This work of Gunther was many years after succeeded by another of the great mathema→ tician Weigelius; wherein he endeavours to deduce the art of demonstration from mathematical principles; and farther to explain, illustrate, and shew the use of Aristotle's analytics*.

The celebrated M. Des Cartes wrote an express treatise de Methodo; wherein he reduces the whole art to four rules, that seem contained in Aristotle's analytics; and will be found to coincide with the Medicina Mentis, mentioned below. This method of Des Cartes, is delivered, with considerable improvements, in the, fourth part of the Art de Penser; where the author constitutes to method; viz. the analytical, and synthetical; the former discovering, and the latter for teaching And he illustrates the whole with a variety of examples, as well physical as mathematical.

Upon the same foundation, in the year 1687, the excellent M. Tschirnhaus, a member of the Royal Academy of Sciences at Paris, published an essay towards a genuine logic, or method of discovering unknown truths +. This is an extra

* See Analysis Aristotelica, ex Euclide restituta. Jenæ. 1658.

† Medicina Mentis, sive Artis Inveniendi Præcepta Generalia: or what explains the design better, Tantamen ge

ordinary performance, that proceeds entirely upon the mathematical, or rather algebraical method; and deserves to be read with care and attention.

M. Tschirnhaus, reflecting that mathematicians being the only set of men, who either maintained no controversies, or at least soon came to a determination of them; hence apprehended, that mathematicians alone were possessed of the right method of enquiry. Upon this, he applied himself to mathematical studies; in order to see whether, by making the proper alterations, the mathematical method could not also be accommodated to other subjects.

In particular, he applied himself to algebra; and found that this art performs even more than it promises; and with the highest degree of certainty: when, having acquired a habit of solving the greatest difficulties therein, and examining the secret nature of its method, or manner of procedure; he says, he observed that unknown truths may be discovered, after the same manner, not only in mathematics, but in every other science.

He makes the foundation of human certainty to lie in those things wherein the operation of

nuinæ Logicæ, ubi disseritur de Methodo detegendi incognitas Veritates. The second edition is corrected and enlarged by the author.

the understanding is most manifest; or those which may be conceived, without any possibili

ty of a contradiction; as that the whole is greater than a part; that the radii of a circle are equal, &c. whence numerous other truths may be drawn: and, on the other hand, he lays it down for certain, that those things which cannot be conceived, are false.

But here the author cautions us against being deceived by the imagination; for, according to him, many things are perceived by the imagination only; of which things, no distinct notion, or conception, can, by words, be communicated to another; as in the case of pain, light, colour, sound, &c. Hence he recommends two ways of distinguishing between the perceptions of the imagination, and the conceptions of the understanding. The first is, by large and frequent experience, and especially, by the help of mathematics, to acquire a habit of finding the difference betwixt them; and the second, is to consider the equality there is in the human understanding, which all men have equally alike: for what a man truly conceives, he can communicate to another; as we see in mathematical demonstrations, which are equally understood

by all men; whereas those things which are perceived by the imagination, as he calls it, are perceived unequally, or more by some, and less

by others. And by justly distinguishing betwixt these two powers, or faculties of the mind, he supposes many great errors may be avoided.

We next proceed to the author's method of discovering new truths; wherein he supposes that any one may continually advance to an'indeterminate length, without danger of falling into error. And here he advises us first to procure, with great diligence, a stock of all the possible conceptions with the mind, in the common course of studies and occasions, takes cognizance of. For, from these conceptions, definitions, in his method, are to be immediately formed; then properties to be deduced from these definitions; which properties he calls by the name of axioms: and from the definitions, combined all manner of ways, he discovers secondary truths, or theorems; thus making the whole process algebraical.

He determines it to be in the power of men to form scientifical definitions; and in order to form them justly, advises us to consider the manner wherein the thing to be defined is itself actually formed: or, as he calls it, still in allusion to mathematics, generated; and from this consideration, he directs us to derive the definition. Thus, for example, he defines virtue to be the power which men have of preserving their own nature, according to the laws of just rea

son; or of procuring to themselves all the real perfections both of body and mind; or, again, the perfection, or melioration of human nature, according to the laws of just reason.

To facilitate this business of forming definitions, he lays down three general rules. The first is, for reducing things, in thought, under their ultimate kinds, or most general concep

tions. And these highest mental kinds, or classes, he makes to be three; relating to things imaginary, mathematical, and physical: under which heads all things that exist may be ranged.

The second rule is, when things are thus reduced under their highest kinds, or classes, to observe, either by reason or experience, what things those are which remain continually present in every conception.

The third rule is, that all the formed conceptions be so ordered, as to succeed each other, according to what he calls the number of possibilities, or elements; or according as one thing supposes the existence of another: beginning with the simple cases, and proceeding gradually to the more complex.

The first elements of imaginary things, perceived by the sense, he makes to be fluidity, and solidity; the first of the mathematical, he makes to be points, strait lines, and curves; and the first of the physical, matter, motion,