A and B are the antecedents, C and D the consequents; and the proportions B+A:D+C:: A Cor:: B: D, B-A: D-C:: A: Cor:: B: D, answer to the following enunciation; The sum of the antecedents of a proportion is to the sum of the consequents, and the difference of the antecedents is to the difference of the consequents, as one antecedent is to its consequent ; Whence it follows, that the sum of the antecedents is to their difference, as the sum of the consequents is to their difference. If we have a series of equal ratios A: B::C: D:: E: F by considering only the two first, which form the proportion A: B:: C: D, we obtain by what precedes A+ C: B+D::A: B; and, since the third ratio E : F, is equal to the first A: B, we have A+C:B+ D : : E : F. If we take the sum of the antecedents and that of the in this last proportion, the result will be consequents A+C+E: B+D+F:: E: For: A: B. By proceeding in the same manner with any number of equal ratios, it will be seen, that the sum of any number whatever of antecedents is to the sum of their consequents, as one antecedent is to its consequent. V. Let there be any two proportions A: B:: C: D, E: F:: G: H, if we multiply them in order, that is, term by term, the products will form a proportion, thus A× E:BXF :: CxG: D× H, This is evident, since the new ratios BxF DX H are respec whence it follows, that the squares of four proportional quantities that is, the cubes of four proportional quantities form a new proportion. VI. When a proportion is said to exist among certain magnitudes, these magnitudes are supposed to be represented, or to be capable of being represented by numbers; if, for example, in the proportion A: B:: C: D, A, B, C, D, denote certain lines, we can always suppose one of these lines, or a fifth, if we please, to answer as a common measure to the whole, and to be taken for unity; then A, B, C, D, will each represent a certain number of units, entire or fractional, commensurable or incommensurable, and the proportion among the lines A, B, C, D, becomes a proportion in numbers. Hence the product of two lines 4 and D, which is called also their rectangle, is nothing else than the number of linear units contained in A multiplied by the number of linear units contained in B; and we can easily conceive this product to be equal to that which results from the multiplication of the lines B and C. The magnitudes A and B in the proportion A: B:: C: D, may be of one kind, as lines, and the magnitudes C and D of another kind, as surfaces; still these magnitudes are always to be regarded as numbers; A and B will be expressed in linear units, C and D in superficial units, and the product A× D will be a number, as also the product B× C. Indeed, in all the operations, which are made upon proportional quantities, it is necessary to regard the terms of the proportion as so many numbers, each of its proper kind; then we shall have no difficulty in conceiving of these operations and of the consequences which result from them. ELEMENTS OF GEOMETRY. Definitions and Preliminary Remarks. 1. GEOMETRY is a science which has for its object the measure of extension. Extension has three dimensions, length, breadth, and thick ness. 2. A line is length without breadth. The extremities of a line are called points. A point, therefore, has no extension. 3. A straight or right line is the shortest way from one point to another. 4. Every line, which is neither a straight line nor composed of straight lines, is a curved line. Thus AB (fig. 1) is a straight line, ACDB is a broken line, or Fig. 1. one composed of straight lines, and AEB is a curved line. 5. A surface is that which has length and breadth, without thickness. 6. A plane is a surface, in which any two points being taken, the straight line joining those points lies wholly in that surface. 7. Every surface, which is neither a plane nor composed of planes, is a curved surface. 8. A solid is that which unites the three dimensions of ex tension. 9. When two straight lines, AB, AC, (fig. 2), meet, the quan- Fig. 2. tity, whether greater or less, by which they depart from each other as to their position, is called an angle; the point of meeting or intersection A, is the vertex of the angle; the lines AB, AC, are its sides. An angle is sometimes denoted simply by the letter at the vertex, as A; sometimes by three letters, as BAC, or CAB, the letter at the vertex always occupying the middle place. Geom. 1 |