DEMONSTRATION, (from the Latin) in mathematics, a method of reasoning, whereby the truth of an assertion is shown by two, or a series of propositions, whose truth is already established.
Thus the 47th proposition of the first book of Euclid demonstrates a certain property of a right-angled triangle, on the supposition :-1. that all the preceding propositions are true ; 2, that the axioms used in geometry, whether expressed or implied, are true also. It makes the consequence as cer tain as the premises, by means of the indubitable character of the connecting process. This strict use of the term demonstration belongs to the science of logic, which is the art of demonstrating from premises, without reference to the truth or falsehood of the premises themselves. In effect, the demonstrations of mathematicians are no other than series of enthymemes ; everything is concluded by three of syllogism, only omitting the premises, which either occur of their own accord, or are recollected by means of quota tions. To have the demonstration perfect, the premises of the syllogisms should be proved by new sN llogisins, till at length you arrive at a syllogism, wherein the premises are either definitions, or identical !impositions.
Indeed, it might be demonstrated, that there cannot be a genuine demonstration, i. e., such a one as shall give full con viction, unless the thoughts he directed therein according to the rules of syllogism. Clavius, it is well known, resolved the demonstration of the first proposition of Euclid into syllogism : Herlinus and Dasipodius demonstrated the whole first six books of Euclid, and Henischus, all arithmetic, in the syllogistic form.
Yet the generality of persons, and sometimes even mathe maticians, imagine, that mathematical demonstrations are conducted in a manner far remote from the laws of syllogism ; so far are they from allowing that those derive all their force and conviction from these. But men of the greatest ability
have taken our view of the question. M. Leibnitz, for instance, declares that demonstration to be firm and valid, which is in the form prescribed by logic ; and Dr. Wallis confesses, that what is proposed to he proved in mathematics is deduced by means of one or more syllogisms ; the great Huygens, too, observes, that paralogisms frequently happen in mathematics, through want of observing the syllogistic form.
Problems consist of three parts : a proposition, resolution, and demonstration.
In the proposition is indicated the thing to be done.
In the resolution, the several steps are orderly rehearsed, whereby the thing proposed is performed.
Lastly, in the demonstration, it is shown, that the things enjoined by the resolution being done, that which was required in the proposition is effected. As often, therefore, as a pro blem is to be demonstrated, it is converted into a theorem ; the resolution being the hypothesis, and the proposition the thesis ; for the general tenor of all problems to be demon strated is this : that the thing prescribed in the resolution being performed, the thing required is clone.
The schoohnen make two kinds of demonstration : the one ris otort, or propter quod ; wherein an effect is proved by the next cause ; as when it is proved, that the moon is eclipsed because the earth is then between the sun and moon. The second, 78 ors, or quia ; wherein the cause is proved from a remote effect ; as when it is proved that fire is hot, because it burns ; or that plants do not breathe, because they are not animals; or that there is a God, from the works of creation. The former is called demonstration a priori, and the latter demonstration a posteriori.