A cast of Dase's head, made when he was 15 years old.Johann Dase
A cast of Dase's head, made when he was 15 years old.
Full name: Johann Martin Zacharias Dase (Dahse)
Born 23rd June 1824 in Hamburg, Germany. Died 11th September 1861.
Excelled at written arithmetic from an early age, devouring "every book in Hamburg" on the subject. His mental arithmetic developed later and proceeded as it would if written down.
Started giving public demonstrations of his calculating ability at the age of 15. During these he would write on a blackboard a problem given him by a member of the audience and then write a provisional solution beneath it. After checking his answer, he would announce it as fact. He was not disturbed by noise and could even hold conversation with spectators while calculating.
Attributed his calculating ability to an early preoccupation with dominoes.
Possessed very vivid mental imagery and excellent spatial awareness.
Claimed to be never fatigued by calculating, which he could continue all day long.
Had epilepsy, suffering attacks since early childhood.
At the age of 16, over the course of two months, Dase computed - though not mentally as is sometimes assumed - the first 205 digits of pi using the following formula, discovered by Austrian mathematician Lutz von Strassnitzky,
pi/4 = arctan(1/2) + arctan(1/5) + arctan(1/8)
combined with the Maclaurin series for arctan(X),
arctan(X) = X - X^3 / 3 + X^5 / 5 - X^7 / 7 + X^9 / 9 - ...
Strassnitzky's formula is better suited to human calculation than the other Machin-like formulae that were known at the time because it can be re-expressed as,
pi/4 = arctan(1/2) + arctan(2/10) + arctan([1/2]3)
so that all the terms in the Maclaurin series involve powers of 2 only.
The first few digits of pi are calculated below using the Strassnitzky/Maclaurin method, as used by Dase.
First, some powers of 2:
| n | 2^n | 2^(-n) |
| 0 | 1 | 1 |
| 1 | 2 | 0.5 |
| 2 | 4 | 0.25 |
| 3 | 8 | 0.125 |
| 4 | 16 | 0.0625 |
| 5 | 32 | 0.03125 |
| 6 | 64 | 0.015625 |
| 7 | 128 | 0.0078125 |
| 8 | 256 | 0.00390625 |
| 9 | 512 | 0.001953125 |
| 10 | 1024 | 0.0009765625 |
arctan(1/2) = 1/2 - (1/2)^3 / 3 + (1/2)^5 / 5 - (1/2)^7 / 7 + (1/2)^9 / 9 - ...
Truncating to the first five terms shown,
arctan(1/2) = 0.5 - 0.125/3 + 0.03125/5 - 0.0078125/7 + 0.001953125/9
= 0.4637 (rounding to four significant figures)
arctan(2/10) = 2/10 - (2/10)^3 / 3 + (2/10)^5 / 5 - (2/10)^7 / 7 + ...
This time the first four terms will suffice, so that
arctan(2/10) = 0.2 - 0.008/3 + 0.00032/5 - 0.0000128/7
= 0.1974 (to four figures)
arctan([1/2]3) = (1/2)^3 - (1/2)^9 / 3 + ...
This time the first two terms will suffice, the remaining terms being very small. So,
arctan([1/2]3) = 0.125 - 0.001953125/3
= 0.1243 (to four figures)
Hence, pi/4 = 0.4637 + 0.1974 + 0.1243
= 0.7854
So pi = 3.1416
This compares favourably with the true value of pi, 3.14159265...
Dase subsequently calculated the natural logarithms of the first million integers to seven places, a task of three years.
Dase appears several times in the correspondence between German mathematicians Heinrich Christian Schumacher and Carl Friedrich Gauss, the latter seemingly unimpressed by Dase's feats. Gauss did, however, recommend Dase to a publisher of mathematical tables. As a result, Dase set about extending the published tables of factors of the integers to ten million and, by the time of his death, had completed a substantial part of it.
Calculations performed by Zacharias Dase
354783293 x 5423957 = 1924329325550401 in 1.5 minutes
684028396281753 / 6541325 = 104570312 with remainder 138353 in 2.5 minutes
423339075240048565 / 708346795 = 597643807 in 5 minutes
19th root of 7093585369945932256195429028464404423 = 87 in 3 minutes
In performing the above calculations, Dase had the question in view (written down on a blackboard or a sheet of paper) and, in the first three problems, would begin writing the answer before the calculation had been completed.
In one of his performances, Dase extracted the 52nd root from a 97-digit number. This appears to have been misreported in some texts as the much more impressive feat of extracting the square root from a 100-digit number in 52 minutes.
Dase was also a 'lightning counter' and could at a glance give the number of spots on 10-20 dominoes laid out in a line. Similarly he could scan a bookshelf and give the number of books it contained.
Dase sometimes attempted extremely large multiplications. He could multiply a pair of 20-digit numbers in 6-8 minutes, and a pair of 60-digit numbers in 3 hours. His largest was the multiplication of two 300-digit numbers, spread over five days.
References/Links
1. The Anthropological Review 1863, Vol. 1, No. 3, p492
2. The Phrenological Journal and Magazine of Moral Science 1841, Vol. XIV, No. 67, p153