as is the case in the problem of the trisection of an angle. We see at once that the dependence between _x_ on the one side, and _ab_ on the other, is completely determined; but, so long as the equation preserves its primitive form, we do not at all perceive in what manner the unknown quantity is derived from the data. This must be discovered, however, before we can think of determining its value. Such is the object of the algebraic part of the solution. When, by a series of transformations which have successively rendered that derivation more and more apparent, we have arrived at presenting the proposed equation under the form _x_ = ∛(_b_ + √(_b²_ + _a³_)) + ∛(_b_ - √(_b²_ + _a³_)), the work of _algebra_ is finished; and even if we could not perform the arithmetical operations indicated by that formula, we would nevertheless have obtained a knowledge very real, and often very important. The work of _arithmetic_ will now consist in taking that formula for its starting point, and finding the number _x_ when the values of the numbers _a_ and _b_ are given.] We thus see that the _algebraic_ calculus and the _arithmetical_ calculus differ essentially in their object. They differ no less in the point of view under which they regard quantities; which are considered in the first as to their _relations_, and in the second as to their _values_. The true spirit of the calculus, in general, requires this distinction to be maintained with the most severe exactitude, and the line of demarcation between the two periods of the solution to be rendered as clear and distinct as the proposed question permits. The attentive observation of this precept, which is too much neglected, may be of much assistance, in each particular question, in directing the efforts of our mind, at any moment of the solution, towards the real corresponding difficulty. In truth, the imperfection of the science of the calculus obliges us very often (as will be explained in the next chapter) to intermingle algebraic and arithmetical considerations in the solution of the same question. But, however impossible it may be to separate clearly the two parts of the labour, yet the preceding indications will always enable us to avoid confounding them. In endeavouring to sum up as succinctly as possible the distinction just established, we see that ALGEBRA may be defined, in general, as having for its object the _resolution of equations_; taking this expression in its full logical meaning, which signifies the transformation of _implicit_ functions into equivalent _explicit_ ones. In the same way, ARITHMETIC may be defined as destined to _the determination of the values of functions_. Henceforth, therefore, we will briefly say that ALGEBRA is the _Calculus of Functions_, and ARITHMETIC the _Calculus of Values_.