3. A man is to lay off a rectangular lot of ground; so that the length shall exceed 10 rods, as much as the breadth falls short of 10. He is offered for it, as many dollars per square rod, as the length exceeds 10 rods. How must he lay it out, that he may receive for it the greatest possible sum? 4. A body is projected upwards with a velocity, in feet per second, equal the square root of the whole time of ascent in seconds. What must be the velocity, in order that the space through which the body ascends, may be the greatest or least possible? 5. Find a geometrical series of four terms, descending, having the first term 10; and, such that the difference between the second and fourth terms may be the greatest possible. 6. There is a number, such that if of it be subtracted from 10, and the remainder be multiplied by the number, the result will be greater than if any other number were used. What is the number? 7. A is to lay off a quantity of land. The sum of the length and breadth is to be a rods. B agrees to give him a number of dollars, equal to the difference of the number of rods in the length and breadth. How must the land be laid out, so that A may receive the greatest or least possible sum for the land? 8. What number is that, which is such, that if its second power be subtracted from 200, and the root of the remainder be multiplied by the number, the product will be the greatest possible? 9. How many seconds must a body fall, so that the space described, subtracted from the velocity acquired, may leave the greatest possible remainder? 10. A man took a certain number and subtracted it from 20, and multiplied the remainder by the number; he then added the number to the product, and multiplied the sum by the number, and found that he had obtained a greater result than if he had performed the same operations on any other number. What was the number? SECTION CXVIII. Development of ath and a3, or aa. It is required to find the differential of a, when a is constant, and the exponent x, is variable. Let x become x+h, and a becomes at. Now, if we can find a series of terms containing the different powers of h, which is equivalent to ath, that term which contains the first power of h, is the differential required. Collecting all of this series, multiplied by the first power of h, we have We will represent this series, omitting h, by c; and, the other terms in the value of (1+b)", by sh2+ph+qh, &c. And (1+6)=1+ch+sh2+ph3+qh*, &c. Or a=1+ch + sh2 + ph3 + q h1, &c. Multiply both sides of this last equation by at, we have a2+= a + c a* h + sah2+pa h3 + q a* h*, &c. cah, then, is the differential of a*; but, the c is a constant quantity, depending upon the value of a. In order, then, to find the differential coefficient of a, multiply it by a constant c depending upon a. Multiply this product by h, or d x, and we have the differential of a. In order to find s, p, q, &c., take the differential coefficient of cat, and divide it by 2, and it will be equal to s. Take the differential coefficient of s, and divide it by 3, and it will be =p, &r. See Taylors Theorem. Y Since the differential coefficient of a is a multiplied by a constant c, the differential coefficient of cat is ca* multiplied by c, or c2 at. The two members of equation (A) are equivalent, and are equal, whatever the letters may be. equation (A) by a*, and it becomes Divide both members of Until we supposed c= 1, a would represent any number; but, c 1 gives a such a value, that (a-1) will be 1. It is necessary to find what this value of a is. Make h = 1 in equation (C,) and we have 1 a = 2 + +++!, &c. Continue this series until the last term amounts to less than rõõō55, and we shall have 2.7182818. This number is the base of the Naperian system of logarithms; and, is generally designated by e. Put e in place of a, in equation (B,) it becomes (D) h2 h3 h1 hs e'=1+h+-+ + + &c. The base of the common system of logarithms is 10. It is proposed to find what c is, when a 10. In equation (B,) =1+1+¦+i+js + vla, &c., ¤ e a is α But, we see in equation (E,) that when c is 6561 c = 1% +30% +370200 + 10000%, &c. Hence, we shall find by calculation This last number is called the modulus of the system of logarithms whose base is 10. SECTION CXIX. Logarithms. Let us now see the application of the principles explained in the preceding section, in the calculation of the logarithms of numbers. We have found the differential of a to be ca dx; and, when c=1, which gives a = e = 2.7182818. The differential of e is edx. Let e be the base of a system of logarithms, and x is the logarithm of e; and, consequently, d x, is the differential of the logarithm of e*. Hence, to find the differential of the logarithm of any function to the base e, or 2.7182818, RULE.-Divide the differential of the function by the function itself. Again, the differential of a is ca* dx; but, x is the logarithm of a2, when a is the base. Hence dx is the differential of the logarithm of a*. But, 1 dx=- X diff. of a Therefore, the differential of the logarithm of any function, when the base of the system is any number, a, is the differential of the function, divided by the function; and, 1 the quotient multiplied by, c being a constant number, de C pending upon a, the base of the system. We have seen that when a = 10, 1 =0.434294484 Hence, the differential of a logarithm to the base 10, is equal to the differential of the function divided by the function, multiplied by .434294484; consequently, we can change the differential of a logarithm to the base e, or 2.7182818, into the differential of a logarithm to the base 10, by multiplying the former by the constant number .434294484; and, therefore, multiply the logarithm of any number to the base 2.7182818, by the number .434294484, and it will give the logarithm of the same number to the base 10. 1. It is required to find the logarithm of (a+x,) to the base 2.7182818; and, from it, the logarithm of (a+x) to the base 10. Assume the logarithm of a +x: thus, Log. (a+x)=A+B x+Cx2+Dx3+ Ex1+Fx3, &c. As it is the logarithm (a+x) to the base 2.7182818, the |