a similar construction by making beam, and dan; for a2 + amc + and it will then become refers itself to the preceding Now the factor constructions, as also the values of m, n. Having found the value of this factor, if we represent it by p, we have only to construct a X p, that is, to make a rhomboid whose altitude is a and base p. a2 + ab 4. Lastly, if the dimensions of the numerator exceed the dimensions of the denominator by three, the quantity expresses a solid, the construction of which may always be reduced to a parallelopiped. If, for example, we were to construct a2 b + a2 b2 , we might consider this quantity as the same as a + c a2 + ab ab x ; and, having constructed a + c ner already explained, if we represent by m, the line given by this construction, the question will be reduced to this, namely, to construct ab X m. Now a b represents, as we have seen, a rhomboid; if, therefore, we conceive a parallelopiped, having for its base this rhomboid, and for its altitude the line m, the solidity of this parallelopiped will represent a b × m, that is, a3 b + a2 b2 a+c 5. What has been said will suffice for constructing any rational quantity; we proceed now to rational quantities of the second degree. D B 1st. In order to construct ab, let us draw an indefinite line AB, upon which we may take the part CA, equal to a, and the part BC, equal to b; upon the whole AB as a diameter, describe a semicircle, cutting in D, the perpendicular CD, raised upon AB at the point C; then CD will be the value of Vab; that is, the value of ab is obtained by finding a mean proportional between the two quantities represented by a, b. Indeed, we have 2nd. If we were to construct √3 ab + b2, or which is the same thing, ✔(3 a + b) b, we should find a mean proportional between 3 a + b and b. 3d. In like manner, if the quantity to be constructed were √ a2 — b2, we might consider this the same as √(a+b) (a—b); and then find a mean proportional between a + b and a-b. If the quantity were va2 + b c, and we make bc = am, then we shall have a + am, or √(a + m) a, which is constructed by finding a mean proportional between a +m and a after having bc constructed the value of m= by the rules already given. α C 4th. To construct va2 + b2, we can in like manner make b2 = am, and construct a + am, in the manner just explained. But the property of a right angled triangle furnishes a more simple construction. If we draw the line AB, equal to a, and at its extremity A erect a perpendicular AC, equal to b, joining BC, we shall have BC ABAC = a + b2, and, consequently, BC = √a2 + b2. A B B 5th. We can also, by means of a right angled triangle, construct ✔a b2 in a manner different from that above given; draw a line, AB, equal c to a, and having described upon AB, as a diameter, the semicircle ACB, draw from the point A a chord AC, equal to b; then, if we draw BC, this line will be the value of Va-b; for the triangle ABC being right angled, we shall have AB AC + BC; consequently, BC AB' AC' ab2; therefore BC = = √ a2 - b2. - A D 6th. Hence, also, ✔a2 + be admits of a different construction from the above. Make bc = m2, and construct va2 + m2, as just shown, first finding for m a mean proportional between b and c, as indicated by the equation be = m2, which gives m = √be 7th. If there are more than two terms under the radical sign, the construction is to be reduced to one of the preceding methods by means of transformations. If, for example, we have ✔a2 + bc + ef we may make be = am, ef =an, and we have va2 + am + an, or √(a + m + n) a, which may be constructed by finding a mean proportional between a and a+m+n, after having constructed the values of m and n, ef namely, m = bc a' n = a We might, moreover, make and then we should have to construct √a2 + m2 + n3. Now, when there are several positive squares contained under the radical sign, as va2+m2 +n2+p2+ &c. we may make √ a2+m2=h, E √h2+n2 = i, √i+p2=k, and so on; and, as each of the quantities is determined by the preceding, the last will give the value of va2+m2+n2+p2+ &c. In order to construct these quantities in the most simple manner, each hypothenuse is to be regarded successively as a side; having, for example, taken AB = a, and raised the perpendicular AC = c, we may join BC, which will be h; then at the point C if we raise upon BC the perpendicular CD = n; and having drawn BD, which will be i, at the extremity D, we may raise upon BD the perpendicular DE = p, and BE will be k, and equal to √ a2 + m2 +n2+p3. c If some of the squares are negative, we may combine the method just given with A that for constructing Vaa- ba B 8th. Lastly, if the quantity to be constructed be of this form multiplying by ✓d+e, will change it into a √(b+c) (d+e) ; d + e then, by finding a mean proportional between b + c and d + e, and calling it m, we have am d + e which is easily constructed. The construction often becomes much more simple by setting out always from the same principles; but these simplifications are derived from certain considerations which are peculiar to each question, and consequently can be made known only as the occasion presents itself. We will merely remark, in concluding, that although the construction of the radical quantities, which we have been considering, reduces itself to finding fourth proportionals, mean proportionals, and constructing right-angled triangles, still we can arrive at constructions more or less simple or elegant by the method employed for finding these mean proportionals; we shall now, therefore, introduce two other methods of finding a mean proportional between two given lines. The first consists in describing upon the greater AB (see 3d diagram to Art. 5) of two given lines a semicircle ACB, and, having taken a part AD equal to the less, raising a perpendicular DC and drawing the chord AC, which will be a mean proportional between AB and AD; for, by drawing CB, the triangle ACB is right angled, (Prop. XIX, Cor. 2, B. III El. Geom.) and consequently AC is a mean proportional between the hypothenuse AB and the segment AD. (Prop. XVII, * Cor. 5, B. IV, El. Geom.) The second method consists in drawing a line AB, equal to the greater given line, and having taken a part AC equal to the less, describing upon the remainder BC a semicircle CDB, to which if we draw the tangent AD; this tangent is a mean proportional between AB and AC. (Prop. XXVII, B. IV, El. Geom.) B C A It is evident, therefore, that rational quantities may always be constructed by means of straight lines, and radical quantities of the second degree may be constructed by means of the circle and straight line united. As to radical quantities of higher degrees, their construction depends upon the combination of different curved lines. We will now proceed to the consideration of questions, the solution of which depends either upon rational quantities or radical quantities of the second degree. Geometrical questions, and modes of forming equations therefrom, and their solutions. 6. The precepts usually given in algebra for putting questions into equations, are equally applicable to questions in geometry. Here, also, the thing sought is to be represented by some symbol; and the equation is to be constructed in such manner, as to express the relations of the quantity represented by such symbol, in quantities that are known, or in those whose values are attainable; and the reasoning is to be conducted by the aid of this symbol, and of those which represent the other quantities, algebraically, as if the whole were known, and we were proceeding to verify it; this method of proceeding is called analysis. Although in expressing geometrical questions by algebraic equations, we have more resources and more facilities according as we are acquainted with a greater number of the properties of lines, surfaces, &c., still, as algebra itself furnishes the means of discovering these properties, the number of propositions really necessary is very limited. The two proposi tions that similar triangles have their homologous sides proportional; and, that, in a right angled triangle the square of the hypothenuse is equivalent to the sum of the squares of the two other sides, are the fundamental propositions and the basis of the application of algebra to geometry. But there are many ways of making use of these propositions, according to the nature of the question, and there is always a discretion to be exercised in the choice of the means, and manner of applying them; and this discertion can only be acquired by practice. When a geometrical question is to be resolved algebraically, it will be necessary to construct a figure that shall represent the several parts or conditions of the problem under consideration, and, if possible, get such expressions for the unknown quantities in terms of those that are known, as may easily be determined, according to the known properties of the figure. But if it so happens, that the required quantity can have no expression which will render it available under the present construction, we may, frequently, by drawing lines having certain relations to the known parts of the figure and also to the unknown, so connect the known to those that are required, as to ⚫get available expressions for their values. Having proposed a figure as above, we may, by means of the proper geometrical theorems, proceed to make out as many independent equations as there are unknown quantities; and the resolutions of these will give the solution. PROBLEM I. The base BC, and the sum of the hypothenuse AB and perpendicular AC, of a right angled triangle being given, to determine the triangle. Let BC = b, and AC =x, and if AB + AC A be represented by s, then will the hypothenuse AB be represented by s―x. Therefore, by the properties of the right angled triangle (Prop. XXIV, B. IV, El. Geom.) we have AC2 + BC2 = AB2 Or, x2 + b2 = s2— 2 sx + x2, B omitting 2 which is common to both sides of the equation, and transposing the other numbers we have, which is the value of the perpendicular AC; where s and b may be any numbers whatever, provided s be greater than b. (Prop. X, B. II, El. Geom.) |