DECIMAL FRACTIONS 319 The dots appearing here are simply the punctuation marks written after (sometimes also before) a number which appears in the running text of most medieval manuscripts and many early printed books on mathematics. For example, Clavius 1 wrote in 160G: “Deinde quia minor est 47. quam 35. erit per propos .8. minutarium libri 9. Euclid, minor proportio 4. ad 7. quam 3. ad 5.” Pitiscus makes extensive use of decimal fractions. In the first five books of his Trigonometria the decimal fractions are not preceded by integral values. The fractional numerals are preceded by a zero; thus on page 44 he writes 02679492 (our 0.2679492) and finds its square root which he writes 05176381 (our 0.5176381). Given an arc and its chord, he finds (p. 54) the chord of one-third that arc. This leads to the equation (in modern symbols) 3x — x3 = .5176381, the radius being unity. In the solution of this equation by approximation he obtains successively 01, 017, 0174 .... and finally 01743114. In computing, he squares and cubes each of these numbers. Of 017, the square is given as 00289, the cube as 0004913. This proves that Pitiscus understood operations with decimals. In squaring 017 ap¬ pears the following: “ 001.7 2 7 1 89 002 89.4 ” What rôle do these dots play? If we put a = r V, & = T -Jy, then (o+ 5) 2 = a 2 +(2a+5)6; 001 = a 2 , 027 = (2a+6), 00189= (2a+6)6, 00289= (a+6). 2 The dot in 001.7 serves simply as a separator be¬ tween the 001 and the digit 7, found in the second step of the approxi¬ mation. Similarly, in 00289.4, the dot separates 00289 and the digit 4, found in the third step of the approximation. It is clear that the dots used by Pitiscus in the foreging approximation are not decimal points.
The part of Pitiscus’ Trigonometria (1612) which bears the title “Problematvm variorvm .... libri vndecim” begins a new pagina¬ tion. Decimal fractions are used extensively, but integral parts appear and a vertical bar is used as decimal separatrix, as (p. 12) where he says, “pro .... 13|00024. assumo 13. fractione scilicet 24100000 neglecta.” (“For 13.00024 I assume 13, the fraction, namely, 24100,000 being neglected.”) Here again he displays his understanding of decimals, and he uses the dot for other purposes than a decimal separatrix. The writer has carefully examined every appearance of 1 Christophori Clavius .... Geometria praclica (Mogvntiae, 1606), p. 343.
