The gnomons for pentagonal numbers are the terms after 1 in the arithmetical progression 1, 4, 7, 10 ... (with 3, or 5-2, as the common difference) and so on; the common difference of the successive gnomons for an _a_-gonal number is _a-2_. From the series of gnomons for squares we easily deduce a formula for finding square numbers which are the sum of two squares. For, the gnomon _2n+1_ being the difference between the successive squares _n²_ and _(n+1)²_, we have only to make _2n+1_ a square. Suppose that _2n+1=m²_; therefore _n=½(m²-1)_, and _{½(m²-1)}²+m²={½(m²+1)}²_, where _m_ is any odd number. This is the formula actually attributed to Pythagoras. Pythagoras is said to have discovered the theory of proportionals or proportion. This was a numerical theory and therefore was applicable to commensurable magnitudes only; it was no doubt somewhat on the lines of Euclid, Book VII. Connected with the theory of proportion was that of _means_, and Pythagoras was acquainted with three of these, the arithmetic, geometric, and sub-contrary (afterwards called harmonic). In particular Pythagoras is said to have introduced from Babylon into Greece the 'most perfect' proportion, namely: _a:(a+b)/2=2ab/(a+b):b_, where the second and third terms are respectively the arithmetic and harmonic mean between _a_ and _b_. A particular case is 12:9=8:6. This bears upon what was probably Pythagoras's greatest discovery, namely that the musical intervals correspond to certain arithmetical ratios between lengths of string at the same tension, the octave corresponding to the ratio 2:1, the fifth to 3:2 and the fourth to 4:3. These ratios being the same as those of 12 to 6, 8, 9 respectively, we can understand how the third term, 8, in the above proportion came to be called the 'harmonic' mean between 12 and 6. The Pythagorean arithmetic as a whole, with the developments made after the time of Pythagoras himself, is mainly known to us through Nicomachus's _Introductio arithmetica_, Iamblichus's commentary on the same, and Theon of Smyrna's work _Expositio rerum mathematicarum ad legendum Platonem utilium_. The things in these books most deserving of notice are the following. First, there is the description of a 'perfect' number (a number which is equal to the sum of all its parts, i.e. all its integral divisors including 1 but excluding the number itself), with a statement of the property that all such numbers end in 6 or 8. Four such numbers, namely 6, 28, 496, 8128, were known to Nicomachus. The law of formation for such numbers is first found in Eucl. IX. 36 proving that, if the sum (S_{n}) of _n_ terms of the series 1, 2, 2², 2³ ... is prime, then S_{n}.2^{n-1} is a perfect number.