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THEOR. 18. Rectangles of equal altitude are to one another in the same ratio as their bases.
Let AC, BD be two rectangles of equal altitude on the bases AO, BQ,
then shall the rectangle AC be to the rectangle BD as the base AO is to the base BQ.
Since the rectangles are of equal altitude, QD is equal to OC, and if DQ be placed on CO, D upon C, and therefore Q on O, and B on the same side of CO as A, Q B will fall along OA. Take OM m.OA and ON = n.OB,
and complete the rectangles MC, NC.
Then if OM were divided into m parts, each equal to OA, and rectangles of altitudes equal to OC described on each part, these rectangles would be all equal,
II. I, Cor. 2.
and they would together make up the rectangle MC,
so is the rectangle MC> or < NC;
II. 1, Cor. 2.
that is, according as m.OA > = or < n.OB,
so is m. the rectangle CA> or <n. the rectangle CB; therefore, m and ʼn being any two numbers,
the rectangle CA: the rectangle CB :: base OA: base OB; Def. 4. or the rectangle AC : the rectangle BD :: base AO : base BQ.
In the same circle or in equal circles angles at the centre and sectors are to one another as the arcs on which they stand.
Let O, O' be the centres of two equal circles; AB, A'B' any two arcs of these circles included between the radii OA, OB and O'A', O'B' respectively:
then, if the arc AB be that described by a point which moves on the circumference in any manner from A to B, and the angle AOB be that turned through by a radius drawn to the point, and the arc A'B' and angle A'O'B' be similarly estimated,
the angle AOB: the angle A'O'B': : arc AB : arc A'B', and the sector AOB: the sector A'O'B': : arc AB: arc A'B'.
Let AM be an arcm. the arc AB,
then, observing that Theors. 4 and 5 of Book III. are true for arcs of unlimited length (not necessarily less than one circumference), provided that the corresponding angles are reckoned as those turned through by a radius drawn to a point which moves along the whole arc from one extremity to the other (and therefore not necessarily less than two straight angles),
because the arc AM is made up of m arcs, each equai to AB, and on each of these arcs stands an angle equal to AOB, III. 5. therefore the angle AOM = m. the angle AOB,
and the sector AOM = m. the sector AOB,
In like manner if A'M' = n.A'B',
III. 4, Cor.
the angle A'O'M' n. the angle A'O'B',
and the sector A'O'M' = n. the sector A'O'B'.
Then according as the arc AM >= or < the arc A'M',
so is the angle AOM > = or < the angle A'O'M', and the sector AOM > = or < the sector A'O'M', that is, according as m.AB>
< n. the angle A'O'B',
so is m. the angle AOB >= or
Whence the angle AOB: the angle A'OB' :: arc AB : arc A'B', and the sector AOB: the sector A'OB' :: arc AB : arc A'B',
Def. 4. Q.E.D.
DEFINITIONS OF BOOK IV.
DEF. I. One magnitude is said to be a multiple of another magnitude, when the former contains the latter an exact number of times. According as the number of times is 1, 2, 3 multiple said to be the 1st, 2nd, 3rd . . . mth.
m, so is the
One magnitude is said to be a measure or part of another magnitude, when the former is contained an exact number of times in the latter.
The ratio of one magnitude to another of the same kind is the relation of the former to the latter in respect of quantuplicity. DEF. 4. The ratio of one magnitude to another is equal to that of a third magnitude to a fourth (whether of the same or of a different kind from the first and second), when any equimultiples whatever of the antecedents of the ratios being taken and likewise any equimultiples whatever of the consequents, the multiple of one antecedent is greater than, equal to, or less than that of its consequent, according as that of the other antecedent is greater than, equal to, or less than that of its consequent.
DEF. 5. The ratio of one magnitude to another is greater than that of a third magnitude to a fourth, when equimultiples of the antecedents and equimultiples of the consequents can be found such that, while the multiple of the antecedent of the first is greater than, or equal to, that of its consequent, the multiple of the antecedent of the other is not greater, or is less, than that of its consequent.
DEF. 6. When the ratio of A to B is equal to that of P to Q the four magnitudes are said to be proportionals or to form a proportion. The proportion is denoted thus:
A: B:: P:Q,
which is read "A is to B as P is to Q." A and Q are called the extremes, B and P the means, and Q is said to be the fourth proportional to A, B, and P. The antecedents A, P are said to be homologous, and so are the consequents B, Q.
Three magnitudes (A, B, C) of the same kind are said to be proportionals, when the ratio of the first to the second is equal to that of the second to the third; that is, when A: BB: C.
In this case C is said to be the third proportional to A and B, and B the mean proportional between A and C.
DEF. 8. The ratio of any magnitude to an equal magnitude is said to be a ratio of equality. If A be greater than B, the ratio A: B is said to be a ratio of greater inequality, and the ratio B : A a ratio of less inequality. Also the ratios A: B and B: A are said to be reciprocal to one another.
DEF. 9. If there are any number of magnitudes of the same kind, the first is said to have to the last the ratio compounded of the ratios of the first to the second, of the second to the third, and so on to the last magnitude.
DEF. 10. If there are any number of ratios, and a set of magnitudes is taken such that the ratio of the first to the second is equal to the first ratio, and the ratio of the second to the third is equal to the second ratio, and so on, then the first of the set is said to have to the last the ratio compounded of the original ratios.
DEF. II. When two ratios are equal, the ratio compounded of them is called the duplicate ratio of either of the original ratios.
DEF. 12. When three ratios are equal, the ratio compounded of them is called the triplicate ratio of any one of the original ratios.