GEOM § 16. The theory of parallels as such may be said to be finished with Props. 33 and 34, which state properties of the parallelogram, i.e., of a qll:l.(ll'lli1lCl'fll formed by two pairs of parallels. They arc—- Prop. 33. The straight lines which join the extremities of two equal and parallel straight lines towards the same parts are themselres equal and parallel ; and Prop. 34. The opposite sides and angles of a parallelogram are equal to one another, and the diameter (diagonal) bisects the parallelo- gram, that is, ¢li'L'ir.lcs it into two equal parts. § 17. The rest of the first book relates to areas of figures. 'l‘he theory is made to depend upon the theorems- l’rop_ 35. Parallelograms on the same base and between the same parallels are equal to one another ; and l‘rop. 36. Parallelograms on equal bases, and between the same parallels, are equal to_ one another. .-s each parallclogi'ain is bisected by a diagonal, the last theorems hold also if the word parallelogram be replaced by “triangle,” as is done in Props. 37 and 38. It is to be remarked that Euclid proves these propositions only in the case when the parallclograins or triangles have their bases in the same straight line. - The theorems converse to the last form the contents of the next three propositions, viz. :- 'l‘ni-:or.i-'..i (Props. 40 and 41).—Equal triangles, on the same or on rqa-_(l bases, in the same straight line, and on the same side of it, are lwhreen the same parallels. 'l‘liat the two cases here stated are given by Euclid in two sepa- 1'a1h- propositions proved separately is characteristic of his method. 3' 18. To compare areas of other figures, Euclid shows first, in l’|'op. 42, hov to draw a parallelogram which is equal in area to a. _q:'rr_'n triangle, and has one of its angles equal to a given angle. If the given angle is right, then the problem is sol'ed to draw a “ -rect- angle " equal in area to a given triangle. Next this parallelogram is trausfornicd into another parallelo- grani, 7l'lLl(.'lL has one of its sides equal to a given straight line, whilst its angles remain unaltered. This may be done by aid of the theorem in Prop. 43. The complements of the parallelograms which are about the diameter of any parallelogram are equal to one another. Thus the problem (Prop. 44) is solved to constructaparallclogram nil a given line, which is equal in area to a gi-vcn triangle, and which l1'(.' one angle equal to a given, angle (gciiernlly a right angle). As every polygon can be divided into a number of triangles, we can now construct a parallelogram having a given angle, say a right angle, and being equal in area to a given polygon. For each of the triangles into which the polygon has been divided, 3. parallelo- gram may be constructed, having one side equal to a given straight line, and one angle equal to a given angle. If these parallelograins l».- placed side by side, they may be added together to form a single parallclograni, having still one side of the given length. This is done in Prop. 43. llercwitli a means is found to compare areas of different polygons. We need only construct two rectangles equal in area to the given polygons, and having each one side of given length. By comparing the uiicqnal sides vc are enabled to jirlgc. whether the areas are equal, or which is the greater. Euclid does not state this con- scqiiciicc, but the problem is taken up again at the end of the second book, where it is shown how to construct a square equal in area to a given polygon. § 19. The first book concludes with one of the most important theorems in the whole of geometry, and one which has been cele- brated since the earliest times. It is stated, but on doubtful authority, that Pythagoras discovered it, and it has been called by his na_mc. If we call that side in a. right-angled triangle which is opposite the right angle the hypotenuse, we may state it as follows :— TlIEOl‘.E.[ or Pr'rii.-(:on.s (Prop. 47).—In every right-angled tri- magi» the square on the h _1/poten-use is equal to the sum of the squares of the other sides. .-iid conversely- Prop. 48. If the square described on one of the sidesof alriangle be equal to the squares described on the other sides, then the angle contained by thrse two sides is a right angle. On this theorem (Prop. 47) almost all geometrical measurement dc-pciids, 'liicli cannot be directly obtained. sooxs 1. 11.] BOOK II. § 20. The propositions in the second book are vcrv different in cliaracter from those in the first; they all rclaté to areas of rectangles and squares. Their truc significance is best seen by stating them in an algebraic foriu. This is often done by expressing the lengths of lines by aid of numbers, which tell how many tlllltg a chosen _unit is contained in the lines. If there is a unit to be found whi«-_h is contained an exact number of times in caclisidc of a rc«_:tanglc., it is easily seen, and generally shown in the teaching of arithmetic, that the rectangle contains a number of unit squares ETRY equal to the product of the numbers which measure the sides, a unit square being the square on the unit line. If, however, no such unit can be found, this process requires that connexion between lines and numbers which is only established by aid of ratios of lines, and which is therefore at this stage altogether inadmissible. But there exists another way of connecting these propositions with algebra, based on modern notions which seem destined greatly to change and to simplify niathcmatics. Ve shall introduce here as much of it as is required for our present purpose. At the beginning of the second book we find a. definition accord- ing to which “ a rectangle is said to be ‘ contained’ by the two sides which contain one of its right angles”; in the text this phrase- ology is extended by speaking of rectangles contained by any two straight lines, meaning the rectangle which has two adjacent sides equal to the two straight lines. We shall denote a finite straight line by a single small letter, a, b, e, . . . :e, and the area of the rectangle contained by two lines a and b by ab, and this we shall call the product of the two lines a and b. it will be understood that this definition has nothing to do with the definition of a product of numbers. Vc define as follows :— The sum of two straight lines a and b means a straight line e which may be divided in two parts equal respectively to a and b. This sum is denoted by a+ b. The diflbrenee of two lines a and b (in symbols, a—b) means a line e which when added to b gives a ; that is, 379 a- b=e if b+c=-a. The product of two lines a and b (in symbols, ab) means the area of the rectangle contained by the lines a and b. F or act, which means the square on the line a, we write a'~’. § 21. The first ten of the fourteen propositions of the second book may then be written in the form of forinulae as follows :— Prop. 1. a(b+c+d+ . . .)=ab+ac+ad+ . . . . ab+ac=a'3 if b+c=a. a(a+b)=a9+ab. (a+b)‘3=a'-'+2ab+b'-’. . (a+b)(a—b)+b'-'=a'-’. . (a+b) (a—b)+b'-’=a2. . a'-'+(a-—b)'-'-=2a(a—b)+b'-’. , .4(a+b)a+b"=('2a+b)'3. H 9, (a+b)‘-’+(a-b)2=‘.2a‘-’+2b‘-’. ,, 10. ((t+b):+((t—lI)3==2(t2+2lI2. It will be seen that 5 and 6, and also 9 and 10, are identical. In El1cll1(l’1S_;;tatC1fi€]1t they do not look the same, the figures being arrancrci (1 ercn y. If {the letters a, b, c, . . . denoted numbers", it folloavs from alrrcbra that each of these formulze is true. But this oes not 1)I%'C them in our case, where the letters denote lines, and their products areas without any reference to inimbcrs. _To prove them we have to discover the laws which rule the operations introduced, viz., addition and multiplication of segments. This we shall do now; and vc shall find that these laws are the same with those which hold in algcbraical addition and multiplication. § 22. In a sum of numbers we may change the order in which the numbers are added, and we may also add the numbers together in groups, and then add these groups. tut this also holds for the sum of segments and for the sum of rectangles, as a little considera- tion shows. That the sum of rectangles has always a meaning follows from the Props. 43-45 in the first book. These laws about addition are reducible to the two— 77 3! 9) 0°~lOt.n5&s§'2Dl~:> a + b = b + a . . . (1), a+(b+c)=a +c . . ('2); or, when expressed for rectangles, ab+ed=-ed+ab . . (3), ab+(cd+e_'f)-=ab+cd+lf . . (4). The brackets mean that the terms in the bracket have been added together before they are added to another term. The more general cases for more terms may be deduced from the above. For the product of two numbers we have the lav that it remains unaltcrcd if the factors be interchanged. This also holds for our geometrical product. For if ab denotes the area of the rectangle which has a as base and b as altitude, then ba will denote the area. of the rectangle which has b as base and a as altitude. But in a rectangle we may take either of the two lines which contain it as base, and then the other will be the altitude. This gives ab = ba . . . . (5)- In order further to multiply a sum by a number, we have in algebra the rulc:—'.lultiply each term of the sum, and add the products thus obtained. That this holds for our geometrical products IS shown by Euclid in his first proposition of the second book, where he proves that the area of a rectangle whose base is the sum of a
nninber of segments is equal to the sum of rectangles which have