will be that of $AB$.

Or thus: Draw the diameter $GH$ perpendicular to $DE$; then from either extremity, as $D$, of this diameter, draw lines through the extremities of the arch intended to be measured; and the intercepted portion of the primitive applied to the line of chords will give the measure of the required arch. Thus $IK$ applied to the line of chords will give the measure of $AB$.

3. To measure an arch of an oblique circle: draw lines from its pole through the extremities of the arch to meet the primitive, then the intercepted portion of the primitive applied to the line of chords will give the measure of the arch of the oblique circle. Thus, let $AB$ (fig. 17.) be an arch of an oblique circle to be measured, and $P$ its pole; from $P$ draw the lines $PAD$, $PBE$ meeting the primitive in $B$ and $E$; then the arch $DE$ applied to the line of chords will give the measure of the arch of the oblique circle $AB$.

Let $DEG$ (fig. 18.) be the given less circle, and $DE$ the arch to be measured: find its internal pole $P$; and describe the circle $AFI$ parallel to the primitive, and whose distance from the projecting point may be equal to the distance of the given less circle from its pole $P$: then join $PD$, $PE$, which produce to meet the parallel circle in $A$ and $F$. Now $AF$ applied to a

line of chords will give the measure of the arch $DE$ of the given less circle.

1. If the angle is at the centre of the primitive, it is measured as a plane angle.

2. When the angular point is in the circumference of the primitive; let $A$ (fig. 19.) be the angular point, and $ABE$ an oblique circle inclined to the primitive. Through $P$, the pole of $ABE$, draw the line $AP\backslash rho$ meeting the circumference in $\backslash rho$: then the arch $E\backslash rho$ is the measure of the angle $BAD$, and the arch $AF\backslash rho$ is the measure of its supplement $BAF$: also $\backslash rho\; F$ is the measure of the angle $BAC$, and $\backslash rho\; ED$ that of its supplement.

3. If the angular point is neither at the centre nor circumference of the primitive. Let $A$ (fig. 20.) be the angular point, and $DAH$, or $GAF$, the angle to be measured, $P$ the pole of the oblique circle $DAF$, and $\backslash rho$ the pole of $GAH$: then from $A$, through the points $P\backslash rho$, draw the straight lines $APM$, $A\backslash rho\; N$, and the arch $MN$ will be the measure of the angle $DAH$; and the supplement of $MN$ will be the measure of the angle $HAF$ or $DAG$.

Find the pole of the given circle, then a great circle described through that pole and the given point will be perpendicular to the given circle. Hence if the given circle be the primitive, then a diameter drawn through the given point will be the required perpendicular. If the given circle is a right one, draw a diameter at right angles to it; then though the extremities of this diameter and the given point describe an oblique circle, and it will be perpendicular to that given. If the given circle is inclined to the primitive, let it be represented by $BAD$ (fig. 21.), whose pole is $P$, and let $A$ be the point through which the perpendicular is to be drawn: then, by Prob. I. describe a great circle through the points $P$ and $A$, and it will be perpendicular to the oblique circle $BAD$.

Let the given circle be the primitive, and let $A$ (fig. 19.) be the angular point. Draw the diameters $AE$, $DF$ perpendicular to each other; and make the angle $CAG$ equal to that given, or make $CG$ equal to the tangent of the given angle; then from the centre $G$, with the distance $GC$, describe the oblique circle $ABE$, and it will make with the primitive an angle equal to that given.

If the given circle be a right one, let it be $APB$ (fig. 22.) and let $P$ be the given point. Draw the diameter $GH$

GH perpendicular to AB; join GP, and produce it to $a$; make $Hb$ equal to twice $Aa$; and $Gb$ being joined intersects AB in C. Draw CD perpendicular to AB, and equal to the cotangent of the given angle to the radius PC; or make the angle CPD equal to the complement of that given; then from the centre D, with the radius DP, describe the great circle FPE, and the angle APE, or BPE, will be equal to that given.

If APB (fig. 23.) is an oblique circle. From the angular point P, draw the lines PG, PC through the centres of the primitive and given oblique circle. Through C, the centre of APB, draw GCD at right angles to PG; make the angle GPD equal to that given; and from the centre D, with the radius DP, describe the oblique circle FPE, and the angle APE, or BPE, will be equal to that proposed.

Any great circle cutting the primitive being given, to describe another great circle which shall cut the given one in a proposed angle, and have a given arch intercepted between the primitive and given circles.

If the given circle be a right one, let it be represented by APC (fig. 24.); and at right angles thereto draw the diameter BPM; make the angle BPF equal to the complement of the given angle, and PF equal to the tangent of the given arch; and from the centre of the primitive with the secant of the same arch describe the arch GG. Through F draw FG parallel to AC, meeting GG in G; then from the centre G, with the tangent PF, describe an arch $no$, cutting APC in I, and join GI. Through G, and the centre P, draw the diameter HK; draw PI perpendicular to HK, and IL perpendicular to GI, meeting PL in L; then L will be the centre of the circle HIK, which is that required.

But if the given great circle be inclined to the primitive, let it be ADB (fig. 25.), and E its centre; make the angle BDF equal to the complement of that given, and DF equal to the tangent of the given arch, as before. From P, the centre of the primitive, with the secant of the same arch, describe the arch GG, and from E, the centre of the oblique circle, with the extent EF, describe an arch intersecting GG in G. Now G being determined, the remaining part of the operation is performed as before.

When the given arch exceeds $90^\backslash circ$, the tangent and secant of its supplement are to be applied on the line DF the contrary way, or towards the right; the former construction being reckoned to the left.

Any great circle in the plane of projection being given, to describe another great circle, which shall make given angles with the primitive and given circles.

Let ADC (fig. 26.) be the given circle, and Q its pole. About P the pole of the primitive, describe an arch $mn$, at the distance of as many degrees as are in the angle which the required circle is to make with the primitive. About Q the pole of the circle ADC, and at a distance equal to the measure of the angle which the required circle is to make with the given circle ADC, describe an arch $on$, cutting $mn$ in $n$. Then

about $a$ as a pole, describe the great circle EDF, cutting the primitive and given circle in E and D, and it will be the great circle required.

It will hence be an easy matter to construct all the various spherical triangles. The reader is, however, referred to the article Spherical Trigonometry, for the method of constructing them agreeably to this projection; and also for the application to the resolution of problems of the sphere. For the method of projecting the sphere upon the plane of the meridian, and of the horizon, according to the stereographic projection, see the article Geography.

THE orthographic projection of the sphere, is that in which the eye is placed in the axis of the plane of projection, at an infinite distance with respect to the diameter of the sphere; so that at the sphere all the visual rays are assumed parallel, and therefore perpendicular to the plane of projection.

Hence the orthographic projection of any point is where a perpendicular from that point meets the plane of projection; and the orthographic representation of any object is the figure formed by perpendiculars drawn from every point of the object to the plane of projection.

This method of projection is used in the geometrical delineation of eclipses, occultations, and transits. It is also particularly useful in various other projections, such as the analemma. See Geography, &c.

Every straight line is projected into a straight line. If the given line be parallel to the plane of projection, it is projected into an equal straight line; but if it is inclined to the primitive, then the given straight line will be to its projection in the ratio of the radius to the cosine of inclination.

Let AB (fig. 27.) be the plane of projection, and let CD be a straight line parallel thereto: from the extremities C, D of the straight line CD, draw the lines CE, DF perpendicular to AB; then by 3. of xi. of Eucl. the intersection EF, of the plane CEFD, with the plane of projection, is a straight line; and because the straight lines CD, EF are parallel, and also CE, DF; therefore, by 34. of i. of Eucl. the opposite sides are equal; hence the straight line CD, and its projection EF, are equal. Again, let GH be the proposed straight line, inclined to the primitive; then the lines GE, HF being drawn perpendicular to AB, the intercepted portion EF will be the projection of GH. Through G draw GI parallel to AB, and the angle IGH will be equal to the inclination of the given line to the plane of projection. Now GH being the radius, GI, or its equal EF, will be the cosine of IGH; hence the given line GH is to its projection EF as radius to the cosine of inclination.

1. 1. A straight line perpendicular to the plane of projection is projected into a point.
2. 2. Every straight line in a plane parallel to the primitive is projected into an equal and parallel straight line.
3. 3. A plane angle parallel to the primitive is projected into an equal angle.
4. 4. Any plane rectilinear figure parallel to the primitive is projected into an equal and similar figure.
5. 5. The area of any rectilinear figure is to the area of its projection as radius to the cosine of its inclination.

Every great circle, perpendicular to the primitive, is projected into a diameter of the primitive; and every arch of it, reckoned from the pole of the primitive, is projected into its fine.

Fig. 28. Let BFD (fig. 28.) be the primitive, and ABCD a great circle perpendicular to it, passing through its poles A, C; then the diameter BED, which is their line of common section, will be the projection of the circle ABCD. For if from any point, as G, in the circle ABC, a perpendicular GH fall upon BD, it will also be perpendicular to the plane of the primitive; therefore H is the projection of G. Hence the whole circle is projected into BD, and any arch AG into EH equal to GI its fine.

1. 1. Every arch of a great circle, reckoned from its intersection with the primitive, is projected into its versed fine.
2. 2. Every less circle perpendicular to the primitive is projected into its line of common section with the primitive, which is also its own diameter; and every arch of the semicircle above the primitive, reckoned from the middle point, is projected into its fine.
3. 3. Every diameter of the primitive is the projection of a great circle; and every chord the projection of a less circle.
4. 4. A spherical angle at the pole of the primitive is projected into an equal angle.

A circle parallel to the primitive is projected into a circle equal to itself, and concentric with the primitive.

Fig. 29. Let the less circle FIG (fig. 29.) be parallel to the plane of the primitive BND. The straight line HE, which joins their centres, is perpendicular to the primitive; therefore E is the projection of H. Let any radii HI and IN perpendicular to the primitive be drawn. Then IN, HE being parallel, are in the same plane; therefore IH, NE, the lines of common section of the plane IE, with two parallel planes, are parallel; and the figure IHEN is a parallelogram. Hence NE = IH, and consequently FIG is projected into an equal circle KNL, whose centre is E.

The radius of the projection is the cosine of the distance of the parallel circle from the primitive, or the

Orthographic projection of the Sphere.

An inclined circle is projected into an ellipse, whose transverse axis is the diameter of the circle.

Fig. 30. 1. Let ELF (fig. 30.) be a great circle inclined to the primitive EBF, and EF their line of common section. From the centre C, and any other point K, in EF, let the perpendicular CB, KI be drawn in the plane of the primitive, and CL, KN, in the plane of the great circle, meeting the circumference in L, N. Let LG, ND be perpendicular to CB, KI; then G, D are the projections of L, N. And because the triangles LCG, NKD are equiangular, $CL\; :\; CG\; ::\; NK\; :\; DK$; or $EC^2\; :\; CG^2\; ::\; EK^2\; :\; DK^2$; therefore the points G, D are in the curve of an ellipse, of which EF is the transverse axis, and CG the semiconjugate axis.

1. 1. In a projected great circle, the semiconjugate axis is the cosine of the inclination of the great circle to the primitive.
2. 2. Perpendiculars to the transverse axis intercept corresponding arches of the projection and the primitive.
3. 3. The eccentricity of the projection is the fine of the inclination of the great circle to the primitive.

Fig. 31. Case 2. Let AQB (fig. 31.) be a less circle, inclined to the primitive, and let the great circle LBM, perpendicular to both, intersect them in the lines AB, LM. From the centre O, and any other point N in the diameter AB, let the perpendiculars TOP, NQ, be drawn in the plane of the less circle, to meet its circumference in T, P, Q. Also, from the points A, N, O, B, let AG, NI, OC, BH, be drawn perpendicular to LM; and from P, Q, T, draw PE, QD, TF, perpendicular to the primitive; then G, I, C, H, E, D, F, are the projections of these points. Because OP is perpendicular to LBM, and OC, PE, being perpendicular to the primitive, are in the same plane, the plane COPE is perpendicular to LBM. But the primitive is perpendicular to LBM; therefore the common section EC is perpendicular to LBM, and to LM. Hence CP is a parallelogram, and $EC\; =\; OP$. In like manner, FC, DI, are proved perpendicular to LM, and equal to OT, NQ. Thus ECF is a straight line, and equal to the diameter PT. Let QR, DK be parallel to AB, LM; then $RO\; =\; NQ\; =\; DI\; =\; KC$, and $PR\; \backslash times\; RT\; =\; EK\; \backslash times\; KF$. But $AO\; :\; CG\; ::\; NO\; :\; CI$; therefore $AO^2\; :\; CG^2\; ::\; QR^2\; :\; DK^2$; and $EC^2\; :\; CG^2\; ::\; EKF\; :\; DK^2$.

1. 1. The transverse axis is to the conjugate as radius to the cosine of the circle's inclination to the primitive.
2. 2. Half the transverse axis is the cosine of half the sum of the greatest and least distances of the less circle from the primitive.
3. 3. The extremities of the conjugate axis are in the line of measures, distant from the centre of the primitive by the cosines of the greatest and least distances of the less circle from the primitive.

4. If from the extremities of the conjugate axis of any elliptical projection perpendiculars be drawn (in the same direction if the circle do not intersect the primitive, but if otherwise in opposite directions), they will intersect an arch of the primitive, whose chord is equal to the diameter of the circle.

The projected poles of an inclined circle are in its line of measures distant from the centre of the primitive the fine of the inclination of the circle to the primitive.

Let ABCD (fig. 32.) be a great circle, perpendicular both to the primitive and the inclined circle, and intersecting them in the diameters AC, MN. Then ABCD passes through the poles of the inclined circle; let these be P, Q; and let Pp, Qq, be perpendicular to AC; p, q are the projected poles; and it is evident that $pO\; =\; \backslash text\{fine\; of\; BP,\; or\; MA,\; the\; inclination.\}$

1. The centre of the primitive, the centre of the projection, the projected poles, and the extremities of the conjugate axis, are all in one and the same straight line.

2. The distance of the centre of projection from the centre of the primitive, is to the cosine of the distance of the circle from its own pole, as the fine of the circle's inclination to the primitive is to the radius.

To describe the projection of a circle perpendicular to the primitive, and whose distance from its pole is equal to a given quantity.

Let PA $\backslash rho$ B (fig. 33.) be the primitive circle, and P, $\backslash rho$ the poles of the right circle to be projected. Then if the circle to be projected is a great circle, draw the diameter AB at right angles to the axis P$\backslash rho$, and it will be that required. But if the required projection is that of a less circle, make PE, PF each equal to the chord of the distance of the less circle from its pole; join EF, and it will be the projection of the less circle required.

Through a given point in the plane of the primitive to describe the projection of a great circle, having a given inclination to the primitive.

1. When the given inclination is equal to a right angle, a straight line drawn through the centre of the primitive, and the given point, will be the projection required.

2. When the given inclination is less than a right angle, and the given point in the circumference of the primitive. Let R (fig. 34.) be a point given in the circumference of the primitive, through which it is required to draw the projection of a great circle, inclined to the primitive in an angle measured by the arch QP of the primitive.

Through the given point R draw the diameter RCS, and draw GCg at right angles to it. Make the arch

GV of the primitive equal to QP, and draw VA at right angles to GC; and in Gg, towards the opposite parts of C, take CB equal to AC; then, with the greater axis RS, and less axis AB, describe an ellipse, and it will be the projection of the oblique circle required.

3. When the distance of the given point from the primitive is equal to the cosine of the given inclination.

Every thing remaining as in the preceding case; let A be the given point, and AC the cosine of an arch GV, equal to the given arch QP; then drawing the diameter RCS at right angles to ACB, the ellipse described with the given axis RS, AB will be the projection of the inclined circle.

4. When the distance of the given point from the centre of the primitive is less than the semidiameter of the primitive, but greater than the cosine of the given inclination.

Let D be the given point, through which draw the diameter IC; and at the point D draw DL perpendicular to DC meeting the primitive in L; also draw LK, making with LD the angle DLK equal to the complement of the given inclination. Let LK meet DC in K; then will DK be less than DC. On DC as a diameter describe a circle, and make DH equal to DK; through H draw a diameter of the primitive RCS, and describe an ellipse through the points R, D, S, and it will be the projection of the inclined circle.

Through two given points in the plane of the primitive to describe the projection of a great circle.

1. If the two given points and the centre of the primitive be in the same straight line, then a diameter of the primitive being drawn through these points will be the projection of the great circle required.

2. When the two given points are not in the same straight line with the centre of the primitive; and one of them is in the circumference of the primitive.

Let DR (fig. 34.) be the two given points, of which R is in the circumference of the primitive. Draw the diameters RCS, and GCg, FDH perpendicular to it, meeting the primitive in GgF. Divide GC, gC, in A, B, in the same proportion as FH is divided in D; and describe the ellipse whose axes are RS, AB, and centre C; and it will be the projection required.

3. When the given points are within the primitive, and not in the same straight line with its centre.

Let D, E (fig. 35.) be the two given points; through C the centre of the primitive draw the straight lines ID, KE; draw DL perpendicular to I, and EO perpendicular to K, meeting the primitive in L, O. Through E, and towards the same parts of C, draw EP parallel to DC, and in magnitude a fourth proportional to LD, DC, OE. Draw the diameter CP meeting the primitive in R, S, and describe an ellipse through the points D and R, or S, and it will also pass through E. This ellipse will be the projection of the proposed inclined circle.

To describe the projection of a less circle parallel to the primitive, its distance from the pole of the primitive being given.

Orthographic Projection of the Sphere. From the pole of the primitive, with the fine of the given distance of the circle from its pole, describe a circle, and it will be the projection of the given left circle.

About a given point as a projected pole to describe the projection of an inclined circle, whose distance from its pole is given.

Let $P$ (fig. 36.) be the given projected pole, through which draw the diameter $Gg$, and draw the diameter $Hb$ perpendicular thereto. From $P$ draw $PL$ perpendicular to $GP$ meeting the circumference in $L$; through which draw the diameter $Ll$. Make $LT$, $LK$ each equal to the chord of the distance of the left circle from its pole, and join $TK$, which intersects $Ll$ in $Q$. From the points $T$, $Q$, $K$, draw the lines $FA$, $QS$, $KB$, perpendicular to $Gg$; and make $OR$, $OS$, each equal to $QT$, or $QK$. Then an ellipse described through the points $A$, $S$, $B$, $R$ will be the projection of the proposed left circle.

To find the poles of a given projected circle.

1. 1. If the projected circle be parallel to the primitive, the centre of the primitive will be its pole.
2. 2. If the circle be perpendicular to the primitive, then the extremities of a diameter of the primitive drawn at right angles to the straight line representing the projected circle, will be the poles of that circle.
3. 3. When the projected circle is inclined to the primitive.

Let $ARBS$ (fig. 36, 37.) be the elliptical projection of any oblique circle; through the centre of which, and $C$ the centre of the primitive, draw the line of measures $CBA$, meeting the ellipse in $B$, $A$; and the primitive in $G$, $g$. Draw $CH$, $BK$, $AT$ perpendicular to $Gg$, meeting the primitive in $H$, $K$, $T$. Biseft the arch $KT$ in $L$, and draw $LP$ perpendicular to $Gg$; then $P$ will be the projected pole of the circle, of which $ARBS$ is the projection.

To measure any portion of a projected circle, and conversely.

1. 1. When the given projection is that of a great circle.

Let $ADBE$ (fig. 38.) be the given great circle, either perpendicular or inclined to the primitive, of which the portion $DE$ is to be measured, and let $Mm$ be the line of measures of the given circle. Through the points $D$, $E$, draw the lines $EG$, $DF$ parallel to $Mm$; and the arch $FG$ of the primitive will be the measure of the arch $DE$ of the great circle, and conversely.

1. 2. When the projection is that of a left circle parallel to the primitive.

Let $DE$ (fig. 39.) be the portion to be measured, of the left circle $DEH$ parallel to the primitive. From the centre $C$ draw the lines $CD$, $CE$, and produce them to meet the primitive in the points $B$, $F$. Then the

intercepted portion $BF$ of the primitive will be the measure of the given arch $DE$ of the left circle $DEH$.

1. 3. If the given left circle, of which an arch is to be measured, is perpendicular to the primitive.

Let $ADEB$ (fig. 40.) be the left circle, of which the measure of the arch $DE$ is required. Through $C$, the centre of the primitive, draw the line of measures $Mm$, and from the intersection $O$ of the given right circle, and the line of measures, with the radius $OA$, or $OB$, describe the semicircle $AFGB$; through the points $D$, $E$, draw the lines $DF$, $EG$ parallel to the line of measures, and the arch $FG$ will be the measure of $DE$, to the radius $AO$. In order to find a similar arch in the circumference of the primitive, join $OF$, $OG$, and at the centre $C$ of the primitive, make the angle $mCH$ equal to $FOG$, and the arch $mH$ to the radius $Cm$ will be the measure of the arch $DE$.

1. 4. When the great projection is of a left circle inclined to the primitive.

Let $RDS$ (fig. 41.) be the projection of a left circle inclined to the primitive, and $DE$ a portion of that circle to be measured. Through $O$ the centre of the projected circle, and $C$ the centre of the primitive, draw the line of measures $Mm$; and from the centre $O$, with the radius $OR$, or $OS$, describe the semicircle $RGFS$; through the points $D$, $E$ draw the lines $DF$, $EG$ parallel to the line of measures, and $FG$ will be the measure of the arch $DE$ to the radius $OR$, or $OS$. Join $OF$, $OG$, and make the angle $mCH$ equal to $FOG$, and the arch $mH$ of the primitive will be the measure of the arch $DE$ of the inclined circle $RDS$.

The converse of this proposition, namely, to cut off an arch from a given projected circle equal to a given arch of the primitive, is obvious.

The above operation would be greatly shortened by using the line of fines in the sector.

It seems unnecessary to insist farther on this projection, especially as the reader will see the application of it to the projection of the sphere on the planes of the Meridian, Equator, and Horizon, in the article GEOGRAPHY; and to the delineation of Eclipses in the article ASTRONOMY. The Analemma, Plate CCXXXV. in the article GEOGRAPHY, is also according to this projection; and the method of applying it to the solution of astronomical problems is there exemplified.

In this projection the eye is in the centre of the sphere, and the plane of projection touches the sphere in a given point parallel to a given circle. It is named gnomonic, on account of its being the foundation of dialling: the plane of projection may also represent the plane of a dial, whose centre being the projected pole, the semiaxis of the sphere will be the stile or gnomon of the dial.

As the projection of great circles is represented by straight lines, and left circles parallel to the plane of projection are projected into concentric circles: therefore many problems of the sphere are very easily resolved. Other problems, however, become more intricate on account of some of the circles being projected into ellipses, parabolas, and hyperbolas.

Every great circle is projected into a straight line perpendicular to the line of measures; and whose distance from the circle is equal to the cotangent of its inclination, or to the tangent of its nearest distance from the pole of the projection.

Fig. 42. Let BAD (fig. 42.) be the given circle, and let the circle CBED be perpendicular to BAD, and to the plane of projection; whose intersection CF with this last plane will be the line of measures. Now since the circle CBED is perpendicular both to the given circle BAD and to the plane of projection, the common section of the two last planes produced will therefore be perpendicular to the plane of the circle CBED produced, and consequently to the line of measures; hence the given circle will be projected into that section; that is, into a straight line passing through $d$, perpendicular to $Cd$. Now $Cd$ is the cotangent of the angle $CdA$, the inclination of the given circle, or the tangent of the arch CD to the radius AC.

1. A great circle perpendicular to the plane of projection is projected into a straight line passing through the centre of projection; and any arch is projected into its correspondent tangent.

2. Any point, as D, or the pole of any circle, is projected into a point $d$, whose distance from the pole of projection is equal to the tangent of that distance.

3. If two great circles be perpendicular to each other, and one of them passes through the pole of projection, they will be projected into two straight lines perpendicular to each other.

4. Hence if a great circle be perpendicular to several other great circles, and its representation pass through the centre of projection; then all these circles will be represented by lines parallel to one another, and perpendicular to the line of measures, for representation of that first circle.

If two great circles intersect in the pole of projection, their representations will make an angle at the centre of the plane of projection, equal to the angle made by these circles on the sphere.

For since both these circles are perpendicular to the plane of projection, the angle made by their intersections with this plane is the same as the angle made by these circles.

Any less circle parallel to the plane of projection is projected into a circle whose centre is the pole of projection, and its radius is equal to the tangent of the distance of the circle from the pole of projection.

Let the circle PI (fig. 42.) be parallel to the plane GF, then the equal arches PC, CI are projected into the equal tangents GC, CH; and therefore C the point of contact and pole of the circle PI and of the projection, is the centre of the representation G, H.

If a circle be parallel to the plane of projection, and 45 degrees from the pole, it is projected into a circle equal to a great circle of the sphere; and therefore may be considered as the primitive circle, and its radius the radius of projection.

A less circle not parallel to the plane of projection is projected into a conic section, whose transverse axis is in the line of measures; and the distance of its nearest vertex from the centre of the plane of projection is equal to the tangent of its nearest distance from the pole of projection; and the distance of the other vertex is equal to the tangent of the greatest distance.

Any less circle is the base of a cone whose vertex is at A (fig. 43.); and this cone being produced, its intersection with the plane of projection will be a conic section. Thus the cone DAF, having the circle DF for its base, being produced, will be cut by the plane of projection in an ellipse whose transverse diameter is $df$; and $Cd$ is the tangent of the angle CAD, and $Cf$ the tangent of CAF. In like manner, the cone AFE, having the side AE parallel to the line of measures $df$, being cut by the plane of projection, the section will be a parabola, of which $f$ is the nearest vertex, and the point into which E is projected is at an infinite distance. Also the cone AFG, whose base is the circle FG, being cut by the plane of projection, the section will be a hyperbola; of which $f$ is the nearest vertex; and GA being produced gives $d$ the other vertex.

1. A less circle will be projected into an ellipse, a parabola, or hyperbola, according as the distance of its most remote point is less, equal to, or greater than, 90 degrees.

2. If H be the centre, and K k, l the focus of the ellipse, hyperbola, or parabola; then $HK\; =\; \backslash frac\{A\; d\; -\; A\; f\}\{2\}$ for the ellipse; $Hk\; =\; \backslash frac\{A\; d\; +\; A\; f\}\{2\}$ for the hyperbola; and $fn$ being drawn perpendicular to AE $f\; l\; =\; \backslash frac\{n\; E\; +\; F\; f\}\{2\}$ for the parabola.

Let the plane TW (fig. 44.) be perpendicular to the plane of projection TV, and BCD a great circle of the sphere in the plane TW. Let the great circle BED be projected into the straight line $bck$. Draw CQS perpendicular to $bk$, and Cm parallel to it and equal to CA, and make QS equal to Qm; then any angle QSt is the measure of the arch Qs of the projected circle.

Join AQ; then because Cm is equal to CA, the angle Qcm equal to QCA, each being a right angle, and the side QC common to both triangles; therefore Qm, or its equal QS, is equal QA. Again, since the plane ACQ is perpendicular to the plane TV, and bQ

Gnomonic Projection of the Sphere. to the intersection $CQ$; therefore $bQ$ is perpendicular both to $AQ$ and $QS$: hence, since $AQ$ and $QS$ are equal, all the angles at $S$ cut the line $bQ$ in the same points as the equal angles at $A$. But by the angles at $A$ the circle $BED$ is projected into the line $bQ$. Therefore the angles at $S$ are the measures of the parts of the projected circle $bQ$; and $S$ is the dividing centre thereof.

1. 1. Any great circle $bQt$ is projected into a line of tangents to the radius $SQ$.
2. 2. If the circle $bC$ passes through the centre of projection, then the projecting point $A$ is the dividing centre thereof, and $Cb$ is the tangent of its correspondent arch $CB$ to $CA$ the radius of projection.

Fig. 44. Let the parallel circle $GLH$ (fig. 44.) be as far from the pole of projection $C$ as the circle $FNI$ is from its pole; and let the distance of the poles $C,\; P$ be bisected by the radius $AO$; and draw $bAD$ perpendicular to $AO$; then any straight line $bQt$ drawn through $b$ will cut off the arches $hI,\; Fn$ equal to each other in the representations of these equal circles in the plane of projection.

Let the projections of the less circles be described. Then, because $BD$ is perpendicular to $AO$, the arches $BO,\; DO$ are equal; but since the less circles are equally distant each from its respective pole, therefore the arches $FO,\; OH$ are equal; and hence the arch $BF$ is equal to the arch $DH$. For the same reason the arches $BN,\; DL$ are equal; and the angle $FBN$ is equal to the angle $LDH$; therefore, on the sphere, the arches $FN,\; HL$ are equal. And since the great circle $BNLD$ is projected into the straight line $bQnI$, &c. therefore $n$ is the projection of $N$, and $I$ that of $L$; hence $fn,\; hI$, the projections of $FN,\; HL$ respectively, are equal.

Fig. 45. If $Fnk,\; hI$, (fig. 45.) be the projections of two equal circles, whereof one is as far from its pole $P$ as the other from its pole $C$, which is the centre of projection; and if the distance of the projected poles $C,\; P$ be divided in $o$, so that the degrees in $Co,\; oP$ be equal, and the perpendicular $oS$ be erected to the line of measures $gh$. Then the line $pn,\; C/$ drawn from the poles $C,\; P$, through any point $Q$ in the line $oS$, will cut off the arches $Fn,\; hI$ equal to each other, and to the angle $QCp$.

The great circle $Ao$ perpendicular to the plane of the primitive is projected into the straight line $oS$ perpendicular to $gh$, by Prop. i. cor. 3. Let $Q$ be the projection of $q$; and since $pQ,\; CQ$ are straight lines, they are therefore the representations of the arches $Pq,\; Cq$ of great circles. Now since $PqC$ is an isosceles spherical triangle, the angles $PCQ,\; CPQ$ are therefore equal; and hence the arches $Pq,\; Cq$ produced will cut off equal arches from the given circles $FI,\; GH$, whose representations $Fn,\; hI$ are therefore equal: and since the angle $QCp$ is the measure of the arch $hI$, it is also the measure of its equal $Fn$.

VOL. XVII. PART II.

Hence, if from the projected pole of any circle a perpendicular be erected to the line of measures, it will cut off a quadrant from the representation of that circle.

Let $Fnk$ (fig. 45.) be the projection of any circle $FI$, Fig. 45. and $p$ the projection of its pole $P$. If $Cg$ be the cotangent of $CAP$, and $gB$ perpendicular to the line of measures $gC$, let $CAP$ be bisected by $Ao$, and the line $oB$ drawn to any point $B$, and also $pB$ cutting $Fnk$ in $d$; then the angle $gOB$ is the measure of the arch $Fd$.

The arch $PG$ is a quadrant, and the angle $gOA\; =\; PA\; +\; oAP\; =\; gAC\; +\; oAP\; =\; gAC\; +\; CAo\; =\; gAO$; therefore $gA\; =\; gO$; consequently $o$ is the dividing centre of $gB$, the representation of $GA$; and hence, by Prop. v. the angle $gOB$ is the measure of $gB$. But since $pg$ represents a quadrant, therefore $p$ is the pole of $gB$; and hence the great circle $pdB$ passing through the pole of the circles $gB$ and $Fp$ will cut off equal arches in both, that is, $Fd\; =\; gB\; =\; \backslash text\{angle\; \}\; gOB$.

The angle $gOB$ is the measure of the angle $g\backslash rho\; B$. For the triangle $g\backslash rho\; B$ represents a triangle on the sphere, wherein the arch which $gB$ represents is equal to the angle which the angle $\backslash rho$ represents; because $g\backslash rho$ is a quadrant: therefore $gOB$ is the measure of both.

To draw a great circle through a given point, and whose distance from the pole of projection is equal to a given quantity.

Let $ADB$ (fig. 46.) be the projection, $C$ its pole or Fig. 46. centre, and $P$ the point through which a great circle is to be drawn: through the points $P,\; C$ draw the straight line $PCA$, and draw $CE$ perpendicular to it: make the angle $CAE$ equal to the given distance of the circle from the pole of projection $C$; and from the centre $C$, with the radius $CE$, describe the circle $EFG$: through $P$ draw the straight line $PIK$, touching the circle $EFG$ in $I$, and it will be the projection of the great circle required.

To draw a great circle perpendicular to a great circle which passes through the pole of projection, and at a given distance from that pole.

Let $ADB$ (fig. 46.) be the primitive, and $CI$ the given circle: draw $CI$ perpendicular to $CI$, and make the angle $CIJ$ equal to the given distance: then the straight line $KP$, drawn through $I$ parallel to $CI$, will be the required projection.

At a given point in a projected great circle, to draw another great circle to make a given angle with the former; and, conversely, to measure the angle contained between two great circles.

Let $P$ (fig. 47.) be the given point in the given great circle Fig. 47.

circle PB, and C the centre of the primitive: through the points P, C draw the straight line PCG; and draw the radius of the primitive CA perpendicular thereto; join PA; to which draw AG perpendicular: through G draw BGD at right angles to GP, meeting PB in B; bisect the angle CAP by the straight line AO; join BO, and make the angle BOD equal to that given; then DP being joined, the angle BPD will be that required.

If the measure of the angle BPD be required, from the points B, D draw the lines BO, DO, and the angle BOD is the measure of BPD.

To describe the projection of a less circle parallel to the plane of projection, and at a given distance from its pole.

Let ADB (fig. 46.) be the primitive, and C its centre: let the distance of the circle from its pole, from B to H, and from H to D; and draw the straight line AED, intersecting CE perpendicular to BC, in the point E: with the radius CE describe the circle EFG, and it is the projection required.

To draw a less circle perpendicular to the plane of projection.

Let C (fig. 48.) be the centre of projection, and TI a great circle parallel to the proposed less circle: at C make the angles ICN, TCO each equal to the distance of the less circle from its parallel great circle TI; let CL be the radius of projection, and from the extremity L draw LM perpendicular thereto; make CV equal to LM; or CF equal to CM: then with the vertex V and asymptotes CN, CO describe the hyperbola WVK; or, with the focus F and CV describe the hyperbola, and it will be the perpendicular circle described.

To describe the projection of a less circle inclined to the plane of projection.

Draw the line of measures $d\_p$ (fig. 49.); and at C, the centre of projection, draw CA perpendicular to $d\_p$, and equal to the radius of projection: with the centre A, and radius AC, describe the circle DCFG; and draw HAE parallel to $d\_p$: then take the greatest and least distances of the circle from the pole of projection, and let them from C to D and F respectively, for the circle DF; and from A, the projecting point, draw the straight lines AF, and AD; then $d\_f$ will be the transverse axis of the ellipse: but if D fall beyond the line RE, as at G, then from G draw the line GAD, and $d\_f$ is the transverse axis of an hyperbola: and if the point D fall in the line RE, as at E, then the line AE will not meet the line of measures, and the circle will be projected into a parabola whose vertex is $f$: bisect $d\_f$ in H, the centre, and for the ellipse take half the difference of the lines Ad, Af, which laid from H will give K the focus: for the hyperbola, half the sum of Ad, Af being laid from H, will give k its focus: then with the transverse axis $d\_f$, and focus K, or k, describe the ellipse dMf, or hyperbola fm, which will be the projection of the inclined circle: for the parabola, make EQ equal to Ff, and draw fn perpendicular to AQ, and make fk equal to

one half of nQ: then with the vertex f, and focus k, describe the parabola fm, for the projection of the given circle FE.

To find the pole of a given projected circle.

Let DMF (fig. 50.) be the given projected circle whose line of measures is DF, and C the centre of projection; from C draw the radius of projection CA, perpendicular to the line of measures, and A will be the projecting point: join AD, AF, and bisect the angle DAF by the straight line AP; hence P is the pole. If the given projection be an hyperbola, the angle fAG (fig. 49.), bisected, will give its pole in the line of measures; and in a parabola, the angle fAE bisected will give its pole.

To measure any portion of a projected great circle, or to lay off any number of degrees thereon.

Let EP (fig. 51.) be the great circle, and IP a portion thereof to be measured: draw ICD perpendicular to IP; let C be the centre, and CB the radius of projection, with which describe the circle EBD; make IA equal to IB; then A is the dividing centre of EP; hence AP being joined, the angle IAP is the measure of the arch IP.

Or, if IAP be made equal to any given angle, then IP is the correspondent arch of the projection.

To measure any arch of a projected less circle, or to lay off any number of degrees on a given projected less circle.

Let Fn (fig. 52.) be the given less circle, and P its pole: from the centre of projection C draw CA perpendicular to the line of measures GH, and equal to the radius of projection; join AP, and bisect the angle CAP by the straight line AO, to which draw AD perpendicular: describe the circle G/H, as far distant from the pole of projection C as the given circle is from its pole P; and through any given point n, in the projected circle Fn, draw Dn/, then H/ is the measure of the arch Fn.

Or let the measure be laid from H to l, and the line D/ joined will cut off Fn equal thereto.

To describe the gnomonic projection of a spherical triangle, when three sides are given; and to find the measures of either of its angles.

Let ABC (fig. 53.) be a spherical triangle whose three sides are given: draw the radius CD (fig. 54.) perpendicular to the diameter of the primitive EF; and at the point D make the angles CDA, CDG, ADI, equal respectively to the sides AC, BC, AB, of the spherical triangle ABC (fig. 53.), the lines DA, DG intersecting the diameter EF, produced if necessary in the points A and G: make DI equal to DG; then from the centre C, with the radius CG, describe an arch; and from A, with the distance AI, describe another arch, intersecting the former in B; join AB, CB, and ACB will be the projection of the spherical triangle (fig. 53.); and the rectilinear angle ACB is the measure of the spherical angle ACB (fig. 53.).

The three angles of a spherical triangle being given, to project it, and to find the measures of the sides.

Let ABC (fig. 55.) be the spherical triangle of which the angles are given: construct another spherical triangle EFG, whose sides are the supplements of the given angles of the triangle ABC; and with the sides of this supplemental triangle describe the gnomonic projection, &c. as before.

It may be observed, that the supplemental triangle EFG has also a supplemental part EFG₁; and when the sides GE, GF, which are substituted in place of the angles A, B, are obtuse, their supplements $\backslash bar\{g\}E$, $\backslash bar\{g\}F$ are to be used in the gnomonic projection of the triangle.

Given two sides, and the included angle of a spherical triangle, to describe the gnomonic projection of that triangle, and to find the measures of the other parts.

Let the sides AC, CB, and the angle ACB (fig. 56.), be given; make the angles CDA, CDG (fig. 56.) equal respectively to the sides AC, CB (fig. 53.); also make the angle ACB (fig. 56.) equal to the spherical angle ACD (fig. 53.), and CB equal to CG, and ABC will be the projection of the spherical triangle.

To find the measure of the side AB: from C draw CL perpendicular to AB, and CM parallel thereto, meeting the circumference of the primitive in M; make LN equal to LM; join AN, BN, and the angle ANB will be the measure of the side AB.

To find the measure of either of the spherical angles, as BAC: from D draw DK perpendicular to AD, and make KH equal to KD: from K draw KI perpendicular to CK, and let AB produced meet KI in I, and join HI: then the rectilinear angle KHI is the measure of the spherical angle BAC. By proceeding in a similar manner, the measure of the other angle will be found.

Two angles and the intermediate side given, to describe the gnomonic projection of the triangle; and to find the measures of the remaining parts.

Let the angles CAB, ACB, and the side AC of the spherical triangle ABC (fig. 53.), be given: make the angle CDA (fig. 56.) equal to the measure of the given side AC (fig. 53.); and the angle ACB (fig. 56.) equal to the angle ACB (fig. 53.); produce AC to H, draw DK perpendicular to AD, and make KH equal to KD; draw KI perpendicular to CK, and make the angle KHI equal to the spherical angle CAB: from I, the intersection of KI, HI, to A draw IA, and let it intersect CB in B, and ACB will be the gnomonic projection of the spherical triangle ABC (fig. 53.). The unknown parts of this triangle may be measured by last problem.

Two sides of a spherical triangle, and an angle opposite to one of them given, to describe the projection

of the triangle; and to find the measure of the remaining parts.

Let the sides AC, CB, and the angle BAC of the spherical triangle ABC (fig. 53.) be given: make the angles CDA, CDG (fig. 56.) equal respectively to the measures of the given sides AC, BC: draw DK perpendicular to AD, make KH equal to DK, and the angle KHI equal to the given spherical angle BAC: draw the perpendicular KI, meeting HI in I; join AI; and from the centre C, with the distance CG, describe the arch GB, meeting AI in B, join CB, and ABC will be the rectilinear projection of the spherical triangle ABC (fig. 53.) and the measures of the unknown parts of the triangle may be found as before.

Given two angles, and a side opposite to one of them, to describe the gnomonic projection of the triangle, and to find the measures of the other parts.

Let the angles A, B, and the side BC of the triangle ABC (fig. 53.) be given: let the supplemental triangle EFE be formed, in which the angles E, F, G, are the supplements of the sides BC, CA, AB, respectively, and the sides EF, FG, GE, the supplements of the angles C, A, B. Now at the centre C (fig. 56.) make the angles CDA, CDK equal to the measures of the sides GE, GF respectively, being the supplements of the angles B and A; and let the lines DA, DK intersect the diameter of the primitive EF in the points A and K: draw DG perpendicular to AD, make GH equal to DG, and at the point H make the angle GHI equal to the angle E, or to its supplement; and let EI, perpendicular to CH, meet HI in I, and join AI: then from the centre C, with the distance CG, describe an arch intersecting AI in B; join CB, and ABC will be the gnomonic projection of the given triangle ABC (fig. 53.): the supplement of the angle ACB (fig. 56.) is the measure of the side AB, (fig. 55.); the measures of the other parts are found as before.

It has already been observed, that this method of projection has, for the most part, been applied to dialling only. However from the preceding propositions, it appears that all the common problems of the sphere may be more easily resolved by this than by either of the preceding methods of projection; and the facility with which these problems are resolved by this method has given it the preference in dialling. It may not perhaps be amiss, in this place, to give a brief illustration of it in this particular branch of science.

In an horizontal dial, the centre of projection Z (fig. 57.) represents the zenith of the place for which the dial is to be constructed; ZA the perpendicular height of the style: the angle ZPA, equal to the given latitude, determines the distance ZP of the zenith from the pole; and AP the edge of the style, which by its shadow gives the hour: the angle ZAP, equal also to the latitude, gives the distance of the equator EO from the zenith: let E a be equal to EA, and a will be the dividing point of the equator. Hence if the angles E a I, E a II, &c. E a XI, E a X, &c. be made equal to $15^\backslash circ$, $30^\backslash circ$, &c. the equator will be divided into hours; and

Geometric Projection of the Sphere. and lines drawn from P to these points of division will be hour lines.

If the dial is either vertical, or inclined to the horizon, then the point Z will be the zenith of that place whose horizon is parallel to the plane of the dial: ZE will be that latitude of the place; and the hours on the former dial will now be changed into others, by a quantity equal to the difference of longitude between the given place and that for which the dial is to be constructed. Thus, if it is noon when the shadow of the style falls on the line PX, then the difference of meridians is the angle E a X, or 30°. Hence, when a dial is to be constructed upon a given plane, either perpendicular or inclined to the horizon, the declination and inclination of that place must be previously found.

In an erect direct south dial, its zenith Z is the south point of the horizon, ZP is the distance of this point from the pole, and ZE its distance from the equator. If the dial is directed to the north, Z represents the north point of the horizon; PZ the distance of Z from the pole under the horizon; and ZE the elevation of the equator above the horizon.

If the dial is an erect east or west dial, the zenith Z is the east or west points of the horizon accordingly, and the pole P is at an infinite distance, for the angle ZAP is a right angle; and therefore the line AP will

not meet the meridian PZ. The line ZA produced is Geometric Projection of the Sphere. the equator, and is divided into hours by lines perpendicular to it.

If the plane of the dial is parallel to the equator, its zenith Z coincides with one of the poles of the equator P; and hence the hour lines of this dial are formed by drawing lines from the point Z, containing angles equal to 15°.

In the preceding methods of projection of the sphere, equal portions of a great circle on the sphere are represented by unequal portions in the plane of projection, and this inequality increases with the distance from the centre of projection. Hence, in projections of the earth, those places towards the circumference of the projection are very much distorted. In order to avoid this inconvenience, M. de la Hire * proposed, that the eye should be placed in the axis produced at the distance of the sine of 45° beyond the pole: In this case arches of the sphere and their projections are very nearly proportional to each other. Hence in a map of the earth agreeable to this construction, the axis, instead of being divided into a line of semitangents, is divided equally, in like manner as the circumference. The map of the world is constructed agreeable to this method of projection.

Projection Prolate. PROJECTION, in Perspective, denotes the appearance, or representation of an object on the perspective plane.

The projection of a point is a point through which an optic ray passes from the objective point through the plane to the eye; or it is the point wherein the plane cuts the optic ray.

And hence may be easily conceived what is meant by the projection of a line, a plane, or a solid.

PROJECTION, in Alchemy, the casting of a certain imaginary powder, called powder of projection, into a crucible, or other vessel, full of some prepared metal, or other matter; which is to be hereby presently transmuted into gold.

POWDER OF PROJECTION, or of the philosophers stone, is a powder supposed to have the virtue of changing any quantity of an imperfect metal, as copper or lead, into a more perfect one, as silver or gold, by the admixture of a little quantity thereof.

The mark to which alchemists directed all their endeavours, was to discover this powder of projection. See PHILOSOPHERS Stone, and CHEMISTRY, History of.

PROJECTION, in Architecture, the outsetting and prominence, or embossing, which the mouldings and other members have beyond the naked wall, column, &c.

PROLAPSUS, in Surgery, a prolapsion or falling out of any part of the body from its natural situation: thus we say, prolapsus intestini, "a prolapsion of the intestine," &c. See SURGERY.

PROLATE, in Geometry, an epithet applied to a spheroid produced by the revolution of a semi-ellipse about its larger diameter. See SPHEROID.

PROLEGOMENA, in Philology, certain preparatory observations or discourses prefixed to a book, &c. containing something necessary for the reader to be apprised of, to enable him the better to understand the book, or to enter deeper into the science, &c.

PROLEPSIS, a figure in Rhetoric, by which we anticipate or prevent what might be objected by the adversary. See ORATORY, N° 80.

PROLEPTIC, an epithet applied to a periodical disease which anticipates, or whose paroxysm returns sooner and sooner every time; as is frequently the case in agues.

PROLIFER FLOS, (proles, "an offspring;" and fero, "to bear;") a prolific flower, or a flower which from its own substance produces another; a singular degree of luxuriance, to which full flowers are chiefly incident. See BOTANY.

PROLIFIC, something that has the qualities necessary for generating.

The prolific powers of some individuals among mankind are very extraordinary.—Instances have been found where children, to the number of six, seven, eight, nine, and sometimes sixteen, have been brought forth after one pregnancy. The wife of Emmanuel Gago, a labourer near Valladolid, was delivered, the 14th of June 1779, of five girls, the two first of whom were baptized: the other three were born in an hour after; two of them were baptized; but the last, when it came into the world, had every appearance of death. The celebrated Tarfin was brought to bed in the seventh month of her pregnancy, at Argenteuil near Paris, 17th July 1779, of three boys, each 14 inches and a half long, and of a girl 13 inches: they were all four baptized, but did not live 24 hours.

The

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PROJECTION of the SPHERE.

Plate CCCCXLVI.

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A Ball from the Antipodal point.

PLATE I. GEOMETRICAL CONSTRUCTIONS. BY J. H. B. ...

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