Now, it is known that the weight _W_, acting by gravity in the direction _ab_, may be taken as proportional to the length of the line _ab_, and the portion of the pressure _P_ in the direction _ac_, which will be effective to turn the wheel, may be taken as proportional to _P_ _ac_ _ac_ the length of the line _ac_; that is, ——— = ————, or _P_ = ———— _W_, _W_ _ab_ _ab_ _ac_ where ———— is evidently always less than unity. Now, if the angle _bac_ _ab_ _ac_ is thirty degrees, and _W_ = 150 pounds, _W_ times ———— is 130 pounds. _ab_ Or, by trigonometry, the weight _W_, acting in the direction _ab_, by gravity as in working a cycle, will have a resultant in the direction _ac_ representing the power acting to turn the wheel equal to _W_ cos _bac_. If the angle _bac_ is thirty degrees and _W_ = 150 pounds, then _W_ cos _bac_ = 130 pounds. Now, in order to still get one hundred and fifty pounds of force on the wheel, a pull on the handle-bars would have to be given sufficient to make up the lost twenty pounds, which the rider would get without any pull on the bars if placed directly over the work. This pull, while not fatiguing to the legs beyond the necessary requirement of power, is an entire loss of work in the arms, and must tell on the system. This is all an additional loss to that which ensues from the fact that nature has fitted us to stand upright and not to work in an angular position; our every-day experience in walking gives us practice in a direct vertical strain on the muscles of the body, and we should make it a point to apply our force as nature intended, in so far as it is applicable to our wheel method. These conditions apply more or less to any form of locomotion, and particularly to the cycle. From the foregoing remarks we are amply justified in drawing the conclusion that the resultant force available in the application of the physical power of man is in proportion to the cosine of the angle at which he exercises this force. We are well aware that many apparent variations will occur when so rigid a mathematical fact comes to be applied to the exercise of man’s energy in driving a bicycle; but all we care for is to lead the reader well up to the point by means of reasoning, which we hope will give at least a partial hypothesis for a conclusion well demonstrated by practical experience. We assert that when we consider the application of the _gravity_ of the body to work on either a bicycle, or to other work of similar requirements, our mathematical demonstration is strictly true. It is justifiable, therefore, from a purely theoretical stand-point, to say that the rider of a bicycle wants to get directly over the work; let us see how our experience demonstrates this conclusion.