19. To inscribe an equilateral triangle in a given square; (1) when the vertex is on the middle of a side; (2) when the vertex is on an angle of the square. 20. To inscribe a circle in a given trapezium, of which two opposite sides are together equal to the other two sides taken together. 21. Given the angles of a triangle, and the radius of the inscribed circle, to construct the triangle. 22. A circle being given, to describe six other circles, each of them equal to it, and touching it, and each touching the two contiguous. 23. In a given circle to inscribe six equal circles in mutual contact, and touching the given circle. 24. Upon a given finite straight line to describe (1) A regular pentagon. (4) A regular decagon. 28. The area of a regular octagon, inscribed in a circle, is equal to the area of the rectangle contained by the sides of the inscribed and circumscribed squares. 29. Inscribe a regular dodecagon in a given circle, and shew that its area equals that of a square described on the side of an equilateral triangle inscribed in the same circle. 30. PQ and RS are two straight lines inclined to one another, and crossed obliquely by a third line VT, cutting PQ in A and RS in B; C is the centre of a circle touching AP, BR, and AB; D the centre of a circle touching AQ, BS, and AB; shew that the four points A, C, B, D must lie on the circumference of a circle. [Vide III. 19.1 BOOK V. DEFINITIONS. 1. A less magnitude is said to be a measure, part, or submultiple of a greater, when the less is contained a certain number of times exactly in the greater. 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. Schol. 1.-The line RO (I. App., schol. 1) is a measure of the line AB, and also of the line AD, and AB and AD are multiples of RO. So also the square AL is a measure of the rectangle AC; and the line AB and rectangle AC are multiples of RO and AL respectively. Schol. 2.--Equi-multiples, or like multiples, of magnitudes are multiples that contain these magnitudes the same number of times. Thus the line AG (I. App. B, cor. 3), and the triangle AGH, are equi-multiples of the line AD and triangle ADE; and the line BA (I. App., schol. 1), and rectangle BH, like multiples of the line RO and square AL. 3. Magnitudes which can be compared in respect of quantity, that is, of which it can be affirmed that they are either equal to another, or unequal, are said to be of the same kind. Thus lines, whether straight or curved, can be compared with one another in respect of length; and so are magnitudes of the same kind; but a line cannot be compared with a surface, or a surface with a solid, in respect of magnitude. 4. Ratio is the relation which one magnitude has to another of the same kind in respect of quantity. Schol.-In using the word "quantity" here, reference is made to what multiple the one magnitude is of the other, or to the number of times the one is contained in the other.—The first term of a ratio is called the antecedent, the second the consequent. 5. If there be four magnitudes, and if any like multiples whatsoever be taken of the first and third, and any whatsoever of the second and fourth; and if, according as the multiple of the first is greater than the multiple of the second, equal to it, or less, the multiple of the third is also greater than the multiple of the fourth, equal to it, or less; then the first of the magnitudes is said to have to the second the same ratio that the third has to the fourth.* Schol. The constructions in Props. 1 and 2, Book VI., may be referred to as exemplifying this method of taking equi-multiples in the case of lines and figures, and applying the test of equality or sameness of ratio. They may be read *The definition may be illustrated by numbers (I. App., schol. 2) as follows:-If the numbers 9, 6, 12, and 8 be taken in order to apply this test of sameness of ratio among them, the following sets of equi-multiples may be formed: Here it appears that if the multiple of the first is greater than that of the second, the equi-multiple of the third is greater than that of the fourth, and if equal, equal, and if less, less. Care must be taken to observe the special condition, "any equi-multiples whatever," and not to employ only particular equi-multiples. This will appear from the following example, in which the test is applied to the numbers 7, 4, 8, 5: 7485 5 35 32 40 40 7485 7 48 5 6 10 6 10 42 40 48 50 6 11 6 11 42 44 48 55 7485 7 97 9 49 36 56 45 Here the multiples, when compared with one another are, in the first and last sets, greater together and less together, as the definition requires; but in the second and third sets not equal together and less together, but greater, equal, greater, less; so that the conditions are not fulfilled. Def. 7 shews that 7 to 4 is a greater ratio than 8 to 5. immediately after this definition, and their perusal will shew that, in the cases there discussed, what is true of one set of multiples must hold for others, without trying all the combinations of them which it is possible to form.-It is manifest that the first and second magnitudes must be of the same kind (def. 4), else they could not be compared; and also that the third and fourth must be of the same kind, though they may be of a different kind from the first and second; plainly all the four may be of the same kind. 6. Magnitudes which have the same ratio are called proportionals. Schol. 1.-Proportion or analogy is the equality of ratios; and the relation of the magnitudes is expressed thus:-The first is to the second as the third is to the fourth, as the fifth is to the sixth, and so on to any number of magnitudes. Schol. 2.-In a proportion consisting of four terms, the first and last terms are called the extremes, the second and third the means. 7. When of the equi-multiples of the four magnitudes, taken as in the fifth definition, the multiple of the first is greater than that of the second, but the multiple of the third is not greater than that of the fourth; then the first is said to have to the second a greater ratio than the third to the fourth; or, which is the same thing, the ratio of the third to the fourth is said to be less than that of the first to the second. 8. When there is any number of magnitudes of the same kind, such that the ratios of the first to the second, second to the third, third to the fourth, and so on, are all equal, the magnitudes are said to be continual proportionals. 9. The second of three magnitudes in continued proportion is said to be a mean proportional between, and the third a third proportional to, the other two. Schol. Thus a proportion must consist of three terms at least. 10. When the consequent of one ratio is the antecedent of another, and also the antecedent of the first the consequent of the second, the two ratios are said to be reciprocal of one another. Thus, if we have the ratio A to B, its reciprocal is B to A. 11. When there is any number of magnitudes of the same kind more than two, the first is said to have to the last the ratio compounded of the ratio which the first has to the second, which the second has to the third, the third to the fourth, and so on to the last magnitude. Thus, if we have any ratio A to B, and another B to C, the ratio of A to C is said to be compounded of the ratios of A to B and B to C; and if we have a third ratio, C to D, the ratio of A to D is said to be compounded of the ratios of A to B, B to C, and C to D. Or, again, if we have two proportions A to B K to M {c to D, A to B, the ratio of K to M is said to be compounded of the ratios of A to B and C to D. So also if we have the three proportions on the margin, the ratio of A to D is said to be compounded of the ratios of E to F, G to H, and K to L. E to F A to D G to H K to L Cor. Hence it follows that the ratio compounded of any given ratio and its reciprocal, is a ratio of equality. 12. When three magnitudes are continual proportionals, the ratio of the first to the third is said to be duplicate of the ratio of the first to the second. Thus, if A is to B as B is to C, then the ratio of A to C is said to be duplicate of that of A to B or that of B to C. 13. When four magnitudes are continual proportionals, the ratio of the first to the fourth is said to be triplicate of the ratio of the first to the second. Thus, if A is to B as B to C, as C to D, then the ratio of A to D is said to be triplicate of the ratio of A to B, or of that of B to C, or of that of C to D. 14. In proportionals the antecedents are said to be homologous to one another, as are also the consequents. Thus in the proportion as A is to B, so is C to D, so is E to F, A, C, and E are homologous terms, as are also B, D, and F. |