PROPOSITION V. THEOREM. If four quantities are proportional, they are also proportional If four quantities are proportional, they are also proportional . or, (A+B)×C=A×(C+D). Therefore, by Pr. 2, A+B:A::C+D: C. PROPOSITION VII. THEOREM. If four quantities are proportional, they are also proportional by division. Let then, by division, For, since by Pr. 1, then, or, Subtract each of these equals from AxC; AXC-BXC-A×C-A×D, Therefore, by Pr. 1, A-B: A:: C-D: C. If four quantities are proportional, the sum of the first and second is to their difference as the sum of the third and fourth is to their difference. then will By Pr. 6, A+B:A-B::C+D:C-D. and by Pr. 7, A-B: A:: C-D: C. By alternation (Pr. 3), these proportions become If any number of quantities are proportional, any one antecedent is to its consequent as the sum of all the antecedents is to the sum of all the consequents. or, A×B+A×D+A×F=A×B+B×C+B×E; But B+D+F may be regarded as a single quantity, and A+C+E as a single quantity. Therefore, by Pr. 1, A:B::A+C+E:B+D+F. PROPOSITION X. THEOREM. Equimultiples of two quantities have the same. ratio as the quantities themselves. Let A and B be any two quantities of the same kind, and m any number, entire or fractional, we have the equality If four magnitudes are proportional, we may multiply the ante cedents or the consequents, or divide them by the same quantity, and the results will be proportional. PROPOSITION XI. THEOREM. If four quantities are proportional, their squares or cubes are or, multiplying each of these equals by itself (Ax. 1), we have If there are two sets of proportional quantities, the products of the corresponding terms are proportional. Multiplying together these equal quantities, we have A×D×EXH=BxCxFxG; (A× E) × (D×H)=(B×F)×(C×G); or, PROPOSITION XIII. THEOREM. If three quantities are proportional, the first is to the third as the square of the first to the square of the second. BOOK III THE CIRCLE, AND THE MEASURE OF ANGLES. Definitions. 1. A circle is a plane figure bounded by a line, all the points of which are equally distant from a point within, called the centre. The fine which bounds the circle is called its circumference. 2. Any straight line drawn from the centre of the circle to the circumference is called a radius of the circle, as CA, CD. Any straight line drawn through the centre, and terminated each way by the circumference, is called a diam eter, as AB. Cor. All the radii of a circle are equal; also all the diameters are equal, and each is double the radius. 3. An arc of a circle is any portion of its circumference, as EGF. E F The chord of an arc is the straight line which joins its two extremities, as EF. The arc EGF is said to be subtended by its chord EF. 4. A segment of a circle is the figure included between an arc and its chord, as EGF. Since the same chord EF subtends two arcs EGF, EHF, to the same chord there correspond two segments EGF, EHF. By the term segment, the smaller of the two is always to be understood, unless the contrary is expressed. 5. A sector of a circle is the figure included between an arc and the two radii drawn to the extremities of the arc, as BCD. 6. A straight line is said to be inscribed in a circle when its extremities are on the circumference, as AB. An inscribed angle is one whose vertex is on the circumference, and which is formed by two chords, as BAC. B 7. A polygon is said to be inscribed in a circle when all its angles have their vertices on the circumference, as ABC. The circle is then said to be described about the polygon. 8. An angle is said to be inscribed in a segment when it is con |