Logarithmo-technia: sive methodus construendi logarithmos nova, accurata, & facilis; scripto antehàc communicata, anno sc. 1667. Nonis Augusti: cui nunc accedit vera quadratura hyperbolæ, & inventio summæ logarithmorum. Huic etiam jungitur Michaelis Angeli Riccii Exercitatio geometrica de maximis & minimis, hîc ob argumenti præstantiam & exemplarium raritatem recusa

The Mercator Series. First edition of the book that put the logarithmic series into print - the thirty-four pages in which Mercator divided 1 by 1 + a, wrote down the quotient 1 ? a + aa ? a3 + a4 and so on without end, summed it term by term to square the hyperbola, and recognised the area he had found as a logarithm - complete with the London reprint of Michelangelo Ricci's tract on maxima and minima that the publisher appended to it, and with the folding plate of Ricci's figures. John Wallis came upon a copy on 7 July 1668 and had a report of it before the Royal Society nine days later. James Gregory answered it from the same printing house within months. Isaac Newton, who had held the same series privately since 1665, let Barrow send his De analysi to Collins a year later because of it. Leibniz bought it in London in the winter of 1672-73 and annotated it, in a copy that survives at Hannover. The present copy is entirely uncut and, at 206 by 156 mm, among the largest recorded at auction in the last decade. The problem it solved had been open for twenty years. Gregory of St Vincent had shown in the Opus geometricum of 1647 that the area between a rectangular hyperbola and its asymptote cut off between abscissae in a fixed ratio is always the same, whatever the abscissae; his pupil Alfonso Antonio de Sarasa saw two years later that this made the area a logarithm of the abscissa, the area from 1 to xy being the sum of the areas from 1 to x and from 1 to y. If the hyperbolic area could be computed, logarithms could be computed from it; but neither Jesuit could compute it, and Pietro Mengoli at Bologna, who in 1659 built an arithmetic of logarithmic ratios on the model of Euclid's fifth book and bounded the logarithm of a ratio between sums of the harmonic series, was working from the other side, on the number rather than the curve. Wallis's Arithmetica infinitorum of 1656 had meanwhile given the rule that the sum of the nth powers of the innumerable parts of a line is the next power divided by n + 1 - the quadrature, in effect, of every curve y = xn - and the one curve on which the rule broke down was the hyperbola y = 1/x, where n + 1 is zero. Mercator's answer was to move the axis so that the ordinate became 1/(1 + x), to turn that ordinate into a power series by division, and to apply Wallis's rule to each power in turn. He approaches the hyperbola only after twenty-seven pages of arithmetic, and the arithmetic is the book's first argument. It opens with a definition: a logarithm is nothing but the number of small equal ratios, ratiunculae, contained in the ratio that a number bears to unity. Mercator fixes the size of the ratiuncula by decree - ten million of them make up the ratio of 10 to 1 - so that his logarithms are Briggs's common logarithms by construction, and then shows how to measure any ratio in that unit without extracting a single root: he squares 1.005 and 0.995 repeatedly until the powers straddle 10, interpolates by the rule of false position, and reads off 21,659.7 and 21,769.3 ratiunculae for the two base ratios. Their sum, 43,429, measures the ratio 100.5 to 99.5, and on it the second part turns. Propositions I to XIII build a table of logarithms by continued addition alone, using the differences and second differences of the logarithms of numbers in arithmetical progression, and reduce a seven-place table from 10,000 to 100,000 to the direct computation of twenty-four of its ninety chiliads. 43,429 is ten million times the common logarithm of 100.5/99.5, and because the natural logarithm of that ratio is 0.01 to within a part in a million, Mercator's constant is the modulus 0.4342945 that converts natural into common logarithms, correct to five figures and reached by a man who had no notion of e. The title page announces three things - a method of constructing logarithms, the true quadrature of the hyperbola, and the finding of the sum of logarithms - and the text delivers exactly three: the first thirt.…