G-hlC)Bl hit fioiii this it follows according to the last theorem that ' a : b = c :d. llence we conclude that the quotient and the ratio azb are BOOK vi.] ditl'crcnt forms of the same in.1gnitu(le, only with this important . . a - - ditlercnee that the quotient 5 would have a meaning only if a and I; have a common measure, until we introduce incoinineiisurable numbers, while the ratio a :b has always a meaning, and thus gives rise to the introduction of incoinnieiisurablc numbers. Tlius it is really the theory of ratios in the fifth book which eiiables us to e.'tcnd the geometrical calculus given before in con- n.-'i.,n with Pmok ll. It will also be seen that if we write the ratio; in Book V. as quotients, or rather as fractions, then most of the theorems state properties of quotients or of fractions. 5‘ 64. Prop. 17 contains only a special case of 16. After the pro- l)lL‘lll, Prop. 18. On a given straight line to tlcscribe a rectilincal jigztrc similar anrl .5-imilarly situated to a giren rcrstilincal figure, the-re follows another fundamental tlicoicin: Prop. 19. Similar triangles are to one another in the duplicate ratio 4_:/'th.'ir lzomulogous Sl(l'J‘»‘. In other words, the areas of similar triangles are to one another as the squares on homologous sides. Tlns is generalized in Prop. '30. Sim ilur polygons may be tliritlrtl into the same number (ff similar triangli,-3, huring the same ratio to one another that the 1ml_1/_v/o-is l1.are,- ttnrl the polygons are to one another in the (triplicate ratio of their homologous sides. ,5 (35. Prop. 21. Itectilincal figures zrhich are similar to the same rcclit i ncal figure are also similar to each other, is an immediate con- sequence of the definition of similar figures. As similar tigurcs 1na_v be said to he cql1‘1l in “shape” but not in “size," we may state it also thus: " Figures which are equal in shape to a third are equal in shape t-i each other. ’’ Prop. ‘.22. If four straight lines be ;noportiouals', the similar rcctiliueztl figures similarly (lC.S'I_'I‘lbf.'(l on them. shall also be pro- por/imials-; and if the similar -rcctilineal _/igures sinz ilarl y tlcseribrel ml four straiglzt lines be proportionals, those straight lines shall be pi-upm-ti'~nal.~". This is essentially the same as the following :-— If a :b = c :47, then a"-’ :b'3 = c’ rd". § 66. New follows a proposition “'1liLll has been much discussed with regard to l‘Iu«-lid’s exact meaning in saying that a ratio is com- pomulcil of two other iatios, viz. : Prop. 23. Parutlclograms u-hich are cquiangular to one another, lt'l7'v to one another the ratio irhieh is compomuletl of the ratiosof their .s'i4l»'5'. The pro-nf of the proposition makes its meaning clear. In symbols the ratio lb :e is coiiipoinided of the two ratios a : b and b :c, and if H :b = a’ :b’ , b :c = b” :c", then a :e is compounded of a.': b’ and
- 1’ 1b" .
lf we consider the ratios as numbers, we may say that the one ratio is the product of those of which it is compounded, or in ~syiiib0ls, a_a b_a.' b" .fa a’ b b" c 5 '2 19'3"‘ 5 “ 27 “ml c=‘a" The tlieorein in Prop. 23 is the foundation of all mensuration of areas. F rom it we see at once that two rectangles have the ratio of their areas compounded of the ratios of their sides. If A is the area of a rectangle contained by a and b, and B that of a rectangle contained by c and d. so that A = ab, 13 = at, then A : B = ab 2 ed, and this is, the theorem says, compounded of the ratios a : c and b : (Z. In forms of quotients, a b _ ab c ' d ed‘ This shows how to multiply quotients in our geometrical calculus. Further, Two triangles haze the ratios of their areas compomided of the ratios of their bases and their altitutle. Foratriangle is equal in area to half a parallelograin which has the same base and the same altitude. 'l‘o bring these theorems to the form in which they are usually given, we assume a straight line it as our unit of length (generallv an inch, afoot, a mile, &c.), and determine the number a which expresses how often a is eontained in a line a, so that a. denotes the ratio a : it whether coniinensurable or not, and that a = au. We call this number a the numerical value of a. If in the same manner 8 be the numerical value of a line b we have a : b = a : B ; in words: The ratio of two lines (and of two like quantities in grnrral) is equal to that of their numerical ralues. This is easily proved by observing that a = alt, b = Bu, there- fore it :b = an: an, and this may without difficulty be shown to equal a :8. E T R Y 385 If now a, b be base and altitude of one, a’, b’ those of another parallelograni, a, B and a’, 3’ their numerical values respectively, and A, A’ their areas, then A4: ’:=a B _ _ _ ‘*3 Al “I ' b; a! ‘ B; a/B/' In words : The areas of two parallclogra-ms are to each other as the pro/lucts of the numerical -values of their bases and altitudes. If especially the second parallelograin is the unit square, 'i.e., a square on the unit of length, then a’ = B’ = 1, A’ = u‘-’, and we have A AI This gives the theorem : The number of unit squares contained in a parallclograin equals the product of the numerical values of base and altitude, and similarly the number of unit squares contained in a triangle equals half the product of the numerical values of base and altitude. - This is often stated by saying that the area of a parallelogram is equal to the product of the base and the altitude, meaning by this product the product of the niniierical values, and not the product as defined al)ove in § 20. § 68. Propositions 24 and 26 relate to parallelograms about diagonals, such as are considered in Book I., 43. They are— Prop. ‘.24. Parallclograms about the diameter of any parallelo- gram are similar to the whole parallelogram and to one another; and its converse (Prop. 26), If two similar paralleloy-rams have a common angle, and be similarly situated, they are about the same diameter. Between these is in.serted a problem. Prop. 25. To describe a rcetilincal figure zrhich shall be similar to one gircn rectilinrar figure, and equal to another gircn rcctilincal _fi_r/urc. § 69. Prop. 27 contains a theorem relating to the theory of maxima and ininima. We may state it thus: Pi op 27 Ifa parallclogrtun be tliritletl into two by a straight line cutting the base, amt if on hair" the base another parallelogram be constructed similar to one of those. parts, then this third parallelo- gram. is greater than the other part. Of far greater interest than this general theorem is a special case of it, where the parallelograins are changed into rectangles, and where one of the parts into which the parallelogram is divided is made a square; for then the theorem changes into one which is easily recognized to be identical with the following :- Tlll-ZOllE!l[.—-Of all rectangles ichich have the same perimeter the square has the greatest area. This may also be stated thus:—— 'l‘iii:or.i:.r.—0_/' all rectangles zchich have the same area the square has the least periotetcr. § 70. The next three propositions contain problems which may be said to be solutions of quadratic equations. The first two are, like the last, involved in somewhat obscure language. 'e tran- scribe them as follows :— Problem. ——To describe on a given base a pai'allelograin, and to divide it either internally (Prop. 28) or externally (Prop. 29) from a point on the base into two parallelograins, of which the one has a given size (is equal in area to a given tigure), whilst the other has a. given shape (is similar to a given parallelogram). If we express this again in symbols, calling the given base a, the one part 51', and the altitude y, we have to determine .1: and y in the first case from the equations (a - 96).'/ = 753, =aB . or A = a3. u‘-’. Ir? being the given size of the first, and p and q the base and alti- tude of the parallelogilam which determine the shape of the second of the required paralle ogranis. If we substitute the value of 3/, we get (a - .’l')1?=?,£21 (1 or, . .-2_ - (t.1.—:L — , where a and b- are known quantities, taking b9=?1I.”;=. 0 'l‘lie second case (Prop. 29) gives rise, in the same niaimer, to the quadratic a:c+:c9=b‘~‘. The next problein— Prop. 30 To cut a given straight line in c.rtrcmc and mean ratio, leads to the equation a..c+.L9=a‘-’. This is, therefore, only a special case of the last, and is, besides, an old acquaintance, being essentially the same problem as that proposed in II. 11.
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