Porro, Francesco. "Notize Intorno alla Vita de agli Scritti di Guiseppe Zecchini Leonelli, Matematico Cremonese." Published in Rome, Typografia delle Scienze Matematiche e Fisiche,1886. Offprint: * Bulletino di Bibliografia e di Storia delle Scienze Matematiche e Fisiche,* tomo XVIII, Nov-Dec, 1885. 10x8 inches, 22pp. In the original wrappers. Removed at some distant point from a larger bound volume. Nice condition. $95

I found the following biographical information on Leonelli in *The Academy*, 20 April 1878, page 348:

"In the year XI. of the old French Republic (1802-3) Zecchini Leonelli, born at Cremona in 1776, two years after Flower's death, and hence at twenty-six years of age, printed (he himself «ays "public," but M. Hoiiel says, "Leonelli n'avait pas mis son ouvrage dans le commerce,") his now celebrated *Supplement Logarithmique, *one single copy of which is known to exist, in the library of the city of Bordeaux, bearing the inscription: "Ex dono Autoris." Leonelli became subsequently physical assistant of Prof. Mossotti at Corfu, where he died in 1847. His book, -which is in every respect remarkable, was reprinted in 1875 with a biographical notice, by M. Hoiiel, professor of pure mathematics at Bordeaux, for 4 fr., and I recommend every one who is interested in the subject to order a copy of the •publisher, Gauthier-Villars, in Paris. In the original memoir, which Leonelli submitted to the French Academy,he rediscovered Briggs's method. In one of the paragraphs added to his own impression of that

Je ne l'ai connue qu'au moment oii je cherchais des moyons pour abreger la division, travail unique et assez bien compense de cette decomposition [de Briggs]. Cost le cil. Eveque, membre de l'lnstitut, qui a bien voulu me Conner l'opuscule de Flower, qu'il a acquis en Angleterre. La decomposition, dont Flower so sort, est en quelque sorte differente [de celle de Briggs], et oppose, en certains cas, quelque petit obstacle a la generality de la reglo; mais elle est plus courte que celle que nous avons exposee " (p. 16). In Delambre's report (p. 61) the title of Flower's book is cited from M. Maseres's reprint of Hutton's preface, together with the false appreciation there given.

Leonelli gives both Briggs's method, explained by himself (for Briggs exemplifies, but can scarcely be said to explain), and re-arranges Flower's table, of which he gives only twenty places, adding a table for natural logarithms, also to twenty places. Moreover, he gives a table to fifteen places proceeding by two figures, instead of one, that is, instead of 1-09, l-08, &c, he has 1-099, 1-098, 1-097, &c, up to the insertion of seven zeroes between 1 and the significant figures. The same work also contains a " ThtSorie des logarithmes additionels et de"ductifs," which gave rise to Gauss's celebrated logarithms of addition and subtraction.

Although Leonelli's work became practically unknown, a German translation (very badly printed, and containing numerous changes) was made by Leonhardi in 1806 (the Royal Society has a copy, and it is described in *Report *of Brit. Assoc., 1873, p. 76), from which Gauss obtained the hint for his own logarithms (see Gauss's *Werke, *vol. iii., p. 244, cited by Hoiiel), and from which probably Schron, in his *Interpolations-Tafel, *1861 (translated by Hoiiel, 1873), gave Flower's table to 16 places, both for natural and tabular logarithms with an explanation; but, singularly enough, without mentioning the name of either Flower or Leonelli, an omission which, more strangely still, is not supplied in Hoiiel's translation. But in Hoiiel's *Itecueil des Formules et de Tables Nume'riques *(second edition, 1868), Table V., he gives Leonelli's table, proceeding by two figures, abridged to 15 places, and in the introduction (p. xiii.) mentions Leonelli, but not Flower. However, in his popular tables to 5 places he mentions Flower's name and date (edition 1877, p. xxix., note); hut not the title of his book, and gives Flower's *Radix *as Table V. to 20 places only, following Leonelli. Don V. Vazquez Queipo, of whom I spoke in my former letter as the author from whom my knowledge of Flower was first obtained, re-arranged his table and description, as he has informed me himself, from Hoiiel, but added the 21st place from F6dor Thoman *{Tables de Logarithmes a *27 *Dicimales pour les Calculs de Precision, *Paris, 1867), who, in Table IV., gives Flower's *Radix, *from 1-09 through 13 classes, to 27 places of decimals, but does not mention the name of Flower or of any previous writer. None of these writers but Leonelli had seen the original book.

In England, as already mentioned, Mr. Orchard rediscovered Flower's method, without knowing of Leonelli, and Messrs. Weddle and Hearn discovered another method, essentially of the same character, but avoiding one of the difficulties in Flower's plan. They left a difficulty in starting, which Thoman endeavours to surmount by using approximate reciprocals, but I believe that my "preparation" (part of my own method not yet published) is simpler. Mr. Gray's tables form very convenient accessible means for finding logarithms, but are much more extensive, and his method is of a totally different character, employing, like Briggs's, a continually augmenting divisor, obtained by a process which is also substantially the same though different in appearance. But Briggs obtains his resolution in single digits, in order to suit his own table of one page, equivalent to Flower's *Radix, *whereas Mr. Gray resolves into sets of three digits for which his own tables of 41 pages are constructed. The essentially different methods of finding logarithms are then, Briggs'i by division with an augmenting divisor, Flower's by resolving by addition, and Weddle's by resolvingby subtraction. Gray's belongs to the first, Orchard'j is the second, Thoman's is a variety of Weddle's. To these we may add Namur's *(Tables de logs, rithmes & *12 *Dicimales, *Bruxelles, 1877) depending upon the properties of the modulus, in which the calculations are short, but the method rather troublesome to use. My own method depends upon another property altogether, used in an entirely different way by Koralek *(Nouvclle MMUk pour calculer rapidement les Logarithm da Kombres, *Paris, 1851), but sometimes employing the same radix as Flower, and sometimes in part the negative radix (as it may be called) of Weddle, Observers like Hutton, Horsley, and Raper, who confused Flower's method with Briggs's on account of the similarity of the table, and snubbed him accordingly, might confuse mine with both, or with Koralek's. The confusion in Robert *Flower's *case has been extremely important, bnt I hope that no doubt will hereafter rest on the originality, independence, and value of the method invented by the poor bachelor writing-master of Biahon's Stortford, who published his *Radix *in the sixtieth year of his laborious and obscure life. Honour be to his memory! Alexandeb J. Ems,

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