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the goniometrical cosines and sines of an angle, with which they are 

 ultimately shown to have a close connexion, which can be rendered 

 most evident by assuming as the unit-angle that subtended by a cir- 

 cular arc of the length of its radius. Studying these series quite in- 

 dependently of these relations to angles, we discover that they bear to 

 each other the same relations as the goniometrical cosines and sines, 

 and that if the least tensor value of the variable for which the cosine 

 series becomes null, is known, all its other values can be found by 

 multiplying this by four times any scalar integer. This last product 

 must be added to the least tensor value of the variable for which 

 both the cosine or the sine series become equal to given scalars, in 

 order to find all the solutions of such equations. Supposing the 

 values of such series tabulated by the method of differences for all 

 scalar values of the variable, so that such least tensor values can 

 always be found, we are now able to assign the meaning of any 

 power whose base and index are both clinants, and the logarithm of 

 any clinant. This enables us to invert completely all the simple for- 

 mations, sum, product, power with variable base and constant index, 

 or constant base and variable index ; and hence to solve all equations 

 of four dimensions with clinant coefficients, and to show that every 

 formation consisting of a sum of integral powers with clinant coeffi- 

 cients, can be expressed as a product of as many simple formations as 

 is determined by the highest index of the variable. The cosine and 

 sine series can also be generally inverted. The versor of any clinant 

 having a known angle (which is always equal to the cosine of its 

 angle added to the product of the sine of its angle into a quadrantal 

 versor), can now be shown to equal the cosine series added to the sine 

 series multiplied by a quadrantal versor, when the variable of the 

 series is the scalar ratio of the angle of the clinant to the angle sub- 

 tended by a circular arc equal to its radius, From this the ratio of 

 the circumference to the diameter of a circle is shown to be twice the 

 least tensor value of the variable, for which the cosine series is equal 

 to null ; and as that value can be readily assigned in a convergent 

 series, the former ratio is determined. The same investigation shows 

 the relation already mentioned between the goniometrical cosines 

 and sines, and the cosine and sine series. 



Clinant algebraical geometry allows us to interpret all results of 

 clinant algebra when referred to lines on one plane. It thus fur- 



