IN Book 1, Geometrical magnitudes of the same kind, lines, angles and surfaces, more particularly triangles and parallelograms, are compared, either as being absolutely equal, or unequal to one another. In Book II, the properties of right-angled parallelograms, but without reference to their magnitudes, are demonstrated, and an important extension is made of Euc. I. 47, to acute-angled and obtuse-angled triangles. Euclid has given no definition of a rectangular parallelogram or rectangle: probably, because the Greek expression wapaλλnλóɣpaμμos ὀρθογώνιον, οι ὀρθογώνιον simply, is a definition of the figure. In English, the term rectangle, formed from rectus angulus, ought to be defined before its properties are demonstrated. A rectangle may be defined to be a parallelogram having one angle a right-angle, or a right angled parallelogram; and a square is a rectangle having all its sides equal. а As the squares in Euclid's demonstrations are squares described or supposed to be described on straight lines, the expression "the square on AB," is a more appropriate abbreviation for "the square described on the line AB," than "the square of AB." The latter expression more fitly expresses the arithmetical or algebraical equivalent for the square on the line AB. In Euc. 1. 35, it may be seen that there may be an indefinite numbe of parallelograms on the same base and between the same parallels whose areas are always equal to one another; but that one of them has all its angles right angles, and the length of its boundary less than the boundary of any other parallelogram upon the same base and between the same parallels. The area of this rectangular parallelogram is therefore determined by the two lines which contain one of its right angles. Hence it is stated in Def. 1, that every right-angled parallelogram is said to be contained by any two of the straight lines which contain one of the right angles. No distinction is made in Book 11, between equality and identity, as the rectangle may be said to be contained by two lines which are equal respectively to the two which contain one right angle of the figure. It may be remarked that the rectangle itself is bounded by four straight lines. It is of primary importance to discriminate the Geometrical conception of a rectangle from the Arithmetical or Algebraical representation of it. The subject of Geometry is magnitude not number, and therefore it would be a departure from strict reasoning on space, to substitute in Geometrical demonstrations, the Arithmetical or Algebraical representation of a rectangle for the rectangle itself. It is however, absolutely necessary that the connexion of number and magnitude be clearly understood, as far as regards the representation of lines and areas. All lines are measured by lines, and all surfaces by surfaces. Some one line of definite length is arbitrarily assumed as the linear unit, and the length of every other line is represented by the number of linear units, contained in it. The square is the figure assumed for the measure of surfaces. The square unit or the unit of area is assumed to be that square, the side of which is one unit in length, and the magnitude of every surface is represented by the number of square units contained in it. But here it may be remarked, that the properties of rectangles and squares in the Second Book of Euclid are proved independently of the consideration, whether the sides of the rectangles can be represented by any multiples of the same linear unit. If, however, the sides of rectangles are supposed to be divisible into an exact number of linear units, a numerical representation for the area of a rectangle may be deduced. On two lines at right angles to each other, take AB equal to 4, and AD equal to 3 linear units. Complete the rectangle ABCD, and through the points of division of AB, AD, draw EL, FM, GN parallel to AD; and HP, KQ parallel to AB respectively. Then the whole rectangle AC is divided into squares, all equal to each other. And AC is equal to the sum of the rectangles AL, EM, FN, GC; (II. 1.) also these rectangles are equal to one another, (1.36.) therefore the whole AC is equal to four times one of them AL. Again, the rectangle AL is equal to the rectangles EH, HR, RD, and these rectangles, by construction, are squares described upon thie equal lines AH, HK, KĎ, and are equal to one another. Therefore the rectangle AL is equal to 3 times the square on AH, but the whole rectangle AC is equal to 4 times the rectangle AL, therefore the rectangle AC is 4 x 3 times the square on AH, or 12 square units: that is, the product of the two numbers which express the number of linear units in the two sides, will give the number of square units in the ectangle, and therefore will be an arithmetical representation of its area. And generally, if AB, AD, instead of 4 and 3, consisted of a and b inear units respectively, it may be shewn in a similar manner, that the area of the rectangle AC would contain ab square units; and therefore the product ab is a proper representation for the area of the rectangle AC. Hence, it follows, that the term rectangle in Geometry corresponds to the term product in Arithmetic and Algebra, and that a similar comparison may be made between the products of the two numbers which represent the sides of rectangles, as between the areas of the rectangles themselves. This forms the basis of what are called Arithmetical or Algebraical proofs of Geometrical properties. If the two sides of the rectangle be equal, or if b be equal to a, the figure is a square, and the area is represented by aa or a3. Also, since a triangle is equal to the half of a parallelogram of the same base and altitude; Therefore the area of a triangle will be represented by half the rectangle which has the same base and altitude as the triangle: in other words, if the length of the base be a units, and the altitude be b units; Then the area of the triangle is algebraically represented by tab. The demonstrations of the first eight propositions, exemplify the obvious axiom, that, "the whole area of every figure in each case, is equal to all the parts of it taken together." Def. 2. The parallelogram EK together with the complements AF FC, is also a gnomon, as well as the parallelogram HG together with the same complements. Prop. I. For the sake of brevity of expression, "the rectangle contained by the straight lines AB, BC," is called "the rectangle AB, BC;" and sometimes "the rectangle ABC." To this proposition may be added the corollary: If two straight lines be divided into any number of parts, the rectangle contained by the two straight lines, is equal to the rectangles contained by the several parts of one line and the several parts of the other respectively. The method of reasoning on the properties of rectangles, by means of the products which indicate the number of square units contained in their areas, is foreign to Euclid's ideas of rectangles, as discussed in his Second Book, which have no reference to any particular unit of length or measure of surface. Prop. 1. The figures BH, BK, DL, EH are rectangles, as may readily be shewn. For, by the parallels, the angle CEL is equal to EDK; and the angle EDK is equal to BDG (Euc. 1. 29.). But BDG is a right angle. Hence one of the angles in each of the figures BH, BK, DL, ÈH is a right angle, and therefore (Euc. 1. 46, Čor.) these figures are rectangular. Prop. 1. Algebraically. (fig. Prop. 1.) Let the line BC contain a linear units, and the line A, b linear units of the same length. Also suppose the parts BD, DE, EC to contain m, n, p linear units respectively. Then am + n + p, multiply these equals by b, therefore ab bm + bn + bp. = That is, the product of two numbers, one of which is divided into any number of parts, is equal to the sum of the products of the undivided number, and the several parts of the other; or, if the Geometrical interpretation of the products be restored, The number of square units expressed by the product ab, is equal to the number of square units expressed by the sum of the products bm, bn, bp. Prop. II. Algebraically. (fig. Prop. 11.) Let AB contain a linear units, and AC, CB, m and n linear units respectively. multiply these equals by a, Then m+ n = a, therefore am + an = = a2. That is, if a number be divided into any two parts, the sum of the products of the whole and each of the parts is equal to the square of the whole number Prop. III. Algebraically. (fig. Prop. 11.) Let AB contain a linear units, and let BC contain m, and AC, n linear units. multiply these equals by m, Then a = m + n, therefore ma = m3 + mn. That is, if a number be divided into any two parts, the product of the whole number and one of the parts, is equal to the square of that part, and the product of the two parts. Prop. IV. might have been deduced from the two preceding propositions; but Euclid has preferred the method of exhibiting, in the demonstrations of the second book, the equality of the spaces compared. In the corollary to Prop. XLVI. Book I, it is stated that a parallelogram which has one right angle, has all its angles right angles. By applying this corollary, the demonstration of Prop. IV. may be considerably shortened. If the two parts of the line be equal, then the square on the whole line is equal to four times the square on half the line. Also, if a line be divided into any three parts, the square on the whole line is equal to the squares on the three parts, and twice the rectangles contained by every two parts. Prop. IV. Algebraically. (fig. Prop. IV.) Let the line AB contain a linear units, and the parts of it AC and BC, m and n linear units respectively. Then am + n, squaring these equals, .. a2 = (m + n)*, or a2 = m2 + 2mn + n2. That is, if a number be divided into any two parts, the square of the number is equal to the squares of the two parts together with twice the product of the two parts. From Euc. 11. 4, may be deduced a proof of Euc. 1. 47. In the fig. take DL on DE, and EM on EB, each equal to BC, and join CH, HL, LM, MC. Then the figure HLMC is a square, and the four triangles CAH, HDL, LEM, MBCare equal to one another, and together are equal to the two rectangles AG, GE. Now AG, GE, FH, CK are together equal to the whole figure ADEB; and HLMC, with the four triangles CAH, HDL, LEB, MBC also make up the whole figure ADEB; Hence AG, GE, FH, CK are equal to HLMC together with the four triangles : but AG, GE are equal to the four triangles. wherefore FH, CK are equal to HLMC, that is, the squares on AC, AH are together equal to the square on CH Prop. v. It must be kept in mind, that the sum of two straight lines in Geometry, means the straight line formed by joining the two lines together, so that both may be in the same straight line. The following simple properties respecting the equal and unequal division of a line are worthy of being remembered. I. Since AB = 2BC = 2 (BD + DC) = 2BD + 2DC. (fig. Prop. v.) and AB AD + DB; .. 2CD + 2DB = AD + DB, and by subtracting 2DB from these equals, .. 2CD = AD - DB, and CD (AD – DB). = That is, if a line AB be divided into two equal parts in C, and into two unequal parts in D, the part CD of the line between the points of section is equal to half the difference of the unequal parts AD and DB. II. Here AD = AC + CD, the sum of the unequal parts, (fig. Prop. v.) and DB AC - CD their difference. Hence by adding these equals together, .. AD + DB = 2AC, or the sum and difference of two lines AC, CD, are together equal to twice the greater line. And the halves of these equals are equal, ...AD + . DB = AC, or, half the sum of two unequal lines AC, CD added to half their difference is equal to the greater line AC. III. Again, since AD = AC + CD, and DB by subtracting these equals, .. AD - DB = 2CD, = AC - CD, or, the difference between the sum and difference of two unequal lines is equal to twice the less line. And the halves of these equals are equal, ...AD -. DB = CD, or, half the difference of two lines subtracted from half their sum is equal to the less of the two lines. IV. Since AC CD = DB the difference, .. AC = CD + DB, and adding CD the less to each of these equals, .. AC + CD = 2CD + DB, or, the sum of two unequal lines is equal to twice the less line together with the difference between the lines. Prop. v. Algebraically. Let AB contain 2a linear units, its half BC will contain a linear units. And let CD the line between the points of section contain m linear units. Then AD the greater of the two unequal parts, contains a+m linear units; and DB the less contains a - m units. Also m is half the difference of a + m and a m; .. (a + m) (a — m) a2 - m2, to each of these equals add m2; .. (a + m) (a - m) + m3 = a3. That is, if a number be divided into two equal parts, and also into two unequal parts, the product of the unequal parts together with the square of half their difference, is equal to the square of half the number. Bearing in mind that AC, CD are respectively half the sum and half the difference of the two lines AD, DB; the corollary to this proposition may be expressed in the following form: "The_rectangle contained by two straight lines is equal to the difference on the squares of half their sum and half their difference." The rectangle contained by AD and DB, and the square on BC are each bounded by the same extent of line, but the spaces enclosed differ by the square on CD. A given straight line is said to be produced when it has its length increased in either direction, and the increase it receives, is called the part produced. If a point be taken in a line or in a line produced, the line is said to be divided internally or externally, and the distances of the point from |