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* no lefs; by which they were called either for their Excellency,
or because of all the Sciences they were first taught, or because
they were judged to comprehend πάντα τα Μαθήματα.» those Sciences * " the Art of Reasoning is allowed to reign in its
greatest Perfection. Hence it was that the Antients, who lo “ well understood the Manner of forming the Mind, always began " with Mathematics, as the Foundation of their Philosophical « Stądies. Here the Understanding is by Degrees habituated to " Truthi, contracts insensibly' a certain Fondness for it, and learns
never to yield its Affent to any Propofition, but where the Evi** dence is fufficient to produce full Conviction. For this Reason, "! Plato' has called Mathematical Demonstrations the Cathartics or « Pargatives of the Soul, as being the proper Means to cleanse it “ from Error, and restore that natural Exercise of its Faculties, in " which jut Thinking confilts. And, indeed, I believe it will “ be readily allowed, that no Science furnishes so many Instances,
of a happy Choice of intermediate Ideas, and a dexterous Appli“cation of them, for the Discovery of Truth, and Enlargement of
Knowledge. Hence Matheis has been juftly called by the Reverend Nr. Baker) the Princess of alt Sciences; and hence is + Kings and Princes heretofore have been fo enamoured with “ her Simplicity and Pleasantness, that (forsaking all the Delights, « of their Kingdoms) they have made their Addresses to her “ Shrines, paid Homage to her Altars ; thus redeeming Science at " so great a Price. Should I mention Anacharfis therScythian, and “ Heraclitus the Eph:finn, who preferred the Contemplation of « Philosophy before their hereditary Kingdoms, and chose rather “ (leaving those) to fit at the Feet of Philosophers, than on their. 's Ringly Thrones : Should I recount Atlas King of Mauritania, “ whom (for his Aftronomic Skill, wherein he excelled) Antiquity “ hath fabled to bear up the Heavens on his Shoulders ; or Aga" thocles King of Sicily, Ptolomy of Philadelphia, Alphonfus of Caliile 6 Frederic of Denmark, W'illiam Landgrave of Helle
, &c. Yea, but < fhould I mention Emperors, viz. Cæfar, Adrian, Theodosius, &c. " who (devoting themselves to these Studies, worthy indeed of .. Emperors) rendered themselves more illustrious, by their Writings, is than by their warlike (though many and great) Atchievements; • I should but filently shame and reproach this
oar degenerate Age In which, notwithstanding the Excellency of this Science is fuch, as to make it necessary to be studied as an Introduction to most other Arts; and that it is known from“ Experience that great Genius's “ have surpaffed themselves 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 so universally studied as fome other Sciences, to the most conyincing Arguments of which the Professors cannot fubfix a Quod
erat demonstrandum: And yet their Schools are so ftuffed with * Duncan's Logic, f. 223. † Preface to the Rev. Mr. Baker's Geometrical Key. I Stonebeuze's Arithmetic.
§ Baker's Geometrical Key.
« Profelytes, that they have scarce Room to breathe in ; whilst the
Mathematic (School) only (in which, not some one Truth only is « expanded but even innumerable ; and those not mean and ob« vious, but most high, admirable, and mysterious, are clearly de" monftrated) lies orbate and neglected. From this they fly as from “ a Pest-house; but to those they troop, as to a Delpbic Oracle, " or as Doves to white Dove-houses.- Lastly, though her intrinsic “Worth and Beauty hath compelled others of the lowest Orb (who
(faluting her only at the Threshold) never entered, or had the * least Glimpse of her Arcana's or inner Rooms) to admire her; yet, “ certain it is very few are skilled in her Mysteries ; 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 Eclipse in her Glory and Efteem with the Vulgar,
now-a-days, I cannot divine ; whether it be, the being a liberal
Science, and therefore (on that Account) unsuitable to the Huqmours of those close-fisted Misers (who are scarce to be reckoned
among the Number of Men) who love to have their Purses, enás riched rather than their Minds : Or, whether their Despondency
of ever arriving to any considerable Eminency of Height, it be * ing as good to be nothing, as not a none-Such, or but a Spy, to
to an Art:) Or whether it be the fancied Difficulty and Knottinefs « of the Study itself, (which I have most Cause to suspect.) Or, “ what that supposed Mormo may be, that forestals and prejudiceth ** some newly entered, and scares others, who have tased some of « her Sweets, from farther Essays (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 “ absurd, as strange, viz. That what should recommend this * Study to thy Reason fhould discourage thee; that what should “ animate thy Diligence, and quicken thee to a further Effay, « should decrest and dispirit thee. Real Difficulties (much less “ conceived Prejudices) should be so far from blunting thy Edge, is that they should rather be the Whetstone of Virtue, and sharpen • thy Endeavours: Why may not the fame Things, which (for " the Excellency of them) are the Objects of thy Admiration, be “ (for their Polibility) as well the Object of thy Hope, and the “ Encouragement of thy Industry? The Difficulties of this Art are “ not so infuperable, but (as in War) may be overcome, either " by Industry, or fortune, or both.” But, if the Learner should meet with such Difficulties as he cannot easily surmount by himself,
go through his Studies with more Pleasantness and Dir patch, if he is of Ability, he would do well to call in the Alistance of some able Profeffor; for * " there are few Persons of so
pene“ trating a Genius and fo just a Judgment, as to be capable of “ learning the Arts and Sciences without the Affistance of Teachers, “ There is scarce any Science fo fafely and so speedily learned, · even by the noblest Genius and the best Books, without a Tutor, “ His Alistance is absolutely necessary for most Persons, and it is
af very useful for all Beginners. Books are a sort of dumb Teacher's,
they point out the Way to Learning; but, if we labour under
any Doubt, or Mistake, they cannot answer fudden Questions, or !! explain present Doubes and Difficulties; This is properly the
Work of a living Instructor.” But, to return from this Digress fion, to shew, Secondly, the great Usefulness of those Sciences to all Persons in general, in the Improvement of the Mind.
The principal Advantages which the Mind receives from Mathematical Studies, are 1. The accustoming it to Attention. 3. The treeing it from Prejudice, Credulity, and Superftition. 3. The acquiring a Habit of close and demonftrative Reasoning. Hoc 1: * The Mathematics make the Mind attentive to the Objects ** it confiders. This they do by entertaining it with a great Va.
siety of Truths, which are delightful and evident, but not ob; qs vious. Truth is the same Thing to the Understanding as Music " to the Ear, and Beauty to the Eye. The Pursuit of it does really
as 'much gratify a natyral Faculty implanted in us by our wife ¢¢' Creator, as the Pleasing of our Senses : Only in the former Cafe,
the Object and Faculty are more spiritual, the Delight is more pare, free from the Regret, Turpitude, Laflitude, and Intempe“ rance, that commonly attend sensual Pleasures. The most Part ço of other Sciences consisting only of probable Reasonings, the * Mind has not where to fix ; and, wanting fufficient Principles to " pursue its Searches upon, gives them over as impossible. Again, " as in Mathematical Investigations Truth may be found, so it is « not always abvious: This purs the Mind, and makes it diligent * and attentive. - And Plato (in Repub. Lib. VII.) obferves, that " the Youth, who are furnithed with Mathematical Knowledge, *" are prompt and quick at all other Sciences, ---Youth is generally ď so much more delighted with Mathematical Studies than with qe the unpleasant Tasks that are sometimes imposed upon them, " that I have known some reclaimed by them from Idleness and “ Neglect of Learning, and acquire in Time an' Habit of Thinking, “ Diligence, and Attention (Qualities which we ought to study " by all Means to beget in their desultory and roving Minds.") And this is no Wonder, if we confider, that the Abstractedness of pure Mathematics is a proper Remedy to cure the Lightnefs of their Minds, acting as a Rein to curb the Impetuolity of their Passions. And that the Study of those Science inspires a Love for Truth, the Purftiit of which" + "' will give, the otherwise unem, s ployed, a Diftaste of those vain Occupations that hurry Men into « Libertinism and Dębauchery."
" Secondly, Mathematical Knowledge adds a manly Vigour " to the Mind, frees it from Prejudice, Credulity, and Superstition. ©. Thiş it does two Ways, 1. By accuftoming us to examine, and
not to take things upon Truit. Z. By giving us a clear and "extenfiye Knowledge of the System of the World; which, as it
creates in us the most profound Reverence of the almighty and
* Efav on the Usefulness of Mathematical Learning. f Stoneborfe's Aritlunetic. Eilay on the Ulefulness of Mathematical Learning.
sa-wife Creator, so it frees us from the mean and narrow, Thoughts ** which Ignorance and Superfition are apt to beget.
The third Advantage which the Mind receives from Mathematics is the Habit of clear, demonstrative, and methodical Reasoning,
Mathefis * is “ a Study that tends not only to the Improvement “ of Arts, but also to the Regulation of the Passions is a Study that * will infenfibly bring Men to think methodically, reason correctly, "and feparate Truth from Falhood, and the Disguise of Words, “ 'which it generally wears. a The Writings of the Mathematicians have been conducted by so perfect a Model, as to be + 4 an incontestable Proof of the Firm" nefs and Stability of human Knowledge, when built upon fo lure Fra Foundation. For not only the Propositions of this Science “ ftood the Test of, all Ages, but are found attended with that in"! vincible Evidence, as forces the Alent of all, who duly conlider “ the Proofs upon which they are establised. The Mathema“ticians are universally allowed to have hit upon the right Me* thod of arriving at Truths. They have been the happielt in the « Choice, as well as Application of their Principles.
Ina Word, fome Knowledge in both pare and mixt Mathematics is by Experience found not only necessary in many particular Professions, but also of great Yle to all Men in general, in the Improvement of the Mind, and, therefore, the Study of them is now deservedly thought, not only of the greatest Use, but also a necessary Part of the Education of Gentlemen ; and are accordingly made a Part of it, in our two famous Universities. Not so much to make them Mathematicians, as, by engaging them to observe the Method of Reasoning made Use of in the Mathematical Sciences, they may
acquire something of that Justness and Solidity of Reasoning, for which the Professors of thefe Sciences are so generally, and defery
edly esteemed. Ti Perhaps what has been already faid, may be sufficient to thew the great Usefulness of Mathematical Studies, for acquiring a juft Method of Reasoning: However, that the Reader may himself be able in some Measure to judge of the Truth of the above Assertions, it may not be improper to lay before him a general Account of the Method made Use of by Mathematicians; which is this. They firit begin with Definitions, (from Definitio, Lat.) in which the Meaning of their Words is so distinctly explained, as to prevent any. Ambiguity, (or double Meaning): By which Means, every attentive Reader has che very fame Ideas excited in his Mind, as the Writer has annexed to them.
By this Means, the Mathematicians have secured themselves, and the Sciences which they profess, from Wrangling and Controverfy, and if the Writers of Natural Philosophy, and Morality, had used the same Accuracy and Care in adjusting the Definitions wherefoever necessary, I " they had effectually secluded a Multi", tude of noisy and fruitless Debates out of their several Provinces, * Nor had that facred Theme of Divinity been perplexed with fa Stone boufe's Arithmetic. Dincar's Logic, p. 181, | Waris's Logic,
* many intricate Disputes; nor the Church of Christ been torn to “ Pieces by so many Sects and Factions, if the Words Grace, Faith,
Righteousness, Repentance, Juftification, Worship, Church, Bishop, Pr?fbyter, &c. had been well defined, and their Signifi“ cation adjusted, as near as possible, by the Use of those Words « in the New Testament; or at least, if every Writer had told us mt at first, in what Sense he would use those Words."
The second Step, in Mathematical Writings, is to lay down fome self-evident Truths, which may serve as a Foundation on which to build the future Reasonings. These Propositions are divided into two Sorts, called Axioms and Poftulates.
An Axiom (Axioma, Lat.) is a felf-evident speculative 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 securely laid the Foundation, the Mathematicians begin in their next Step to build their Superstructure of demonstrable Propositions, i. e. Propositions which are not of themselves self-evident. Of demonftrable Propositions there are also two Kinds, fpeculative and practical; a speculative Proposition is called a Theorem (Seenpea); and a practical one, a Problem (wzófumpa). These they demonftrate in a Series of Reasoning, proceeding carefully Step by Step, assuming nothing for Truth, but the Axioms and Poftulates, before laid down; or fome Proposition already demonftrated; and hence it follows, that, as the Principles on which their Reasoning is founded is true, the Consequences (rightly deduced) must be true also.
Mathematicians also make Use of Lemma's, Corollaries, and Scholiums. A Lemma (añurea) is a Proposition premised as introdactory to the demonstrating a subsequent Propofition. Corollaries (from Corollarium, Lat. from Corolla) are subjoined either to Theorems, or Problems ; and differ from them only in flowing so naturally from them, that the Truth of them
almost infantaneously, from the preceding Propofition.
Scholiams (Schol:a, Lat.) are Remarks made occasionally to ex, plain whatever may appear intricate or obscure, in a continued Chain of Reasoning; or, to remove any Objection ; or, to shew the Use and Application of the Subject; or, in short, to acquaint the Reas der with any useful Thing, which could not be inserted in another Place, without interrupting the Series of Reasoning. The fe are annexed indifferently either to Definitions, Propositions, or Coroliaries, aniwering the fame Purposes as Annotations upon Classic Authors.
Thus we have taken a flort View of the Method used by Mathematicians, and certainly it is no Wonder, if a System of Knowa jedge, so uniform and well connected, is recommended by the most celebrated Authors, as á Model; or universal Role, for Reafoning, applicable in other Sciences. Thus Mr. Durican fays *;
* In his