K: but K is equal to GH; because EB is equal to C; therefore M is not greater than GH; but FG is greater than D (by conft.) wherefore FH is greater than M and D together, that is N; therefore FH is greater than N; and K is not greater than N: and FH and K are equimultiples of AB and C and N is a multiple of D; but by the fuppofition any equimultiples of AB and C were fo related to any multiple of D; that whenever the multiple of AB exceeds the multiple of D; the multiple of C also exceeds the multiple of D; but the contrary has been proved when we suppose AB unequal to C; therefore the multiples being according to the fuppofition AB cannot be unequal to C. Which was to be demonstrated. Let there be fix magnitudes, or rather straight lines A, B, C, D, E, F whofe multiples are thus related: fuppofe any equimultiples of A and C to be taken, and also any equimultiples whatever of B and D; and farther whenever the multiple of A exceeds the multiple of B; let the multiple of C always exceed the multiple of D: and again let there be fome equimultiples of C and E and alfo of D and F fuch that the multiple of C exceeds the multiple of D; but the multiple of E does not exceed the multiple of F: I say that there are some equimultiples of A and E; and fome of B and F fuch that the multiple of A exceeds the multiple of B; when the multiple of E does not exceed the multiple of F.. Suppofe G and H to be fuch equimultiples of C and E; and K and L fuch equimultiples of D and F; that G exceeds K but H does not exceed L: now because by the supposition G and K are given, I can find what number of times C and D are contained in them by measuring them, that is by cutting off parts equal to C and D. Take M the fame multiple of A that G is of C; that is, which H is of E: again take N the fame multiple of B which K is of D; that is, which L is of F. DEMONSTRATION. Now by the fuppofition A, B, C, D are fo related to one another that any equimultiples of A and C; and also any equimultiples of B and D being taken; if the multiple of A exceed the multiple of B the multiple of C-exceeds the multiple of D: but M and G are (by conft.) equimultiples of A and C; as alfo N and K of B and D; therefore if M exceed N; G exceeds exceeds K but G does exceed K (by fupp.); therefore M exceeds N: but H does not exceed L (by fupp.); wherefore M exceeds N but H does not exceed L; and M and H are equimultiples of A and E; and N and L of B and F. Wherefore &c. Which was to be demonstrated. If D and E (see the next figure) be such equimultiples of A and B ; and F such a multiple of C, that D exceeds F but E does not exceed F; then I fay that A is greater. than B.. MABN GCDK HEFL For because D exceeds F and E' does not exceed F; therefore of confequence D exceeds E: but because D and E are equimultiples of A and B and D' exceeds E, therefore any part of D will exceed the fame part of E; that is if one of the magnitudes be greater than another, the third &c. part of the one will exceed the third &c. part of the other; therefore A is greater than B. D Again if F be such a multiple of C; and D and E For, because F exceeds E but F does not exceed B I F of E will be less than the fame part of D; but A and B are the. fame parts of D and E therefore B is less than A. Which was to be demonftrated. Containing an explanation of the remaining definitions IF what has been faid, in this differtation, be properly attended to, there will be little difficulty in understanding the meaning of the third definition; which has been the occafion of fo much trouble to the commentators; and which they have been so very unfuccefsful.: unfuccessful in their attempts to explain. For by proceeding according to the fpecimen here given, it is very obvious that all the properties of magnitudes, which are mentioned in the fifth book, may be demonftrated from the two firft definitions, without any mention of the term ratio. But to fave the trouble of fuch circumlocutions; our author gives particular names to the different fuppofitions; and this is the true origin of the following definitions, from the fecond. Now fays he, in the third, any certain mutual habitude of magnitudes of the fame kind, considered according to this quantuplicity or multiplicity; or according to this relation of parts and multiples, I call ratio. And I believe I may fay that this definition has never been rightly understood fince the greek became a dead language. Barrow calls it a metaphyfical definition; but it appears from this, that it is a mathematical one. He fays that fuch definition was equally neceffary in the seventh book, when speaking of the ratio of numbers, but it appears from this explanation, that such a definition could have no place there. When any equimultiples of the first and third, and any equimultiples whatever of the fecond and fourth, have this relation to one another, that when the multiple of the firft exceeds the multiple of the second; the multiple of the third always exceeds the multiple of the fourth; or when equal, equal; or when lefs, lefs: then he says the first has the same ratio to the fecond which the third has to the fourth: But if the multiple of the first exceed the multiple of the fecond, when the multiple of the third does not exceed the multiple of the fourth; he then says that the first has a greater ratio to the second than the third has to the fourth. The fixth, eighth, and ninth definitions are fufficiently obvious. When three magnitudes are proportionals; the first is said to have to the third the duplicate ratio, of that which it has to the second. But this definition and the next I fhall have occafion to explain afterwards. When we say that the first has the fame ratio to the second which the third has to the fourth; the first and third are the antecedents; and the second and fourth are called the confequents. Now the antecedents are called homologous terms, or terms of like ratio and fo are the confequents alfo. This diftinction is necessary for feveral feveral reafons; in the following definitions, and in fixing the equal angles of fimilar triangles &c, and therefore ought to be particularly attended to. The following definitions will be fufficiently understood, when the particular propofitions are read, wherein it is proved that the magnitudes will have the fame ratio, the first to the fecond and the third to the fourth, after they have undergone the changes, mentioned in these definitions. I know there is a way of explaining these definitions by numbers, which is nothing to the purpose; for Euclid is not fpeaking of numbers at prefent but of magnitudes. However as this fupplies the place of all the knowledge which this book contains to many, who are not so scrupulous as to require a demonstration; nor fo attentive as to confider whether they are talking about fomething or nothing, to gratify such indolent readers, I have presented them with the following scheme, ferved up in the true French taste. I hope, from what has been faid, the reader understands the meaning of these definitions. The triangle, rectangle and circle have been confidered as inftruments of investigation; but this doctrine of ratios is to be regarded as a new mode of comparison, very extensive, in its confequences, giving us a wonderful command over the magnitudes, which we have already confidered, by discovering a great many properties far beyond the reach of the former method of comparison. And although a great deal of pains has been taken to render our ideas upon this fubject, confused, indistinct and nothing; yet it is wonderful, when properly examined, what a fimplicity there is in the principles upon which this method of comparison is founded; the conftructions are all performed (by 3. 1.) and requires no fuch apparatus as is neceffary even VOL. I. S to to prove one line to be equal to, or fhorter than another, when we cannot make ufe of the circle. THE reader who has been accustomed to have his head filled with numerical ideas as explanatory of the propofitions contained in this book, will perhaps be surprized to hear me affirm that he has taken a great deal of pains to fill his head with abfurdities. For I am perfuaded that if a fcheme were to be thought of, for depriving a man of his reason, still keeping up an impreffion upon his mind that he was a rational creature, there could not be a more effectual method, than to set him upon reconciling the demonftrations of this book to numerical ideas. Indeed the twisting ropes of fand would be a rational employment when compared with this. All numerical reasoning proceeds upon the fuppofition, that the unit is the fame. But Euclid is not yet prepared for this confinement, which I fhall prove particularly, upon the feventh book. He does not carry on his demonftrations in the first four books, upon the fuppofition that the fides of his triangles, or parallelograms or the radius of his circle, are three or four feet long; or as having a reference to any kind of measure. Nor is his reasoning in this book less general. His ratio is a mutual habitude of magnitudes according to quantuplicity, or according to the doctrine of parts and multiples. The queftion is not; is the part one; but is it a fixt magnitude? and farther, if you mean to reason upon it according to this doctrine of ratios; is it a magnitude that you can multiply? that is, that you can double; for the whole operation confifts only of doubling. It would not be more abfurd, to fuppofe that the nature of things, or human imagination does not allow of any triangle but an equilateral one, and nevertheless to try to prove that the fquare of one fide of a triangle might be equal to the fquares of the other two; than to attempt the proof of fome of the fimpleft properties of triangles, parallelograms and circles upon the fuppofition that all magnitudes can be expreft by numbers. It |