Now, I find that all the contemporary authorities agree that the Saxons did actually fight in this solid order.  For example, in the “Carmen de Bello Hastingensi,” a poem attributed to Guy, Bishop of Amiens, living at the time of the battle, we are told that “the Saxons stood fixed in a dense mass,” and Henry of Huntingdon records that “they were like unto a castle, impenetrable to the Normans;” while Robert Wace, a century after, tells us the same thing.  So in this respect my newly-discovered chronicle may not be greatly in error.  But I have reason to believe that there is something wrong with the actual figures.  Let the reader see what he can make of them.

The number of men would be sixty-one times a square number; but when Harold himself joined in the fray they were then able to form one large square.  What is the smallest possible number of men there could have been?

In order to make clear to the reader the simplicity of the question, I will give the lowest solutions in the case of 60 and 62, the numbers immediately preceding and following 61.  They are 60 x 4 squared + 1 = 31 squared, and 62 x 8 squared + 1 = 63 squared.  That is, 60 squares of 16 men each would be 960 men, and when Harold joined them they would be 961 in number, and so form a square with 31 men on every side.  Similarly in the case of the figures I have given for 62.  Now, find the lowest answer for 61.

130.—­THE SCULPTOR’S PROBLEM.

An ancient sculptor was commissioned to supply two statues, each on a cubical pedestal.  It is with these pedestals that we are concerned.  They were of unequal sizes, as will be seen in the illustration, and when the time arrived for payment a dispute arose as to whether the agreement was based on lineal or cubical measurement.  But as soon as they came to measure the two pedestals the matter was at once settled, because, curiously enough, the number of lineal feet was exactly the same as the number of cubical feet.  The puzzle is to find the dimensions for two pedestals having this peculiarity, in the smallest possible figures.  You see, if the two pedestals, for example, measure respectively 3 ft. and 1 ft. on every side, then the lineal measurement would be 4 ft. and the cubical contents 28 ft., which are not the same, so these measurements will not do.

[Illustration]

131.—­THE SPANISH MISER.

There once lived in a small town in New Castile a noted miser named Don Manuel Rodriguez.  His love of money was only equalled by a strong passion for arithmetical problems.  These puzzles usually dealt in some way or other with his accumulated treasure, and were propounded by him solely in order that he might have the pleasure of solving them himself.  Unfortunately very few of them have survived, and when travelling through Spain, collecting material for a proposed work on “The Spanish Onion as a Cause of National Decadence,” I only discovered a very few.  One of these concerns the three boxes that appear in the accompanying authentic portrait.