216 MATHEMATICAL MODES OF 



cessful in the geometrical analysis of the head, downwards 

 through the thorax and body, and by a series of measurements 

 of individuals, to construct triangles and figures in combina- 

 tion, enclosing within them the great outlines of the body. 



In the first place, dividing the figure into five parts, on 

 each of which an equilateral triangle was produced or formed, 

 doubling the right-angled scalene triangle, whose angles 

 were 30, 60, and 90, he got the breadth of the shoulders 

 and various points of the figure. A remarkable confirmation 

 of Mr. Hay's views was arrived at by the division of the body 

 into five, which might be effected by continuing downwards 

 the equilateral triangle contained within the second oval of 

 the head in a succession of figures. By this succession they 

 had also the breadth of the figure. 



The angles of the first right-angled equilateral triangle 

 were 45, 45, and 90 ; those of the second, or scalene, being 

 half of the equilateral triangle, 30, 60, and 90 ; and those of 

 the third, or right-angled triangle, being the primary angle of 

 the pentagon, as seen at the neck, 18, 72, and 90. The 

 ratio of the angles in the right-angled equilateral triangle was 

 as 1 to 2, in the scalene as 1 to 3, and in the third right- 

 angled triangle as 1 to 5. The angles of the other triangles 

 were 22 30', 67 30', and 90 ; and 11 15', 78 45', and 90, 

 and their ratios 1 to 4, and 1 to 8 ; whilst the angles of the re- 

 maining triangle were 15, 75, and 90 the angle at the 

 apex bearing a ratio to the right angle of 1 to 6. 



They would thus find that, in regard to the various angles, 

 Mr. Hay had arrived at them partly on theoretical views. 

 He was just as much entitled to theorise as any other man of 

 science. No discovery had ever been made without hypothesis. 

 He had started, expecting to find in the proportions of the 

 human frame a musical harmony ; and, by continual measur- 

 ing, had at last been able to construct a figure, partly hypo- 

 thetically, but which was found to correspond with nature. 



