if equals be taken from equals, the remainders will be equal. Mr. Locke has fhewn the infignificance of these axioms in the feventh chapter of his fourth book. In fact, they are only founded upon the induction of particular experiments and obfervations; and until that induction be compleat, we can never be convinced of their truth. They do not prove any thing themselves, but require to be proved; and if a Newton were to devote his powers to the study of axioms for an hundred years, he would not be able to draw from them one fingle conclufion worth notice. In this manner does every demonftration proceed upon the refults of experiments, as the reader will find, in as many inftances as he shall take the pains to examine. And fince the appeal in demonstrative reafoning is always made to what is now exhibited to the fenfes, or to what we have before learned by the exercise of the fenfes, too much pains cannot be taken, at the commencement of the study of I geo metry, metry, to fatisfy the mind of the learner by appealing to his fenfes. The more distinct and deep the impreffions of fenfe are at the beginning, the greater will the power of abstraction afterwards be, when the progrefs of his ftudies fhall have carried him into the higher mathematics. Abstraction is not, in fact, a distinct power, as the metaphyficians, who seem to imagine that they increase the importance of their science, as they multiply diftinctions, teach. We abftract, when we narrow the fphere of fenfations and dwell upon impreffions, or when we recollect the ideas thus acquired. So far is this talent from forming a distinction between man and beaft, that the animals which do not take cognizance of more than two or three objects in this fublunary world may, I think, be fairly reckoned the most abstracted of all living creatures.-It is, at least, evident that of any object, I shall recollect the whole or any part the better, as the original impreffion was more lively. If I am to imagine, or form an image, image, by putting things together in my mind, in an arrangement different from that in which I have beheld them, and thus create a whole which I have not seen, out of parts which I have feen; the diftinctness of the original conceptions will be equally fubfervient to this procefs. By appealing in this manner to his fenfes, and making him feel the firmness of the ground on which he treads, one might probably instruct a boy, at an early age, in the elements of geometry, fo as rarely to give him disgust, and frequently great fatisfaction. He would by imperceptible degrees acquire the power of abftraction, or learn to reconfider each feparate perception, as well as to combine them anew. Suppofing it unneceffary to multiply instances of the experimental reasoning of geometry, fince the instance already quoted fairly represents all the reft, I shall shortly con confider the definitions of the first book of Euclid, except the merely nominal definitions, such as those of a rhomb, trapezium, &c. There is, it seems, fome uncertainty as to the author of the definitions. I fhall take them, as they occur in Dr. Simson's tranflation, occafionally, however, referring to Dr. Gregory's edition of the Greek text *. DEFINITION I. A point is that which hath no parts, or which hath no magnitude. Σημειον ἐξ]ι, ου μέρος εθεν. Here the beginner immediately finds himself transported into the land of won ΕΥΚΛΕΙΔΟΥ ΤΑ ΣΩΖΟΜΕΝΑ. Οxon. 1703. ders; ders; and fuppofing it neceffary to his progrefs to conceive a thing that has no parts, he is apt to furmise that mathematics is a study for which nature never defigned him; and as he proceeds, he looks back from time to time with an eye of regret upon the firft definition, earnestly wishing he had but force of mind enough to comprehend it. Dr. Simfon's demonftration will not afford him any affiftance in his difficulty; and he will still be unable to conceive what that can be, which has no parts or magnitude; if a variety of phrafes be, as ufual, repeated to him; he may reply, it is in vain to utter new founds; what I want is fenfible evidence of the thing; and if he should but have the good fortune to attend to the evidence of his fenfes, and to understand the nature of language, the difficulty will instantly vanish: for a point is first the end of any thing fharp; omne quod |