A Treatise on Infinitesimal Calculus: Differential calculus. 1857University Press, 1857 - Calculus |
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Page 4
... shewn by an example ; as , for instance , 5 is greater than 2 ; let each side be multiplied by +4 , then 20 > 8 ; again , let each side be mul- tiplied by a negative number , as -2 : the > is changed into a < , -10 < -4 , because -10 is ...
... shewn by an example ; as , for instance , 5 is greater than 2 ; let each side be multiplied by +4 , then 20 > 8 ; again , let each side be mul- tiplied by a negative number , as -2 : the > is changed into a < , -10 < -4 , because -10 is ...
Page 14
... shewn that the distance would be 20 billions of miles ; and as it is de- terminable , we may say that it is comparable with finite quan- tities , though on the verge of the infinite . " In such numbers the imagination is lost ; the mode ...
... shewn that the distance would be 20 billions of miles ; and as it is de- terminable , we may say that it is comparable with finite quan- tities , though on the verge of the infinite . " In such numbers the imagination is lost ; the mode ...
Page 15
... shewn in the sequel , enable us to overcome the apparent difficulty . For we shall create our numbers , and our subsidiary numbers , sub- ject to certain laws : and therefore , so long as they are applied within the conditions ...
... shewn in the sequel , enable us to overcome the apparent difficulty . For we shall create our numbers , and our subsidiary numbers , sub- ject to certain laws : and therefore , so long as they are applied within the conditions ...
Page 37
... shewn that the sum of all the lines similar to SR is less than AT ; but the limit of all such lines is the circular arc ; therefore AT is greater than the arc . Again , the chord AP is greater than PM , which is the sine of x , and PQ + ...
... shewn that the sum of all the lines similar to SR is less than AT ; but the limit of all such lines is the circular arc ; therefore AT is greater than the arc . Again , the chord AP is greater than PM , which is the sine of x , and PQ + ...
Page 38
... shewn , always intermediate to these , it is clear that all three are equal to , and therefore may be used indifferently for , each other . Hence , when x is an infinitesimal , sin x x = tan x ; which proposition is frequently expressed ...
... shewn , always intermediate to these , it is clear that all three are equal to , and therefore may be used indifferently for , each other . Hence , when x is an infinitesimal , sin x x = tan x ; which proposition is frequently expressed ...
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Common terms and phrases
a₁ algebraical angles b₁ becomes Calculus change of sign changes sign circle coefficients constant cosec curve d2F d2F d²u d²x d²y d³u d³y derived function determine differential equation dr dr dr dy dx dx dx dy dx² dy dx dy dy dy dz dy² dz dx equal equicrescent explicit function expression F(xo F(xo+h factor finite quantity fraction func given Hence homogeneous function increases increments indeterminate form infinite infinitesimal Infinitesimal Calculus infinity involved logarithms loge Maclaurin's maxima and minima maximum or minimum minimum value negative partial derived-functions positive primitive equation radius replaced result right-hand member roots Similarly sin x singular value Sturm's Theorem substituting suppose symbols Theorem tion vanish variables variation versin whence zero