When learning flourish'd amongst the Romans, in the time of Cicero, mathematics were honoured and esteemed, and that by this great man himself; but afterwards, in worse times, viz. the reign of Nero, &c. when gross ignorance had seized the ftate, and learning became very weak and decrepid, the mathematics were neither understood nor in any repute, and some of the degenerate writers of those times, such 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 said in Tacitus's Annals, that the senate passed a decree for banishing mathematicians out of Italy. As tyrants and arbitrary governours, immersed in pride, and stain’d with madness, are generally too stupid and brutal to be capable of understanding and esteeming these sciences, and their excellencies, and the major part of crafty Jesuits and other fpiritual juglers, retailers of nonsense and spreaders of lies, have been always aspersing and lessening them, because, by their evidence, certain. ty, and use, they are so apt to dispose the mind to discover truth, and so much ftraiten, and clog up the paths and passages of fraud and implicit faith; fo on the other hand, all rational rulers, friends to liberty, and lovers of wisdom and mankind, deçesters of fallhood, 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, spring the principal delights of life, securities of health, increase of fortụne, and convenience of labour; which habituate the mind to a constanç delight in ftudy; which strengthen and subject it to the government of right reason ; which wonderfully fortify it against all kind of imposition; and make it more easily, readilý, and and powerfully encounter and overcome falshood, 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 misapplied, 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 philosophy; as also rational systems of morality, politics, and good government : certainly some or all of these should be known, in a competent degree, more especially by all statesmen, lawyers, soldiers, failors, physicians, surgeons, artificers, &c. whereby they would all be better ficted for their several employments, know more of them, and execute them with much more ease, exactness and certainty. So that as Plato, the philosopher, wrote over the door of his academy, Let none enter in bere but those 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 Matbema. tical science. It would require a volume to distinctly discuss, and fully lay out the beauties, excellencies, and uses of these sciences, as well as an abler hand than mine to do it; however, if any body is suspicious of the truth of these naked affertions or general posi. tions, and should call upon me for a particular proof, I shall at all times be both ready and willing to do it. Being firmly persuaded, that the truth of every one of them can be proved by such arguments, as cannot but be convincing to every rational man. As use has a most powerful influence over the mind, as well as the body, there is one certain ad. vantage derived from the study of geometry especially, if it were no more, for which it always ought to be highly valued and held in the greatest esteem, viz. by constantly searching after and demonstrating geometrical truths. The mind of a geometrician is so 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 constantly seeking it, in all sorts of subjects as well as those of geometry: he is so conversant in truth, that he naturally hates fallhood, tho' perhaps he fomen times is obliged, contrary to his inclination, to make use of it.--Hence, since it is demonstrable that the happiness and preservation of a society is augmented by the greater quantity of veracity dispersed throughout the individuals of that society, for lying and deceit tend to destroy it ; who does not see the great use of the study of geometry in the promotion and maintenance of truth, and consequently its tendency to the preservation of society itself? Even the * author of a discourse, called The Analyst, printed in the year 1734. altho' he is so full of mistakes, and so much lessens and disparages fluxions and the mathematical students therein, and that chiefly because he does not understand them, has done justice to geometry, and spoken truly thereof. He says, “ Geometry is an excellent logic, and it must be own'd, that, when the de finitions are clear; when the postulata 'cannot “ be refused, nor the axioms denied; when from “ che distinct contemplation and comparifon of “ figures, their properties are derived by a per" petual well-connected chain of confequences, che objects being still kept in view, and the atten" tion ever fixed upon them, there is acquired an habit of reafoning, close, and exact, and me" thodical: which habic strengthens and sharpens si the mind, and being transferred to other subjects " is of general use in the inquiry after truth." * The late Bishop of Cloyme, as I have heard. But But enough of this in general ; let us come nearer our present work, and more particularly observe, that we Britons have had amongst us scarcely 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 best, 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 worfe. We think it far more eligible to do nothing at all, than vainly busy ourselves in mat. ters introducing confusion and error into the geometrical world, and, by avoiding falfe praise, have likewise been free from evident disgrace, and real dishonour. For the method and order of Euclid's Elements of Geometry can neither be mended nor altered, but for the worfe. 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, simple, and clear, taken from our primitive and natural conceptions of things, and such as every one can easily apprehend, and no man in his senses can deny. And his method is such, that nothing is taken as true, unless it be demonstrated, and nothing is demonstrated, but from what went before. And the demonstrations, as I said before, are in the main so perfect and compleat, that the most severe, critic could never find a real fault in them. The greatest masters of reasoning have always been captivated and charmed by their beauty and elegance, and, as one may fay, they ravish the reader's aflepi, and force an absolute com mand mand over the mind that dares encounter them. But other element-writers, whether fuch as have alter'd Euclid's order; such as have given other demonstrations of some of his propositions ; fuch as have left out some of the most simple propofitions, or ranked them amongst the class of axioms, are generally faulty in some respect or other; their own demonstrations are oftentimes imperfect, or sophistical, and fometimes obscured and vitiated by the mixture of algebra and algebraical signs. In a word, the Elements of Euclid far exceed them all, especially as to method and elegance of demonstration, 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 requisite to the necesfary mechanic arts, and useful reputable fciences, now exercised and cultivated amongst the several nations of Europe. The seventh, eighth, and ninth books are pretty short elements of arithmetic, and not of geometry; altho' these, with the rest, are altogether usually called Euclid's Elements of Geometry. The tenth book, being the Elements of the doctrine of incommensurability, is indeed fine; but its length, and apparent inutility in any of the favourite mathematical studies of these ages, makes it very unpalatable, and much neglected. The fame may almost be said of the 13th, 14th, and 15th books, containing a theory of the five reguJar solids, or Platonic bodies, as they are called. What divinity the antients found in these bodies I cannot at all imagine; surely there must have been something very extraordinary in them, for Euclid is expresly related by Proclus to have compiled the whole system of his Elements only for the sake of the doctrine of the five regular solids. However it must be owned, that these books, tho* elegant in themselves, and, it may be prefum’d, sufficiently |