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Johann Martin Zacharias - The mental calculator in a class by himself



Type of Spiritual Experience


A description of the experience

The American Journal of Psychology XVIII April 1907 – Mathematical Prodigies – Frank D Mitchell

By a "mathematical prodigy" we shall mean a person who shows unusual ability in mental arithmetic or mental algebra, especially when this ability develops at an early age, and without external aids or special tuition. We shall use the word "calculator" in the sense of "mental calculator," as a synonym for "mathematical prodigy," and shall usually mean by "calculation" "mental calculation," unless the contrary is clearly indicated by the context. A "professional calculator" will be taken to mean a mental calculator who gives public exhibitions of his talent. "Computer," however, will be restricted to mean one who calculates on paper. All problems mentioned as solved by the mathematical prodigies will be understood to be done mentally, unless otherwise indicated.

Johann Martin Zacharias Dase (1824-1861) was born in Hamburg. Concerning his heredity we have no information. He attended school at the age of 2½ years, but attributed his powers to later practice and industry rather than to his early instruction. He seems to have been little more than a human calculating machine, able to carry on enormous calculations in his head, but nearly incapable of understanding the principles of mathematics, and of very limited ability outside his chosen field.

In this respect he resembled Buxton; but in the rapidity and extent of his calculations he was incomparably superior to Buxton, or indeed to any other calculator on record.

He multiplied together mentally two 8-figure numbers in 54 seconds, two 20-figure numbers in 6 minutes, two 40-figure numbers in 40 minutes, and two 100-figure numbers in 8¾ hours; he could extract the square root of a 60-figure number in an "incredibly short time," and the square root of a 100-figure number in 52 minutes.

All these times, with the exception of that for the 100-figure multiplication, are probably more rapid, in some cases much more rapid, than those of a good computer using paper. Buxton, it will be remembered, once succeeded in multiplying two 39-figure numbers; other calculators, however, seem to have been unable to handle multiplications much above 15 figures. But if there was any definite limit to Dase's powers, the experiments of which we have record do not show it. We shall later find reason for believing that the 100-figure multiplication was not really a sever tax upon his powers of mental arithmetic. In short, Dase's achievements so far transcend those of any other recorded calculator that he stands in a class by himself, unapproached by any of his rivals.

At the age of 15, Dase began his public exhibitions, and continued them for a number of years. He soon numbered among his friends several eminent mathematicians, however, and their influence gradually led him more and more to devote his vast powers to the service of science. Among his (non-mental) computations are included the determination of the value of pi to 200 decimal places by the formula


= tan-1


+ tan-1


+ tan-1



a labor of two months; the computation of the 7-place natural logarithms of the numbers from 1 to 1005000; and factor-tables for the 7th and 8th millions (except a small portion) and parts of the 9th and 10th millions. This last task, however, was one in which his patience and perseverance were of more value than his skill in calculation, since, by methods to which Gauss was careful to call his attention, the work was made mainly mechanical. Dase had planned to carry the table through the 10th million, but death cut short his labors. The tables were completed by another hand, and published as far as the 9th million in 1862-5.

Dase had one other notable gift, doubtless related to his calculating power: he could count objects with the greatest rapidity. With a single glance he could give the number (up to thirty or thereabouts) of peas in a handful scattered on a table; and the ease and speed with which he could count the number of sheep in a herd, of books in a case, or the like, never failed to amaze the beholder. Here, again, his powers are so far in advance of those of any other recorded person that be stands in a class by himself.

Sources:  On Dase's life and calculations see Scripture, op. cit., p. 18; Briefwechsel zwischen Gauss und Schumacher, Altona, 1861, III, p. 382; V, pp. 30, 32, 277-8, 295-8, 300-304; VI, pp. 27-8, 78, 112; Crelle's Journal (Journal f.d. reine u. angewandle Mathematik), XXVII, 1844, p. 198; Zacharias Dase, Factoren-Tafeln, Hamburg, Vol. I, 1862, Preface; Schroder, Lexikon d. hamburgischen Schriftsteller, 1851, art. Dase; Preyer, "Counting Unconsciously," Pop. Sci. Monthly, XXIX, 1886, p. 221; Brockhaus's Konversations-Lexikon, 1898, art. Dase.

The source of the experience

Ordinary person

Concepts, symbols and science items



Science Items

Activities and commonsteps