as another has been cut; to find a third, fourth and mean proportional; which being practical principles, should be made habitual. The next four propofitions are all upon the fame subject, and that a very curious one, namely to fix the equality of the parallelograms and triangles from the ratio of the fides and conversly. And here the indolent reader, fhould be required to fhew what use is made of the equality of the angles, or where the demonftration would fail if that circumftance were omitted. The nineteenth and twentieth propofitions are of great importance and should be carefully ftudied it will affift a beginner very much to confider them as problems, omitting the term duplicate ratio entirely: Thus the ninteenth may be exprest in this manner; two fimilar triangles being given, to find two straight lines which fhall have the fame ratio to one another which the triangles have. And BC and BG will be the two ftraight lines required. The reader may know whether he understands the propofition, by taking a third proportional to EF and BC, and compleating the construction and demonftration. The twentieth may be worded thus; to find two ftraight lines which shall have the same ratio to one another which two fimilar polygons have; and then to prove that the fame two straight lines, will have the fame ratio to one another which the fimilar triangles, into which the polygons are divided, have to one another. And it will also be convenient, according to the fame plan, to confider the twenty third as a problem; which may be expreft in this manner; to find two ftraight lines which fhall have the fame ratio to one another which two equiangular parallelograms have; without making any use of the phrase ratio compounded of ratios; and for a proper explanation of this term the reader may confult Simfon's edition. I would propose it to the reader to examine these propofitions first as problems, because this will fix his attention more to the particular steps of the construction and the confequences drawn from them. And alfo the extent and importance of thefe very beautiful theorems is more to be seen by confidering them in this practical point of view. For by the twentieth propofition, figures may be increased or diminished in any proportion. For instance if I want to make a figure the fifth part of a given figure; I take the fifth part of one of the fides of the given figure; and find a mean mean proportional between the line and its fifth part, upon which if a fimilar and fimilarly fituated figure be described, it will be the fifth part of the given figure. Likewife by this propofition the forty seventh of the first book, may be extended to any fimilar rectilineal figures. The twenty third is often quoted to prove, that, if there be four lines which have the fame ratio; and also other four, the rectangles contained by the antecedents and confequents will have the fame ratio, the first to the second which the third has to the fourth. This the reader fhould examine, which he may eafily do by the affistance of the note below. * The subject and arrangement of what remains is so obvious, that it would be paying the reader but a poor complement to suppose that he stood in need of any farther information, efpecially if he read what Simfon has faid upon the 28, and 29 propofitions. I have said nothing of corollaries; and it is doubtful in what fenfe Euclid would have them be understood. In the firft four books, they are all some circumstances which are in fact demonftrated in the propofition, but which, not being expreft in words, the reader might without this notice have overlooked; but he seems afterwards not to confine himself to this fenfe, as will be obvious to the reader. THE ELEMENTS O F EUC C L LI D. BOOK V. I. DEFINITION S. 1.A Part is a magnitude of a magnitude, the less of the when it measures the greater. 2. But a multiple, the of the less, when it is measured by the lefs. greater, greater 3. Ratio is a certain habitude of magnitudes of the same kind to one another, which is only as to quantity. 4. Magnitudes are faid to have a ratio to one another, which may be multiplied fo as to exceed one another. 5. Magnitudes are faid to be in the fame ratio; the first to the fecond and the third to the fourth; when the equimultiples of the first and third, according to any multiplication, are at the fame time less, or at the fame time equal, or are at the same time greater than each of the equimultiples of the second and fourth, compared with one another. 6. And let magnitudes, having the same ratio, be called proportionals. 7. But when of equimultiples; the multiple of VOL. I. *A the Book V. Book V. the first exceeds the multiple of the fecond, but the multiple of the third does not exceed the multiple of the fourth; then the first is faid to have to the second a greater ratio than the third to the fourth. 8. And proportion is a fimilitude of ratios. 9. But proportion confifts in three terms at least. 10. And when three magnitudes are proportionals; the first is said to have to the third a duplicate ratio of that which it has to the fecond. when four magnitudes are proportionals; the firft is faid to have to the fourth a triplicate ratio of that which it has to the fecond; and always in order one more, as long as the proportion continues. 11. And 12. The leading magnitudes are faid to be of like ratio with the leading magnitudes; and thofe that follow are faid to be of like ratio with thofe that follow. OR. The antecedents are faid to be homologous magnitudes to the antecedents ; and the confequents to the confequents. 13. Alternate ratio is the taking of the antecedent to the antecedent; and of the confequent to the confequent. 14. Inverfe ratio is the taking of the confequent as an antecedent to the antecedent as a confequent. 15. The compofition of a ratio is the taking of the antecedent together with the confequent, as one, to the confequent. 16. But the divifion of a ratio is the taking of the excefs, by which the antecedent exceeds the confequent, to the confequent itself. 17. The converfion of a ratio is the taking of the antecedent, to the excefs, by which the antecedent exceeds the confequent. 18, A ratio of equality is, there being several magnitudes, and others equal to them in multitude, and in the fame proportion taken two by two, when it is, as in the first magnitudes; the first to the last, so in the fecond magnitudes the first to the laft. or otherwise. The taking of the extremes by a substraction of the middle terms. 19. Ordinate proportion is, when it is as the antecedent to the confequent, fo is the antecedent to the confequent; and it is as the consequent to fome other, fo is the confequent to fome other. 20. But perturbate proportion is, when there being three magnitudes and others equal to them in number, it is as in the first magnitudes the antecedent to the confequent, fo in the second magni tudes |