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In

"no lefs; by which they were called either for their Excellency "or becaufe of all the Sciences they were firft taught, or because "they were judged to comprehend távra ra Manyara," thofe Sciences the Art of Reasoning is allowed to reign in its greatest Perfection. Hence it was that the Antients, who so "well understood the Manner of forming the Mind, always began "with Mathematics, as the Foundation of their Philofophical Studies. Here the Understanding is by Degrees habituated to "Truth, contracts infenfibly a certain Fondnefs for it, and learns never to yield its Affent to any Propofition, but where the Evidence is fufficient to produce full Conviction. For this Reafon, "Plato has called Mathematical Demonftrations the Cathartics or Purgatives of the Soul, as being the proper Means to cleanse it "from Error, and reftore that natural Exercife of its Faculties, in which juft Thinking confifts. And, indeed, I believe it will "be readily allowed, that no Science furnishes fo many Instances of a happy Choice of intermediate Ideas, and a dexterous Application of them, for the Discovery of Truth, and Enlargement of Knowledge." Hence Mathefis has been juftly called (by the Reverend Mr. Baker) the Princefs of all Sciences; and hence + Kings and Princes heretofore have been fo enamoured with "her Simplicity and Pleafantnefs, that (forfaking all the Delights, of their Kingdoms) they have made their Addreffes to her Shrines, paid Homage to her Altars; thus redeeming Science at "fo great a Price. Should I mention Anacharfis the Scythian, and "Heraclitus the Ephefian, who preferred the Contemplation of Philofophy before their hereditary Kingdoms, and chofe rather "(leaving those) to fit at the Feet of Philofophers, than on their Kingly Thrones: Should I recount Allas King of Mauritania, whom (for his Aftronomic Skill, wherein he excelled) Antiquity hath fabled to bear up the Heavens on his Shoulders; or Agathocles King of Sicily, Ptolomy of Philadelphia, Alphonfus of Caftile, "Frederic of Denmark, William Landgrave of He, &c. Yea, but "fhould I mention Emperors, viz. Cæfar, Adrian, Theodofius, &c. "who (devoting themselves to thefe Studies, worthy indeed of Emperors) rendered themselves more illuftrious, by their Writings, than by their warlike (though many and great) Atchievements "Ifhould but filently fhame and reproach this our degenerate Age" In which, notwithstanding the Excellency of this Science is fuch as to make it neceffary to be studied as an Introduction to most other Arts; and that it is known from " Experience that great Genius's have furpaffed themfelves by cultivating it, and ordinary ones have become great and fublime; and the meanest have thereby acquired a Capacity and Enlargement of Judgment;" yet it is not fo univerfally ftudied as fome other Sciences, to the most convincing Arguments of which the Profeffors" cannot fubfix a Quod erat demonftrandum: And yet their Schools are fo ftuffed with

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*Duncan's Logic, p. 223. + Preface to the Rev. Mr. Baker's Geometrical Stonebeufes Arithmetic. § Baker's Geometrical Key, "Profe

Key.

Profelytes, that they have fcarce Room to breathe in; whilft the Mathematic (School) only (in which, not fome one Truth only is "expanded but even innumerable; and thofe not mean and obvious, but moft high, admirable, and myflerious, are clearly de"monftrated) lies orbate and neglected. From this they fly as from

a Peft-house; but to thofe they troop, as to a Delphic Oracle, "or as Doves to white Dove-houfes.- Laftly, though her intrinfic "Worth and Beauty hath compelled others of the lowest Orb (who

(faluting her only at the Threfhold) never entered, or had the "leaft Glimpse of her Arcana's or inner Rooms) to admire her; yet, "certain it is, very few are fkilled in her Myfteries; by which "Means it comes to pass, that she is as little regarded, as her "Clients rewarded. For what Cause this beautiful Goddess should *thus fuffer an Eclipfe in her Glory and Efteem with the Vulgar, now-a-days, I cannot divine; whether it be, fhe being a liberal *Science, and therefore (on that Account) unfuitable to the Hu"mours of thofe close-fifted Mifers (who are scarce to be reckoned among the Number of Men) who love to have their Purses, enriched rather than their Minds: Or, whether their Defpondency "of ever arriving to any confiderablé Eminency of Height, (it be*ing as good to be nothing, as not a none-fuch, or but a Spy, to s to an Art:) Or whether it be the fancied Difficulty and Knottinefs

of the Study itself, (which I havé moft Caufe to fufpect.) Or, "what that fuppofed Mormo may be, that forestals and prejudiceth fome newly entered, and fcares others, who have tasted some of her Sweets, from farther Effays (which, in fine, would have crowned their Sedulity and Diligence with Evidence and Certainty, I "fhall not undertake to determine. But this (Reader) is as much abfurd, as ftrange, viz. That what should recommend this Study to thy Reafon fhould difcourage thee; that what should "animate thy Diligence, and quicken thee to a further Effay, fhould decreft and difpirit thee. Real Difficulties (much less "conceived Prejudices) fhould be fo far from blunting thy Edge, that they fhould rather be the Whetstone of Virtue, and fharpen thy Endeavours: Why may not the fame Things, which (for "the Excellency of them) are the Objects of thy Admiration, be " (for their Poffibility) as well the Object of thy Hope, and the "Encouragement of thy Induftry? The Difficulties of this Art are "not fo infuperable, but (as in War) may be overcome, either

by Induftry, or Fortune, or both." But, if the Learner should meet with fuch Difficulties as he cannot eafily furmount by himself, or would go through his Studies with more Pleasantnefs and Difpatch, if he is of Ability, he would do well to call in the Affiftance of fome able Profeffor; for 66 there are few Perfons of fo

pene

trating a Genius and fo juft a Judgment, as to be capable of learning the Arts and Sciences without the Affittance of Teachers. "There is fcarce any Science fo fafely and fo fpeedily learned, i. even by the nobleft Genius and the beft Books, without a Tutor. "His Afiflance is abfolutely neceffary for most Perfons, and it is

very useful for all Beginners. Books are a Sort of dumb Teachers, "they point out the Way to Learning; but, if we labour under any Doubt, or Miftake, they cannot answer fudden Questions, or explain prefent Doubts and Difficulties; This is properly the Work of a living Inftructor." But, to return from this Digref fion, to fhew, Secondly, the great Ufefulness of thofe Sciences to all Perfons in general, in the Improvement of the Mind.

The principal Advantages which the Mind receives from Mathematical Studies, are 1. The accuftoming it to Attention. 2. The freeing it from Prejudice, Credulity, and Superftition. 3. The acquiring a Habit of clofe and demonftrative Reafoning."

* The Mathematics make the Mind attentive to the Objects it confiders. This they do by entertaining it with a great Variety of Truths, which are delightful and evident, but not ob vious. Truth is the fame Thing to the Understanding as Mufic to the Ear, and Beauty to the Eye. The Pursuit of it does really as much gratify a natural Faculty implanted in us by our wife Creator, as the Pleafing of our Senses: Only in the former Cafe, as the Object and Faculty are more spiritual, the Delight is more pure, free from the Regret, Turpitude, Laffitude, and Intempe"rance, that commonly attend fenfual Pleafures. The moft Part

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of other Sciences confifting only of probable Reafonings, the *Mind has not where to fix; and, wanting fufficient Principles to "purfue its Searches upon, gives them over as impoffible. Again,

as in Mathematical Investigations Truth may be found, fo it is "not always obvious: This spurs the Mind, and makes it diligent and attentive. And Plato (in Repub. Lib. VII.) obferves, that the Youth, who are furnifhed with Mathematical Knowledge, are prompt and quick at all other Sciences.-Youth is generally fo much more delighted with Mathematical Studies than with the unpleafant Tafks that are fometimes impofed upon them, that I have known fome reclaimed by them from Idlenefs and Neglect of Learning, and acquire in Time an Habit of Thinking, "Diligence, and Attention; (Qualities which we ought to ftudy

by all Means to beget in their defultory and roving Minds.") And this is no Wonder, if we confider, that the Abftractedness of pure Mathematics is a proper Remedy to cure the Lightness of their Minds, acting as a Rein to curb the Impetuofity of their Paffions. And that the Study of thofe Science infpires a Love for Truth, the Purfait of which t "will give, the otherwife unemployed, a Diftafte of thofe vain Occupations that hurry Men into "Libertinifm and Debauchery."

"Secondly, Mathematical Knowledge adds a manly Vigour to the Mind, frees it from Prejudice, Credulity, and Superstition. This it does two Ways, 1. By accuffoming us to examine, and not to take Things upon Truit. 2. By giving us a clear and "extenfive Knowledge of the Syftem of the World; which, as it creates in us the moft profound Reverence of the almighty and

*Effay on the Ufefulness of Mathematical Learning. † Stoneboufe's Arithmetic. Elay on the Ufefulness of Mathematical Learning.

wife Creator, fo it frees us from the mean and narrow Thoughts which Ignorance and Superftition are apt to beget.",

The third Advantage which the Mind receives from Mathematics is the Habit of clear, demonftrative, and methodical Reasoning., Mathefis is a Study that tends not only to the Improvement of Arts, but alfo to the Regulation of the Paffions; a Study that "will infenfibly bring Men to think methodically, reafon correctly. and feparate Truth from Falfhood, and the Difguife of Words, which it generally wears."

The Writings of the Mathematicians have been conducted by fo perfect a Model, as to be an inconteftable Proof of the Firm"nefs and Stability of human Knowledge, when built upon fo fure "a Foundation. For not only, the Propofitions of this Science "ftood the Teft of all Ages, but are found attended with that in"vincible Evidence, as forces the Affent of all, who duly confider "the Proofs upon which they are established. The Mathema"ticians are univerfally allowed to have hit upon the right Method of arriving at Truths. They have been the happiest in the Choice, as well as Application of their Principles."

In a Word, fome Knowledge in both pure and mixt Mathematics is by Experience found not only neceffary in many particular Profeffions, but alfo of great Ufe to all Men in general, in the Improvement of the Mind; and, therefore, the Study of them is now defervedly thought, not only of the greatest Ufe, but also a neceflary Part of the Education of Gentlemen; and are accordingly made a Part of it, in our two famous Universities. Not fo much to make them Mathematicians, as, by engaging them to obferve the Method of Reasoning made Ufe of in the Mathematical Sciences, they may acquire fomething of that Juftnefs and Solidity of Reasoning, for which the Profeffors of thefe Sciences are fo generally, and deferyedly esteemed.

Perhaps what has been already faid, may be fufficient to fhew the great Ufefulness of Mathematical Studies, for acquiring a juft Method of Reafoning However, that the Reader may himself be able in fome Measure to judge of the Truth of the above Affertions, it may not be improper to lay before him a general Account of the Method made Ufe of by Mathematicians; which is this: They first begin with Definitions, (from Definitio, Lat.) in which the Meaning of their Words is fo diftinctly explained, as to prevent any Ambiguity, (or double Meaning) By which Means, every attentive Reader has the very fame Ideas excited in his Mind, as the Writer has annexed to them..

By this Means, the Mathematicians have fecured themselves, and the Sciences which they profefs, from Wrangling and Controverfy; and if the Writers of Natural Philofophy, and Morality, had ufed the fame Accuracy and Care in adjufting the Definitions wherefoever neceffary, they had effectually fecluded a Multi

tude of noify and fruitlefs Debates out of their feveral Provinces, Nor had that facred Theme of Divinity been perplexed with fo

Stoneboufe's Arithmetic. † Duncan's Logic, p. 181.

Watts's Logic. "many

many intricate Difputes; nor the Church of Christ been torn to "Pieces by fo many Sects and Factions, if the Words Grace, Faith, Righteoufnefs, Repentance, Juftification, Worship, Church, Bifhop, Prefbyter, &c. had been well defined, and their Signifi"cation adjusted, as near as poffible, by the Ufe of thofe Words " in the New Teftament; or at leaft, if every Writer had told us at firft, in what Senfe he would ufe thofe Words."

The fecond Step, in Mathematical Writings, is to lay down. fome felf-evident Truths, which may ferve as a Foundation on which to build the future Reafonings. Thefe Propofitions are divided into two Sorts, called Axioms and Poftulates.

An Axiom (Axioma, Lat.) is a felf-evident fpeculative Truth, as," the Whole is greater than its Part." A Poftulate (Poftulatum, Lat.) is a felf-evident practical Propofition, as, "grant that a finite "Right-line may be continued directly forward.

Having thus fecurely laid the Foundation, the Mathematicians begin in their next Step to build their Superftructure of demonftrable Propofitions, i. e. Propofitions which are not of themfelves felf-evident. Of demonftrable Propofitions there are also two Kinds, fpeculative and practical; a fpeculative Propofition is called a Theorem (Sena); and a practical one, a Problem (g). Thefe they demonftrate in a Series of Reafoning, proceeding carefully Step by Step, affuming nothing for Truth, but the Axioms and Poftulates, before laid down; or fome Propofition already demonftrated; and hence it follows, that, as the Principles on which their Reafoning is founded is true, the Confequences (rightly deduced) must be true alfo.

Mathematicians alfo make Ufe of Lemma's, Corollaries, and Scholiums. A Lemma () is a Propofition premised as introductory to the demonftrating a fubfequent Propofition. Corollaries (from Corollarium, Lat. from Corolla) are fubjoined either to Theorems, or Problems; and differ from them only in flowing fo naturally from them, that the Truth of them appears almost infantaneoufly, from the preceding Propofition.

Scholiums (Schola, Lat.) are Remarks made occafionally to explain whatever may appear intricate or obfcure, in a continued Chain of Reasoning; or, to remove any Objection; or, to fhew the Ufe and Application of the Subject; or, in fhort, to acquaint the Reader with any ufeful Thing, which could not be inferted in another Place, without interrupting the Series of Reasoning. These are annexed indifferently either to Definitions, Propofitions, or Coroliaries, antwering the fame Purpofes as Annotations upon Claffic Authors.

Thus we have taken a fhort View of the Method ufed by Mathematicians, and certainly it is no Wonder, if a Syftem of Know Jedge, fo uniform and well connected, is recommended by the moft celebrated Authors, as a Model, or univerfal Rule, for Reafoning, applicable in other Sciences. Thus Mr. Duncan fays *;

*In his Logic, D. 213.