gave the value of the circle, or the circumference to within a few millionths. Adrianus Romanus carried the approximation to 17 figures. But all that is far below what was done by Ludolph Van Ceulen, and which he published in his book de Circulo et adscriptis, of which Snellius published a Latin translation, at Leyden, in 1619. Ceulen, assisted by Petrus Cornelius, found with inconceivable labor a ratio of 32 decimals; see V. II, p. 6. Snellius found the means of shortening this calculation by some very ingenious theorems, and if he did not excel Van Ceulen he verified his result, which he put beyond attack. His discoveries on this subject are found in the book entitled Willebrordi Snelli Cyclometricus de Circuli dimensione, etc. Descartes also found a geometrical construction which, carried to infinity, would give the circular circumference, and from which he could easily deduce an expression in the form of a series. (See his Opera posthuma.) Gregoire de Saint-Vincent is one of those who are most distinguished in this field; true, he claimed incorrectly to have found the quadrature of the circle and of the hyperbola, but the failure in this respect was preceded by so great a number of beautiful geometrical. discoveries, deduced with much elegance according to the method of the ancients, that it would have been unjust to have placed him among the paralogists we have mentioned. He announced, in 1647, his discoveries in a book entitled: Opus Geometricum quadraturae Circuli et Sectionem Coni libris, X, Comprehensum. All the beautiful things contained in this book are admired; only the conclusion is impugned. Gregoire de Saint-Vincent lost himself in the maze of his proofs which he calls proportionalities, and which he introduces in his speculations. It was the subject of quite a lively quarrel between his disciples on the one hand, and his adversaries on the other, Huygens, Mersenne, and Leotand, from 1652 to 1664. If that skillful geometrician had not been mistaken, it would only have followed from his investigations that the quadrature of the circle depends upon logarithms, and consequently on that of the hyperbola. That would still be a handsome discovery, but it did not even have that advantage. This furnished Huygens the occasion of divers investigations on this subject, He demonstrated several new and curious. theorems on the quadrature of the circle: Theoremata de quadratura hyper, ellipsis et Circuli, 1651; De Circuli Magnitude inventa, 1654. He gave several methods of approaching his quadrature much shorter than the usual way. He demonstrated a theorem which Snellius had taken for granted. There are also many very simple geometrical constructions which give lines singularly near any given area. If, for example, the arc is 60° the error is scarcely both. 6000 James Gregory distinguished himself in this controversy, and whatever may be our opinion of his demonstration of the impossibility of the definite quadrature of the circle, he can not be denied the authorship of many curious theorems on the relation of the circle to the inscribed and circumscribed polygons, and their relation to each other. By means of these theorems he gives with infinitely less trouble than by the usual calculations, and even those of Snellius the measure of the circle and of the hyperbola (and consequently the construction of the logarithms) to more than twenty decimal places. Following the example of Huygens, he also gave constructions of straight lines equal to arcs of the circle, and whose error is still less. For example, let the chord of the arc of a circle be a, the sum of the two equal inscribed chords equal to b, then make this proportion: A+B: B::B: C; if you take the following quantity, it does not exceed the 350th for a semicircle, and for 120° it would be less than oth; finally the error for a quarter of a circle will not be goooooth. 1 8c+8 B-A The discoveries of Wallace, found in his Arithmetica infinitum, published in 1655, lead him to a singular expression of the relation of the circle to the square of its diameter; it is a fraction in this 3×3×5×5×7×79×9×11×11 form, 2X4X4X6X6X8X8X10X10X12 etc. 2.4 This fraction, carried to infinity, expresses exactly the above relation, Arithmet. infinit., prop. 191; but if we confine ourselves, as is necessary, to a finite number of terms we have alternately a relation greater or less than the true one according as we take an odd or an even number of terms of the numerator and denominator. Thus & gives too great a relation, and 3x3 give too small a relation. The fraction 3:42:57 is too small, and 3:47:38 too great. But to bring the one near to the other, Wallis directs to multiply this product by the square root of a fraction formed by the unity plus unity, divided by the last figure which ends the series. Then the product, although much nearer, will be too large if the figure is the last of the numerator, and too small if the last of the denominator. The values of 3.3.5.5.7.7 • ·3.5.5.7.7.9 3.3.5.5. 3:8:5: 8 1/1 +3; 3:8:5:5; √1+7; 3:4:5:5:7:31 1+7; 8:8:5:5:7:3 √1+3; 3.3.5 2.4.4.6 2.4.4.6) etc., alternately too small or too great, will fall within the known limits. Here is another expression of the relation of the circle to the square of the diameter, found by Lord Brunker about the same time. The circle being one, the square is expressed by the following fraction carried to infinity: 1 1 + 2 + 9 2+25 2+-49 2+etc. It will be seen that this fraction is such that the denominator is an integer plus a fraction, whose denominator is 2 plus the square of one of the odd numbers 1, 3, 5, 7, etc.; when brought to an end the limits obtained are alternately in excess or too small. Such was the knowledge of geometricians on this famous problem when Newton and Leibnitz appeared on the arena. In 1682, Leibnitz gave out in his Actes de Leipsig what he had discovered as early as 1673, namely, that the square of the diameter being one, the area of the circle is expressed by the infinite series 1-+-+1, etc. It follows from his discovery about the same time that the radius of the circle being unity and the tangent of an arc t, this arc itself is t} t3 +} 15—‡ t7, etc. If then the arc is 45°, the tangent t is equal to the radius or one. Thus the arc of 45° is 1-1++etc.; multiplying by 4 we shall have the semi-circumference, which multiplied by the radius will give the area of the circle equal 4—4—4, etc.; the square of the diam. eter being 4. Thus the square of the diameter being made unity, the area of the circle will be 1-+-, etc. to infinity. 15 6 3 99 The area can also be expressed by 3+2+3+135, etc., viz.: by adding together the two first terms, and the next two by two, or else in this way, 1-2-32, etc., where it is easy to see that the denominators are successively in the first the squares of 2, 6, 10, etc., diminished by unity, and in the second the squares of 4, 8, 12, etc., similarly reduced. But it must be conceded that these different series do not converge rapidly enough to derive from them a value sufficiently accurate without the addition of a prodigious number of terms; but Euler found a remedy. The discoveries made by Newton, even before Leibnitz, had also placed him in possession of various methods of expressing the circumference and the area of the circle, as also of segments by infinite series. Nothing is better known to-day by all those who have any knowledge, even elementary, of the new calculations; but among these series, those that have been used most successively for that purpose are the following: Let C, A, F, be a quarter of a circle, whose radius A Q is unity and QB an abscissa, x, we find the area of the segment Q, B, A, F, represented by this series, x-x3 — 2.1.7 x 5 — 1.1:87 x7, etc. 1 48 1 1 2.3 3. X 4 1 Now, supposing B Q or x equal to 1, this series is reduced to the following: 1280 000000 00000000, etc. Calculating therefore in decimal fractions each of these terms, and taking the sum of the negatives from the first, which is positive, we obtain the value of the segment Q, B, A, F, from which it will be necessary to take the value of the triangle A, B, Q, calculated in decimals, and there will remain the sector F, A, Q, which is the third of the quarter of the circle, or the 12th part of the entire circle. Thus, multiplying this value by 12, we shall obtain that of the circle for a diameter equal to 2; and finally dividing this last one by 4, we shall have that of the circle to the diameter. The first ten terms above given, calculating in this manner, give the near ratio of the diameter to the circumference of 1 to 3.141. The tangent of a small arc as that of 30° might also be employed for the same purpose; for this tangent is, or; besides, if the radius, unity, and the tangent t, the arc is, as already shown, t-1 13— 13. t5t7 19 t 11 Thus t being made y, this series is reduced to 5 7 9 11 9 s 18 3.3√3+5.513 7.27/3+5.8113 11.248/8+18.72918 15.218718 +17.65617319.198837/5, etc., whose progression can easily be seen. 3.3 5.9 7.27 This series, by transferring the radical to the numerator, becomes } Ꮴ −1}+1}−√ §, etc., which can be expressed thus: √} (1—3.3+5.5 -77); thus must be taken in as many decimals as are to be used in the approximation or a little more, to be more certain of the last figures; then this value must be divided successively by 3.3 or 9 by 5.9 or 45, by 7.27 or 189, by 9.81 or 7.29, etc.; this will give the approximate value of each term in as many decimals as there are in the value of 13. Add all the positive terms together, and from the sum subtract that of the negatives. This will give very nearly the arc of 30°, which being multiplied by 12, will be the value of the circumference with the diameter 2, consequently the half will be that of the circumference with the diameter 1. It is by this means and others similar that Samuel Sharp extended to 75 decimals the approximate ratio of Ludolph, which had only 35. Machin, towards the beginning of the present century, carried it as far as 100. Lagny, in 1719, carried it to 128, and another to 155. Thus, the diameter of the circle being 1, followed by 128 zeros, the circumference is according to Lagny, greater than 3.14159, 26535, 89793, 23846, 26433, 83279, 50 | 288, 41971, 69399, 37510, 58209, 74944, 59230, 78164, 06286, 20899, 86280, 34825, 34211, 70679, 82148, 08651, 32723, 06647, 09384, 46, and less than the same number increased by unity (added to the last figure). I have separated by a dash the 32 decimals of Ceulen. The error for a circle with a diameter 100 millions times greater than that of the sphere of the fixed stars. Supposing the parallax of the terrestrial orb to be only one second, would still be several billions of billions times less than the breadth of a hair. The 114th figure of the seven which is underlined, ought to be 8; Mr. Vega ascertained this as appears from his large tables of Logarithms, page 633, where he gives the values of the series. Baron de Zach saw, in a manuscript of the library of Ratcliffe at Oxford, the calculation carried still further, and as far as 155 figures; after 446 add 0955058, 22317, 25359, 40812, 84802. The expedient found by Euler for using the series which the arc by the tangent gives, will bring us nearer the truth and with less trouble. The expedient deserves to be inserted here. It consists in the remark made by that great geometrician that every arc is rational or commensurable with the radius (the arc of 45° for example, whose tangent is 1), can be divided into 2 arcs whose tangents |