In 1770, one of the founders of pure mathematics, Swiss mathematician Leonard Euler (1707-1783), published Elements of Algebra, a mathematics textbook for students. This edition of Euler's classic, published in 1822, is an English translation which includes notes added by Euler's tutor, Johann Bernoulli, and additions by Joseph-Louis Lagrange, both giants in eighteenth-century mathematics, as well as a short biography of Euler. Part 1 begins with elementary mathematics of determinate quantities and includes four sections on simple calculations (adding, subtracting, division, multiplication), and then progresses to compound calculations (fractions), ratios and proportions and algebraic equations. Part 2 consists of 15 chapters on analyses of indeterminate quantities. Here, Euler shows the reader several ways to solve polynomial equations up to the fourth degree. This landmark book showed students the beauty of mathematics, and more significantly, how to do it.

dispute began with a paper by O'ALEMBERT published in 1749 and continued through O'ALEMBERT'S remaining life. HANKINS on page 48 of his biography, Jean D'Alembert, Oxford, Clarendon Press, 1970, states that D' ALEMBERT conceded defeat in a final volume of his Opuscules, which exists in manuscript but was never published. On the whole, the controversy was not resolved during the lifetimes of any of the main disputants but rather just died out. EULER solved all the central problems concerning a

to one of the following expressions : 5a + 1, 5a + 2, 5a + 3, 5a + 4; and in the same manner we may continue, and consider any greater divisor. 63. It is here proper to recollect what has been already said on the resolution of numbers into their simple factors; for every number, among the factors of which is found 2, or 3, or 4, or 5, or 7, or any other number, will be divisible by those numbers. For example; 60 being equal to 2x 2 x 3 x 5, it is evident that 60 is divisible by 2, and by 3, and

704 99800i (999 81 189) 1880 1701 1989) 17901 17901 O. 325. But when there is a remainder after all the figures have been used, it is a proof that the number proposed is not a square; and consequently, that its root cannot be 104 ELEMENTS SECT. II. assigned. In such cases, the radical sign, which we before employed, is made use of. This is written before the quantity, and the quantity itself is placed between parentheses, or under a line; thus, the square root of a2 + b2 is represented

to the power which is required; in the following terms, the powers of a diminish continually by unity, and the powers of b increase in the same proportion; so that the sum of the exponents of a and of b is always the same, and always equal to the exponent of the power required; and, lastly we find the term b by itself raised to the same power. If therefore the tenth power of a + h were required, we are certain that the terms, without their coefficients, would succeed each other in the following

n2, as we have already seen, Article 422; but it is unnecessary to extend our consideration of square numbers any farther, having already treated of them at length. 432. If now we call the difference 3, and take the sums in the same manner as before, we obtain numbers which are called pentagons, or pentagonal numbers, though they cannot be so well represented by points.* * It is not, however, that we are unable to represent, by points, polygons of any number of sides; but the rule which I am