b -- a In this example, & divided by a, the quotient is, the product of into ac is ——=b+➡, which being taken from the dividend a bc ; again, if be divided by a, the quotient will be b, leaves ——; a bc2 aa Thus it appears how the divifion is to be continued. aa aa SURDS. Surds are fuch numbers as cannot be exactly expreffed in figures, and as they arise in the solution of algebraie questions, I shall explain to the young algebräift fe much of them only as is neceffary to the prefent defign. ADDITION OF SURD QUANTITIES. CASE I. When the quantities under the radical figns are alike, add the rational quantities, or thofe which are without the radical figns together, by the rules of addition, and to this join the furd quantities, and this will be the fum required. If there be no rational quantities without the radical fign, then unity, or 1, is always fuppofed to be the rational quantity. In example 1, there being no rational quantities, therefore unity or I is the rational quantity to each. joining the furd Vab, we have 2 Now I added to I makes 2, to which ab, the fum required. 3D CASE 2. When the letters under the radical figns are different, place them down one after the other, with the fame figns they have, in the queftion. Essbivih a 16MNO PREXAMPLES. CASE 1. When the letters under the radical figns are alike, fubtract the rational quantities from the rational quantities, and to the difference join the furd quantities, which will be the remainder required. The truth of these operations are proved as in fubtraction of common numbers. CASE 2. When the letters under the radical figns are d different, fet them down one after the other, but care must be taken to change the figns of those quantities that are to be fubtracted. Thefe operations are proved in the fame manner as in the laft cafe, by adding the remainder to the quantity that was fubtracted. MULTIPLICATION. MULTIPLICATION. CASE i. When there are no rational quantities joined to the furd, multiply the furd quantities together, and to their product prefix the radical figns. √xyd Vax+xb √axn-ayn CASE. 2. If rational quantities be joined to the furds, then multiply the rational into the rational, and the furd into the furd, and join the products together. CASE I. When there are no rational quantities joined with the furd quantities, reject all thofe quantities in the dividend and divifor that are alike, and fet down the remainder, to which prefix the radical figns, and this will be the quotient fought. Divide Vabx By Vx Quotient Vab EXAMPLES. In example 1, becaufe x is in both dividend and divifor, reject it, and put down ab with the fign√ before it, and ✅ab is the quotient required; and fo of others. By Divide √bn+ba √mzmp Quotient Va CASE 2. When there are rational quantities joined with the furds, divide the rational quantities by the rational quantities, and to their quotient join the quotient of the furds found by the laft cafe. EXAMPLES. The truth of these operations are proved by multiplying the quotient by the divifor, for if that produces the dividend, the work is true, other wife it is erroneous. EQUATIONS. EQUATIONSUM An equation is, when two equal quantities, differertly expressed, are compared tagether by means of the fign placed between them. Thus 6-24 is an equation expreffing the equality of the quantities 6-2 and 4 and x=a+b is an equation, fhewing that the quantity represented by x is equal to the fum of the two quantities reprefented by a and b. Equations are the means whereby we come at fuch conclufions as answer the conditions of a problem, wherein, from the quantities given, the unknown ones are determined, and this is called the reduction of equations. REDUCTION OF SINGLE EQUATIONS. Single equations are fuch as contain only one unknown quantity, which must be fo ordered by addition, fubtraction, multiplication, divifion, &c. of equal quantities, that a juft equality between the two parts. thereof may be still preserved, and that there may refult, at laft, an equa tion wherein the unknown quantity stands alone on one fide, and all the known ones on the other; the best manner of doing which will be obtained by the following rules. RULE 1. Any term of an equation may be transposed to the contrary fide, if its fign be changed. EXAMPLE 1. Thus, x+8=18, then will 18—8—10. In this equation x+8=18, which by tranfpofition becomes 18 8=10, by only fubtracting the number 8 from both fides. E. 2. If x+19=107, what is the value of x? By tranfpofition the above equation is changed into this x=107-19, therefore 107-19-88 the value of x. E. 3. Given x-10719, required the value of x? By tranfpofition of 107 to the other fide of the equation, and changing the fign, the equation ftands thus, x=19+107, therefore 19+107=126, the value of x. E. 4. If 20-3x-860-7x, what is the value of x? By tranfpofing 7, we fhall have-3x+7x=60-20+8, or 4x48, therefore x: 48 12 the value of x. 4 For 20-12X3-8-60-12X7-24 proof: RULE 2. If there is any quantity by which all the terms of the equation are multiplied, let them all be divided by that quantity; but if all of them be divided by any quantity, let the common divifor be t way. E. 1. Suppose axab, then by the rule x-b; alfo rox=60, re duced x=6; and by the latter part of the rule b is reduced to x=b. 4.x E. 2. Required the value of x, which 36---8. 9 By multiplying both fides by 9, we have 324472, therefore E. 3. If 6 x20x16x+2x2, what is the value of x? By dividing by 2x we have 3x-10=8+x, and by tranfpofition 18 3x-x=8+10, that is zx 18, therefore x = -=9 anfwer. 2 RULE 3. If there are reduceable fractions, let the whole equation be multiplied by the product of all their denominators, or, which is the fame, let the numerator of every term in the equation be multiplied by all the denominators, except its own, fuppofing fuch terms (if any there be) that stand without a denominator, to have an unit fubfcribed. ༤ E. 1. If %+-+ -= 10, what is ≈ equal to ? 2 4 ལ By multiplying the equation by 8, the product of the two denomina tors 2 and 4, we have 8x + 42 + 22 = 80, or 142 80; therefore 80 145,728. E. 2. Let 5 ÷ 3≈≈— 7, required the value of z? This reduced will become 32+ 5%÷1582÷ 15-7, confe quently 82 152— 105, whence 7% = 105, therefore 105÷7. 15 the answer. RULE. 4. If in your equation there is an irreduceable furd, wherein the unknown quantity enters, let all the other terms be transposed to the contrary fide (by rule 1) and then, if both fides are involved to the power. denominated by the furd, an equation will arife free from radical quan tities, unless there happens to be more furds than one, in which cafe the operation is to be repeated. Thus, √x+6=10, by tranfpofition becomes x10-64, which, by fquaring both fides, gives x16. So likewife aa+xx-c=x becomes √aa+xx=c+x, which, fquared, gives aa+xxcc+2cx+xx, or aa-c=2cx,, per rule 1, By tranfpofition becomes ✓5x=17-12=5, and √ 5x=5×3= 3. 15; then by involving 15 to the power denominated by the furd, we ave 5x=225, therefore x=245 E. 2. What |