The reader is desired to attend to the following Corrections and Amendments. The editor regrets the necessity of such a long list of errors. He is not the publisher of the book, and is not responsible for typographical imperfections and 151 2 10. Two straight lines which intersect each other cannot be both parallel to the same straight line. For 4 Def read Cor. 3 Def. After AF put, For agles read angles For first comma read = After line 1 read AC, BD bisect each other, & For, read. For contains read contain For -- ABDE read =}AB—\DE For time read times Add, Algebraically. Let a denote the greater part, and the less; then a+x is the sum of the parts, and a x is the difference. Hence (a+x)× (a—x)=aa-xx, which is the prop. For DEHG read DEHF For ET read EI For These and the first four lines of p. 111, read, Because the consequents of these two analogies are the same the antecedents are proportional (Propor. A. p. viii); therefore, the triangle ECH: ADL:: base CH : DL. For meet read meet, for if not, the angles B and E would be equal to two right angles (29.1) For; wherefore read, and For 61 read 6 For 2 read 3 For circles read circle For 2 read 3 At the upper right corner of the fig. write E 152 7,10,13,23 For 2 Sup. read 11 For axis read axes For 11 read 12 A. If the antecedents in two proportions be the same, the consequents are proportional; and if the consequents be the same, the antecedents are proportional. If A : B :: C : D, and AE::C:F; then B: E:: D: F. For ACB: D (36), and A:C::E:F; Again, if B: A:: D: C, and E AF: C; : then BE::D: F. The proof is the same as the first. B If the means in two proportions be the same, the extremes are proportional in a cross order; and if the extremes be the same, the means are proportional in a cross order. If A: B::C:D and EB::CF, then A: F:: E: D. For AD = BC, and EF AD EF, = = BC (26); Again, if B: A:: D: C, and B: E:: F: C, it may be proved in the same manner that A: F::E:D. INTRODUCTION. OF PROPORTION. THE doctrine of PROPORTION, in the Fifth Book of Euclid's Elements, is obscure, and unintelligible to most readers. It is not taught either in foreign or American colleges, and is now become obsolete. It has therefore been omitted in this edition of Euclid's Elements, and a different method of treating PROPORTION has been substituted for it. This is the common algebraical method, which is concise, simple, and perspicuous; and is sufficient for all useful purposes in practical mathematics. The method is clear and intelligible to all persons who know the first principles of algebra. The rudiments of algebra ought to be taught before geometry, because algebra may be applied to geometry in certain cases, and facilitates the study of it. Those persons who desire to see the doctrine of PROPORTION treated according to a general method which is plainer than Euclid's, and equally accurate, may consult the geometry of Playfair, Ingram, Leslie, Cresswell, and J. R. Young. cent writers have adopted the their elements of geometry. perly a geometrical subject. B Hutton, and other realgebraical method in Proportion is not pro PROPORTION. 1. A less magnitude is said to be a part of a greater magnitude, when the less measures the greater, that is, when the less is contained in the greater a certain number of times exactly. 2. A greater magnitude is said to be a multiple of a less, when the greater is measured by the less, that is, when the greater contains the less a certain number of times exactly. Thus, if A be exactly three times B, then A is said to be a multiple of B, and B is said to be a part of A. 3. When several magnitudes are multiples of as many other magnitudes, and each magnitude contains its part the same number of times, the former magnitudes are said to be equimultiples of the latter, and the latter are said to be like parts of the former. Thus, if A be triple of B, and C triple of D, then A, C are called equimultiples of B, D; and B, D are called like parts of A, C. 4. Two magnitudes are said to be homogeneous, or of the same kind, when the less can be multiplied so as to exceed the greater. Thus, a minute may be multiplied till the product exceed an hour, a yard till the product exceed a mile, &c. 5. Two quantities are said to be commensurable, when they are divisible by a third quantity without a remainder; and the third quantity is called their common measure. Thus, 4 and 6 are commensurable, and 2 is their common measure. 6. Two quantities are said to be incommensurable, when they are not divisible by a third quantity without a remainder. Thus, 4 and 7 are not commensurable, because they cannot be divided by a third number without a remainder. 7. Between any two finite quantities of the same kind there subsists a certain relation in respect of magnitude, which is called their ratio. 8. When we observe two quantities, one of which is double of the other, we acquire the idea of a particular ratio, or relation, which the greater has to the less; and when we afterward find two other quantities, one of which is also double of the other, we say that they have the same ratio which the two for |