 A gnomon was originally a sort of set-square that could be stood on its edge and was used to measure the lengths of shadows — present-day sundials have a ‘gnomon’ on the top though the shape is more complicated (Note 1). Thales is supposed to have used a gnomon to estimate the height of the Great Pyramid by employing properties of similar triangles and it was data amassed by similar methods that enabled Eratosthenes to estimate the circumference of the Earth by comparing noonday shadows cast at different localities (Note 2). The gnomon thus provided a precious link between three different disciplines : geometry, astronomy and, as we shall see, arithmetic : it was perhaps the first precision instrument of physical science.

Sets of gnomons put together — or drawings of them — became a surprisingly useful early calculator and enabled the early Greek mathematicians to investigate spatial properties of numbers.

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Each coloured inverted L shape border in the above represents an odd number with the unit in the top left hand corner. The early Greek mathematicians deduced the important property that Any square number can be represented by successive odd numbers commencing with unity. As we would put it:  n2  =  1 +  3 + 5 + ……(2n + 1)   For example, 41 +  3 + 5 + 7   And this can be extended to the observation that Every difference between two squares can be represented by a sum of successive odd numbers. Thus, 5–  2=  5 + 7 + 9    (It was in fact this relation which struck me as being quite astounding that instigated my interest in mathematics which up to then I had despised.)
But much more can be got out of the simple diagram above. The Egyptians were certainly aware of certain cases of the the property forever associated with Pythagoras, namely that The Square on the hypotenuse is equal to the sum of the squares on the other two sides of a right-angled triangle since they used stretched ropes with lengths in the ratio 3, 4, 5 to lay out an accurate square corner. However, they may not have realized that this property applied to all right angled triangles. The question provided a fruitful contact between geometry, the science of shape, and arithmetic, the science of exact quantity and the gnomon most likely played an important role here. Greek mathematicians were interested in sets of numbers that were ‘Pythagorean triples’, i.e. numbers a, b, c where   a2 = b +  c2 .
Now,  adding on a gnomon “preserves the square form”  and, more significantly for the present discussion, that the difference of two successive squares is an odd  number.

        +               =           
                                       
                                       
                          

Some sharp sighted mathematician, perhaps Pythagoras himself or one of his disciples, realized that if the gnomon is itself a square we have a Pythagorean triple. (This follows from the observation that adding on the relevant gnomon leads from one square to the next.) So, if we select an odd square number, we can make it the gnomon and thus give an example of a Pythagorean triple. The first odd square is

          (our 9) and to make it into a gnomon we stretch it out into three parts , two equal and the third a unit
                   This provides the outer framework for the two squares :

              The inner square has side 4 and the outer side 5. This gives the simplest
              Pythagorean triple  52 = 42 + 32 .
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   

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However, any odd square will do and, since 49 = 72 we can construct a Pythagorean triple involving it. The gnomon is 24 + 1 + 24  giving 24 for the side of the larger square and 23 for the smaller one. This gives the triple  242 = 23  + 72 .  The series of Pythagorean triples using this procedure is endless : it suffices to find an odd square number.
This procedure can be generalised if we allow a gnomon to be made up of more than one row + column. For example, we might allow the gnomon to have three rows + three columns.

………………       

………………       

………………       

…          
…          
…          

If r is the side of the inner square, the outer square is (r + 3) instead of (r + 1) and the little square in the bottom right hand corner will be 3 x 3 = 9 instead of 1. The gnomon is made up of two rectangles (r x 3)  the little square giving  (6r + 9) = 3(r + 3)   We must thus find a square which is equal to the gnomon or solve  3(2r + 3) =  m2   for some m.   Since  m2  is divisible by 3 this makes m a multiple of 3 as well. We must also have m large enough so that r is at least 1.  The first possibility is m = 21 = 7 x 3  so that  3(2r + 3) =  212    This makes r = 72  This will be the side of the inner square while the outer one will be 72 + 3 = 75.  So, if this reasoning is correct, we should find that 752 = 722 + 21 which is indeed the case (check this). So a rather more spaced out but still unending set of Pythagorean triples can be manufactured where the difference between the sides of the squares is 3 rather than 1.  It is left to the interested reader to concoct other sets.
As a matter of fact we have reason to believe that the early Pythaogoreans knew of such sets of triples and it is plausible that they hit upon them using some such method which has its basis in the manipulation of sets of wooden gnomons and/or actual counters on actual boards. Interestingly enough, the Babylonians a thousand years earlier were aware of Pythagorean triples and seem to have had some method of concocting them (Note 3)The basic formula for all Pythagorean Triples is given in Euclid — or rather can be deduced from the argument given in Euclid which is mainly verbal since the Greeks did not have our algebraic notation. I shall not give it here — you can get it from Wikipedia or some other site by the click of a key — as I am more interested in seeing how such formulae arose in the first place and indeed in (re)discovering them for myself, something that I encourage you to do as well. In the next post I will examine the slightly more complicated problem of an isosceles right-angled triangle, i.e. one where the two smaller sides are equal. This provoked a trmendous rumpus at the time because it raised the issue of so-called ‘incommensurables”. If the short side is set at unity, the square on the hypotenuse comes out at  of  12 + 12 = 2 so the side itself is the square root of 2. But was there such a number? In the ideal world of Platonic forms (not yet elaborated) certainly there was, but in the Pythagorean world of number where number meant ratio between two integers there was apparently no such quantity and thus no such length.     SH   1/1/13

Note 1   “The word gnomon ….literally means an “indicator”, or “one who knows”. Specifically, it was the name of the sundial first brought to Greece from Babylonia by Anaximander, who was probably one of Pythagoras’s teachers. The word also serves to indicate any vertical object like an obelisk which serves to indicate time by means of a shadow.”  Valens, The Number of Things

gnomon : Stationary arm that projects a shadow on a sundial” (Collins)

Note 2  Actually, it seems that Eratosthenes’ data did not depend on gnomons as such but it did rely on the measurement of noonday shadows. Reputedly, Eratosthenes based his remarkably good estimate of the circumference of the Earth on the information. presumably relayed by a traveller, that the sun at noon at midsummer’s day at Syene was directly overhead because it was reflected at the bottom of a deep well. Eratosthenes, as Librarian at Alexandria some 500 miles or so due north of Syene, knew that the shadow of a pillar cast by the sun at the same moment in time was a little more than 7 and a half degrees off the vertical . This enabled him to come up with an estimate of 4,000 miles in our reckoning for the Earth’s radius using geometrical techniques. Current estimates put the Earth’s mean radius at about 3960 miles.

Note 3     “The Babylonian tablet called Plimpton 322 (dating from between 1900 and 1600 B.C.) shows that the Babylonians had studied this problem [of Pythagorean triples] much earlier. The tablet merely lists a series of Pythagorean triples but the order in which they are listed makes us believe that the Babylonians had a general and systematic solution for the problem of finding Pythagorean triples.”

Bunt, Jones & Benient, The Historical Roots of Elementary Mathematics