When learning flourish'd amongst the Romans, in the time of Cicero, mathematics were honoured and efteemed, and that by this great man himself; but afterwards, in worse times, viz. the reign of Nero, &c. when grofs ignorance had feized the ftate, and learning became very weak and decrepid, the mathematics were neither understood nor in any repute, and fome of the degenerate writers of thofe times, fuch as Tacitus, Suetonius, Gellius, feemed even not to know what mathematics or mathematicians were; they ranked mathematicians with conjurers and fortune-tellers. Nay it is faid in Tacitus's Annals, that the fenate paffed a decree for banishing mathematicians out of Italy. As tyrants and arbitrary governours, immersed in pride, and ftain'd with madness, are generally too ftupid and brutal to be capable of understanding and esteeming thefe fciences, and their excellencies, and the major part of crafty Jefuits and other fpiritual juglers, retailers of nonfenfe and fpreaders of lies, have been always afperfing and leffening them, because, by their evidence, certainty, and ufe, they are fo apt to difpofe the mind to discover truth, and so much straiten, and clog up the paths and paffages of fraud and implicit faith; fo on the other hand, all rational rulers, friends to liberty, and lovers of wisdom and mankind, detefters of falfhood, and despisers of fraud, have ever embraced, and always encouraged these sciences; well knowing them to be the plentiful fountains of advantage to human affairs, from whence, in a great measure, fpring the principal delights of life, fecurities of health, increase of fortune, and convenience of labour; which habituate the mind to a conftant delight in ftudy; which strengthen and subject it to the government of right reason; which wonderfully fortify it against all kind of impofition; and make it more eafily, readily, and and powerfully encounter and overcome falfhood, wherever it appears. I have often thought, that the time of too many of the youth of this kingdom has long been, and now is mifapplied, in learning Latin and Greek, efpecially the latter, and believe it would be of much more general advantage for greater numbers of them to be more employed in learning arithmetic, geometry, mechanics, and natural philofophy; as alfo rational fyftems of morality, politics, and good government: certainly fome or all of thefe fhould be known, in a competent degree, more especially by all statesmen, lawyers, foldiers, failors, phyficians, furgeons, artificers, &c. whereby they would all be better fitted for their feveral employments, know more of them, and execute them with much more eafe, exactnefs and certainty. So that as Plato, the philofopher, wrote over the door of his academy, Let none enter in here but thofe that understand Geometry, I would have it wrote over the door of every academy and public school in Britain, Let none depart from hence unskilled in Mathematical Science. It would require a volume to diftinctly difcufs, and fully lay out the beauties, excellencies, and uses of thefe fciences, as well as an abler hand than mine to do it; however, if any body is fufpicious of the truth of these naked affertions or general pofitions, and should call upon me for a particular proof, I fhall at all times be both ready and willing to do it. Being firmly perfuaded, that the truth of every one of them can be proved by fuch arguments, as cannot but be convincing to every rational man. As ufe has a moft powerful influence over the mind, as well as the body, there is one certain advantage derived from the study of geometry efpecially, if it were no more, for which it always ought to be highly valued and held in the greatest efteem, viz. by constantly searching after and demonftrating geometrical truths. The mind of a geometrician is fo much employ'd in, and used to truth, that it becomes, as it were, a part of his geometrical conftitution: he ever loves it, and is conftantly feeking it, in all forts of subjects as well as thofe of geometry: he is fo converfant in truth, that he naturally hates falfhood, tho' perhaps he fometimes is obliged, contrary to his inclination, to make use of it.-Hence, fince it is demonftrable that the happiness and prefervation of a fociety is augmented by the greater quantity of veracity difperfed throughout the individuals of that fociety, for lying and deceit tend to deftroy it; who does not fee the great use of the study of geometry in the promotion and maintenance of truth, and confequently its tendency to the preservation of fociety itfelf? Even the author of a discourse, called The Analyst, printed in the year 1734. altho' he is fo full of mistakes, and fo much leffens and difparages fluxions and the mathematical students therein, and that chiefly because he does not understand them, has done juftice to geometry, and spoken truly thereof. He fays, "Geometry is an excellent "logic, and it must be own'd, that, when the de"finitions are clear; when the poftulata cannot be refused, nor the axioms denied; when from "the diftinct contemplation and comparifon of "figures, their properties are derived by a per"petual well-connected chain of confequences, the "objects being ftill kept in view, and the atten"tion ever fixed upon them, there is acquired "an habit of reafoning, clofe, and exact, and me thodical: which habit ftrengthens and fharpens "the mind, and being transferred to other fubjects "is of general use in the inquiry after truth." • The late Bishop of Cloyne, as I have heard. But But enough of this in general; let us come nearer our prefent work, and more particularly obferve, that we Britons have had amongst us fcarcely any who have ventured to write Elements of Geometry of their own; altho' this nation has produced both the greatest mathematicians, and the greatest number of them too. We have generally liked Euclid's Elements beft, and know them too well to be at the trouble of compiling new Elements of Geometry (as many of the French, and other foreigners have done) and thereby changing better for worse. We think it far more eligible to do nothing at all, than vainly busy ourselves in matters introducing confufion and error into the geometrical world, and, by avoiding falfe praise, have likewise been free from evident difgrace, and real difhonour. For the method and order of Euclid's Elements of Geometry can neither be mended nor altered, but for the worse. And the demonstrations are mostly so very accurate, nervous, and elegant, as not to be exceeded, if equalled, by any geometrical writer whatsoever, either ancient or modern. His method may be defended for ever, and his demonftrations will be approved by all men of found judgment to the end of the world. His first principles or axioms are few, fimple, and clear, taken from our primitive and natural conceptions of things, and fuch as every one can easily apprehend, and no man in his fenfes can deny. And his method is fuch, that nothing is taken as true, unless it be demonftrated, and nothing is demonftrated, but from what went before. And the demonstrations, as I faid before, are in the main fo perfect and compleat, that the most severe critic could never find a real fault in them. The greatest masters of reafoning have always been captivated and charmed by their beauty and elegance, and, as one may fay, they ravish the reader's affent, and force an abfolute com mand over the mind that dares encounter them. But other element-writers, whether fuch as have alter'd Euclid's order; fuch as have given other demonftrations of fome of his propofitions; fuch as have left out fome of the moft fimple propofitions, or ranked them amongst the clafs of axioms, are generally faulty in fome refpect or other; their own demonstrations are oftentimes imperfect, or fophiftical, and fometimes obfcured and vitiated by the mixture of algebra and algebraical figns. In a word, the Elements of Euclid far exceed them all, especially as to method and elegance of demonftration. As to the fifteen books of Euclid's Elements, it is true there are some of more importance and use than others in the geometry requifite to the neceffary mechanic arts, and useful reputable fciences, now exercised and cultivated amongst the feveral nations of Europe. The feventh, eighth, and ninth books are pretty fhort elements of arithmetic, and not of geometry; altho' thefe, with the reft, are altogether ufually called Euclid's Elements of Geometry. The tenth book, being the Elements of the doctrine of incommenfurability, is indeed fine; but its length, and apparent inutility in any of the favourite mathematical ftudies of these ages, makes it very unpalatable, and much neglected. The fame may almost be faid of the 13th, 14th, and 15th books, containing a theory of the five regular folids, or Platonic bodies, as they are called. What divinity the antients found in these bodies I cannot at all imagine; furely there must have been fomething very extraordinary in them, for Euclid is exprefly related by Proclus to have compiled the whole fyftem of his Elements only for the fake of the doctrine of the five regular folids. However it must be owned, that these books, tho' elegant in themselves, and, it may be prefum'd, fufficiently I |