The History of Mathematics: A Brief Course

162 7. ANCIENT NUMBER THEORY and "diagonal." Those labels tell us that we are dealing with dimensions of a rectangle here, and t ...

PLIMPTON 322 163 as factors. Of all right triangles, the 45-45-90 and the 30-60-90 are the two that play the most important ro ...

164 7. ANCIENT NUMBER THEORY already see why we need both ñ and q to be products of 2, 3, and 5. This problem amounts to the qua ...

ANCIENT GREEK NUMBER THEORY 165 of which naturally have 1 as a common divisor. Euclid, on the other hand, does not confine it ...

166 7. ANCIENT NUMBER THEORY into exactly one of these three classes. Such is not the case, however. The property of primeness i ...

ANCIENT GREEK NUMBER THEORY 167 When these have been discovered, 6 among the units and 28 in the tens, you must do the same to ...

168 7 ANCIENT NUMBER THEORY simplest way to denote any integer would be repeating a symbol for 1 an appropri- ate number of time ...

ANCIENT GREEK NUMBER THEORY 169 FIGURE 1. Figurate numbers. Top row: triangular numbers Tn — n(n + l)/2. Second row: square nu ...

170 7. ANCIENT NUMBER THEORY of constructing perfect numbers (Proposition 36), quoted above. No perfect num- ber has yet been fo ...

ANCIENT GREEK NUMBER THEORY 171 You may be asking why it was necessary to use a square number (16) here. Why not separate any ...

172 7. ANCIENT NUMBER THEORY To give another illustration of the same method, we consider the problem fol- lowing the one just d ...

CHINA 173 FIGURE 2. The Luo-shu. gestation is nine months. The problem is to determine the gender of the unborn child. In what ...

174 7. ANCIENT NUMBER THEORY that any number of such congruences can be solved simultaneously if the divisors are all pairwise r ...

INDIA 175 4. India The Sulva Sutras contain rules for finding Pythagorean triples of integers, such as (3,4,5), (5,12,13), (8, ...

176 7. ANCIENT NUMBER THEORY solutions of the equation Ax = 23?/+ 5. First, we carry out the Euclidean algorithm until 1 appears ...

INDIA 177 On that basis Jupiter was given (inaccurately) a sidereal period of 10 years. Again inaccurately, using a year of 36 ...

178 7. ANCIENT NUMBER THEORY Although it is trivial to verify that this rule is correct using modern algebraic notation, one wou ...

THE MUSLIMS 179 more than one unknown. But Bhaskara also asks harder questions. For example (Colebrooke, 1817, p. 202): Find t ...

180 7. ANCIENT NUMBER THEORY profound insight into the divisibility properties of numbers. It is very difficult to imagine how h ...

MEDIEVAL EUROPE 181 round, and the first child will start the following round as number 3. The problem is to see which child w ...