"I am apt to imagine, that if we were to employ the fame Care about all our other Ideas, as Mathematicians have done about thofe of Number and Magnitude, by forming them into exact "Combinations, and diftinguishing these Combinations by parti"cular Names, in order to keep them fteady and invariable, we fhould foon have it in our Power to introduce Certainty and De"monftration into other Parts of human Knowledge" And again," If we would form our Minds to a Habit of Reasoning clofely and in Train, we cannot take any more certain Method, than the exercifing ourselves in Mathematical Demonftrations, "fo as to contract a kind of Familiarity with them." Not that we look upon it as neceffary, (to ufe the Words of the great + Mr. Lock) that all Men fhould be deep Mathematicians, but that, "having got the Way of Reasoning, which that Study neceffarily brings the Mind to, they may be able to transfer it to other "Parts of Knowledge, as they shall have Occafion. For, in all forts of Reafoning, every fingle Argument fhould be managed as a "Mathematical Demonftration, the Connection and Dependence of Ideas fhould be followed, till the Mind is brought to the Source "on which it bottoms, and can trace the Coherence through the "whole Train of Proofs. It is in the general obfervable, that the "Faculties of our Souls are improved and made ufeful to us, juft

after the fame Manner as our Bodies are. Would you have a "Man write or paint, dance or fence well, or perform any other manual Operation, dexterously and with Eafe? Let him have ever fo much Vigour and Activity, Supplenefs and Addrefs naturally, yet no-body expects this from him, unless he has been "ufed to it, and has employed Time and Pains in fashioning and ❝forming his Hand, or outward Parts, to thefe Motions. Juft fo

it is in the Mind; would you have a Man reafon, you must ufe him to it betimes, exercife his Mind in obferving the Con"nection of Ideas, and following them in Train. Nothing does

66

this better than Mathematics; which therefore I think should "be taught all thofe, who have Time and Opportunity; not fo "much to make them Mathematicians, as to make them reafona"ble Creatures; for though we all call ou felves fo, because we are born to it, if we pleafe; yet we may truly fay, Nature gives us but the Seeds of it. We are born to be, if we please, ratio"nal Creatures; but 'tis Ufe and Exercise only that makes us fo, and we are indeed fo, no farther than Industry and Application has carried us.' And in another † Place the fame learned Gentleman fays, "We muft-if we will proceed as Reafon advifes, adapt our Methods of Enquiry to the Nature of Ideas we exa"mine, and the Truth we fearch after. General and certain "Truths are only founded in the Habitudes and Relations of "abstract Ideas. A fagacious and methodical Application of our Thoughts, for the finding out thefe Relations, is the only Way

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*Duncan's Logic, p. 224. + Lock's Conduct of human Understanding.

In his Effay concerning human Understanding, Vol. 2. P. 262.

to

to difcover all that can be put, with Truth and Certainty con. "cerning them, into general Propofitions. By what Steps we are "to proceed in thefe, is to be learned in the Schools of the Ma"thematicians, who from very plain and eafy Beginnings, by gentle Degrees, and a continued Chain of Reafonings, proceed to the Difcovery and Demonstration of Truths, that appear at first Sight beyond human Capacity. The Art of finding Proofs, and the admirable Methods they have invented for the fingling "out, and laying in Order thofe intermediate Ideas, that demonftratively fhew the Equality or Inequality of unapplicable Quanti.. ties, is that which has carried them fo far, and produced fuch "wonderful and unexpected Discoveries."

Having thus fhewn the great Ufefulness of Mathematical Learning, in improving the Mind by giving it an Habit of close and demonftrative Reasoning, we might now proceed to fhew, that the Undertanding is by it greatly enlarged, yea enlarged fo vaftly, that an ingenious Author makes no Scruple of afferting, that whoever s ignorant of thefe Sciences, though they may have had the Ho"nour of the Title, are yet only nominal Mafters of Arts," but this will in a great Meafure appear under the next Head, in which we are to fhew, Thirdly, the Advantages of these Studies in fome particular Profeffions, &c. viz.

1. That it is a useful Study for Children,

2. For Tradefmen.

3. For young Gentlemen,

4. For Phyficians,

5.

For Divines.

1. We are to fhew that it is a ufeful Study for Children There are, fays Dr. Waits, feveral of the Sciences that will more agreeably employ our younger Years, and the general Parts of them may be eafily taken in by Boys The first Principles and "eafier Practices of Arithmetic, Geometry plain Trigonometry, Measuring Heights, Depths, Lengths, Diftances, &c. the Rudiments of Geography and Aftronomy, together with fome-. thing of Mechanics, may be eafily conveyed into the Minds of acute young Perfons from 9 or 10 Years old or upward. "Thefe Studies may be entertaining and ufeful to young Ladies, as well as young Gentlemen; and all who are bred up to the learned Profeffions. The fair Sex may intermingle thefe with "the Operations of the Needle, and the Knowledge of domestic "Life. Boys may be taught to join them with their Rudiments

of Grammar, and their Labour in the Languages. And even those who never learn any Language but their Mother Tongue, may "be taught thefe Sciences with latting Benefit in early Days...

"That this may be done with Eafe and Advantage, take thefe three Reafons.

"1. Because they depend so much upon Schemes and Numbers, Images, Lines, and Figures, and fenfible Things, that the Imagination

*Supplement to hie Logic

66

gination or Fancy will greatly affift the Understanding, and ren-, der the Knowledge of them much more easy.

2. These Studies are fo pleasant that they will make the dry Labour of learning Words, Phrafes, and Languages, more tolerable to Boys in a Latin School by this moft agreeable Mixture, The Employment of Youth in thefe Studies will tempt them to neglect many of the foolish Plays of Childhood, and they will * find fweeter Entertainment for themselves and their leisure Hours, by a Cultivation of these pretty Pieces of alluring Knowledge.

3. The Knowledge of thefe Parts of Science are both eafy and worthy to be retained in Memory by all Children, when they come to manly Years, for they are ufeful through all the Parts "of Human Life: They tend to enlarge the Underftanding early, and to give a various Acquaintance with ufeful Subjects betimes. "And furely it is best as far as poffible to train up Children in the Knowledge of thofe Things which they fhould never forget, rather than to let them wafte Years of Life in Trifles, or in "hard Words which are not worth remembering."

Secondly, That the Mathematical Sciences, are useful to Tradef men, or of the greatest Service in the Conveniencies of Life and Com merce, will appear by only enumerating a few of them; fuch are, the regular Keeping of Accounts, Meafuring and Gauging of Solids and Veffels, &c. for giving every Man his juft Property; the Regulation of Time by Sun-dials, Clocks, Watches, &c. the Feafts of the Church by the Motion of the Sun and Moon; and Chronology for the better Understanding of Hiftory. The Construction of Houses for the Conveniency of Life, and of Fortifications for our better Defence from the Ravages of our Enemies. By these Sciences it is, that we, after the best Manner, conftruct “all forts "of Inftruments to work with, all Engines of War, Ships,

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Bridges, Mills, curious Roofs and Arches, flately Theatres, "Columns, Pendent Galleries, and all other grand Works in Building. Alfo Jacks, Chariots, Carts and Carriages, and even the Wheel-barrow.. —and whatever hath artificial Motion by Air, Water, Wind, or Cords." By thefe Sciences weafcertain the Changes or Alterations in the Air, either as to Weight or Moisture, Heat and Cold; by Barometers, Thermometers, Hygrofcopes, &c. By thefe Sciences we conftruct Globes, Spheres, Orreries, &c. for the readier conveying Ideas of the Motion of the Sun, Stars, and Planetary Bodies. By thefe Sciences we are furnished with Telescopes, Microfcopes, &c. by which the Organ of Sight is furprizingly extended and augmented." How fur"prized had the Antients been, if they had been told that their Pofterity, by the Help of fome Infruments, fhould one Day fee a Heaven that was unknown to them, and Plants and Animals "they did even fufpect it was poffible to exift."

Emerson's Mechanics in the Preface,

Fontenelle's Speech.

Mathef

"

Mathefis how explains all the Phænomena of Motion; "which involve almoft every Branch of Natural Philofophy. She "purfues the Rays of Light in the Immenfity of etherial Space; pervades all Substances through which they penetrate, detects them in all the Mazes of Reflection and Refraction, and com"pels them to discover those Objects which either Distance of "Minutenefs had fo long concealed. She has ftretched her Line "round the World on which we live, has determined its Magnitude, and defcribed its Figure; fhe has pointed out to Geogra phy the true Situation of every Sea, and every Shore, and "and taught the Mariner to purfue his Way without Deviation "wherever bound, though the Wave that clofes after him leaves no Track, and the Land which finks below the Horizon-affords "no Direction.”

In fhort, the Ufefulness of thefe Sciences even in Mechanical Profeffions is very great: For what could the Architect, Shipwright, Optician, Painter, &c. perform without fome Knowledge in the Mathematics? A complete Architect and Shipwright must have confiderable Knowledge in Geometry and Mathematics, for drawing their Defigns, and judging of the Strength, &c. of their Works. The Optician ought to understand the Laws of Refrac tion and Reflection, the Foci of Glaffes, Mirrors, &c. and the Painter, the Principles of Perspective. And to the Mathematics it must be owned the principal Improvements in these Arts have been owing, though the generality of Workmen are ignorant therez of. In a Word, how could we perform the neceffary Helps of Sot ciety? without these inestimable! these admirable Arts!

3. That Mathematical Sciences are proper Studies for young Gentlemen.

Mr. RAY was fo fully perfuaded of the Ufefulness of thefe Studies to young Gentlemen, that, he fays, tdo earnestly "exhort those that are young, especially Gentlemen, to fet upon "these Studies, and take fome Pains in them." For he adds, he does not" fee what more ingenious and manly Employment they "can purfue; tending more to the Satisfaction of their own Minds, and the Illustration of the Glory of God. For He is wonderful in all his Works". Mr. RAY is alfo of Opinion, that," did but young Men fill up that Time with these Studies, "which lies upon their Hands, which they are incumbered with, " and troubled how to pass away, much might be done, even "fo."

1

Dr. WATTS makes this Obfervation « That where Stu dents, or indeed any young Gentlemen, have in their early Years "made themselves Mafters of a Variety of elegant Problems in the "Mathematic Circle of Knowledge, and gained the most easy, "neat, and entertaining Experiments in Natural Philofophy, with "fome short and agreeable Speculations or Practices in any other

In

Mel. Maupertuis's Speech to the Royal Academy at Berlin; fee Gentleman's Magazine, Vol. 21. † In his Wifdom of Ged in the Creation, p. 203. the Spplement to his Logic.

** of

"of the Arts and Sciences, they have hereby laid a Foundation "for the Elteem and Love of Mankind among those with whom they converfe, in higher or lower Rank of Life; they have "been often guarded by this Means from the Temptation of no"cent Pleasures, and have fecured both their own Hours, and the "Hours of their Companions, from, runing to Wafte in Santering, and Trifles, and from a thousand Impertinences in filly Dialogues. Gaming and Drinking, and many criminal and foolish Scenes of Talk and Action, have been prevented by thefe innocent and improving Elegancies of Knowledge."

This learned Gentleman in another Place fays, that," befides the common Skill in Accounts which is needful for a Trader, "there is a Variety of pretty and useful Rules and Practices in Arithmetic, to which a Gentleman fhould be no Stranger; and, "if his Genius lie that Way, a little Infight into Algebra would

be no Difadvantage to him. It is fit that young People of any "Figure in the World fhould fee fome of the Springs and Clues: "whereby fkilful Men, by plain Rules of Reafon, trace out the moft deep, diftant, and hidden Queftions; and whereby they find certain Anfwers to thofe Enquiries, which, at first View, feem to lie without the Ken of Mankind, and beyond the Reach "of human Knowledge. It was for Want of a little more general Acquaintance with Mathematical Learning in the World, that a "good Algebraift and Geometer were accounted Conjurers a Century ago, and People applied to them, to feek for loft Honfes "and ftolen Goods."

Young Gentlemen "fhould know fomething of Geometry, lợ "far at least as to understand the Names of the various Lines and "Angles, Surfaces and Solids; to know what is meant by a right "Line or a Curve, a right Angle, &c. The World is now grown "fo learned in Mathematical Science, that this fort of Language is "often ufed in common Writings and Converfations, far beyond "what it was in the Days of our Fathers. And befides, without "fome. Knowledge of this Kind, we cannot make any farther "Progress toward an Acquaintance with the Arts of Surveying, "Meafuring, Geography, and Aftronomy, which are fo enter"taining and fo ufeful an Accomplishment to Perfons of a polite

Education. Geography, and Aftronomy, are exceeding delightful Studies. The Knowledge of the Lines and Circles of "the Globes of Heaven and Earth is counted fo neceflary in our Age, that no Person of either Sex is now efteemed to have had "an elegant Education without it, Even Tradesmen and the "Actors in common Life fhould, in my Opinion, in their younger "Years, learn fomething of thefe Sciences, inftead of vainly "wearing out feven Years of Drudgery in Greek and Latin,

"It is of confiderable Advantage, as well as Delight, for Man“ kind, to know a little of the Earth on which they dwell, and of "the Stars and Skies that furround them on all Sides. It is almoft "neceffary for young Perfons, who pretend to any thing of In