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the shape, is the beautiful surface known to geometers by the name of 

 the hyperboloid of one sheet. Here you see a number of straight lines, 

 which I shall suppose to represent muscular fibres passing from the bone 

 A B to another bone A' b'. I have made the bones of the same length, 

 but the results would be similar, if I made the bones of different 

 lengths or of any curvature I pleased, and placed them in the 

 same position. I now take this muscle and distort it out of its 

 plane. You see I have now a curved surface in which every portion 

 is made up of straight lines. I can curve the surface in the oppo- 

 site direction, and so I make the hyperboloid of one sheet out of a plane 

 quadrilateral muscle. This is not a mere fiction of geometers. The 

 adductor magnus muscle in the leg of man, and the great pectoral 

 muscle in the wing of every bird, are living examples of the reality of 

 this curious fact, that Nature constructs not merely plane muscular struc- 

 tures, but that she is capable of constructing muscular surfaces belong- 

 ing to the most beautiful and elegant forms that have been studied and 

 invented by abstract geometers. A friend of mine, one of the most dis- 

 tinguished of living geometers, when I informed him that Nature used 

 familiarly the hyperboloid of one sheet in making her muscles, told me 

 that his respect for her was considerably increased. The last forms of 

 muscle to which I shall direct your attention are the sphincter muscle, 

 which represents a number of circular fibres surrounding an orifice, and 

 the ellipsoidal muscle, which represents muscles of greater or less thick- 

 ness surrounding a cavity called ellipsoidal, because the cavity so formed 

 is generally egg-shaped. In the next lecture I shall direct your atten- 

 tion to the most important of all these forms of muscles in the human 

 heart and the hearts of animals. I shall confine our attention to-day 

 to the more elementary muscles represented by the prismatic, penni- 

 form, triangular, quadrilateral, and the hyperboloid muscle of one 

 sheet. 



The prismatic muscle and the penniform muscle possess the remark- 

 able property, which can be demonstrated mathematically, that in their 

 contraction no loss whatever takes place. Nature, therefore, according 

 to my principle, is entitled to employ these two forms of muscles when- 

 ever she pleases. She suffers no loss or injury by using these forms of 

 muscles, and we find, therefore, that both these classes of muscles are 

 constantly employed. "When you come to the triangular, the quadri- 

 lateral, and skew muscles, we can demonstrate by mathematics that in 

 the use of every such muscle there is a necessary loss of force. I may, 

 therefore, be asked — How comes it, if the principle of least action be 

 true, that Nature ever employs muscles involving a necessary loss of 

 force? I answer, because Nature has other problems in view than mere 

 economy of force in a single muscle. She has to consider if she econo- 

 mise force simply, without regard to other circumstances, such as 



