PREFACE.

Besides this person, I encountered another who can also be rightly placed among the ignorant. Though living outwardly quite regulated and modest, he did not hesitate to say straight out, when he thought he was speaking with those through whom his sentiment would not become too well-known, that everything was just as it is due to Nature (this was his expression). When one spoke with him a bit more closely, he brought forward no other reason than that it appeared so to him, and he could not understand it otherwise. Yet in this thick darkness of ignorance, he still displayed a certain grandiosity, as if his intellect reached further than that of others.

§. 9. To set these people right, since the best metaphysical arguments have no power over them because they do not understand them and have no desire to apply the effort to do so, nothing seems more useful to me than to present them with such proofs that have their ground only in common experiences, in what everyone sees with their eyes. I know that this latter man, who was otherwise not easily accustomed to showing any weakness or doubts concerning his sentiment, was there forced to admit that he was brought to unrest in this matter.

§. 10. The fourth cause of godlessness, as far as has become known to me through observation and experience, springs in others from an excessively large imagination of being wise, and a too-blind acceptance as truth of that which they are accustomed to derive from their intellect or ideas. They are accustomed to set these ideas with great pride as an infallible rule of possibility and impossibility, truth and falsehood, good and evil in everything, concerning the Divine Attributes as well as the smallest phenomenon in the creatures, in short, with nothing excepted.

This is the most harmful sort of all. Firstly, because they deny everything they do not understand; and therefore all Divine revelation (which exceeds their comprehension) is rejected and sufficiently mocked by them. Secondly, because these have the most capability to advocate their error with subtle reasonings and to evade the force of the arguments used against them. They immediately make use of this as soon as anyone who contradicts them falls into even the slightest careless or ambiguous expression. Thirdly, because many of these, showing an outward appearance of civility and morality in their social interactions, sometimes gain a certain esteem among the inexperienced, which is dangerous to unstable listeners. All the more since several of them, having learned the principles of Euclides Euclid, Algebra, and other only speculative parts of Mathefis mathematics, are therefore held by the ignorant to be great mathematicians. Yet this belongs to them no more than the name of a great philosopher belongs to someone who understands nothing but Logica logic: since one can be very practiced in these ideal or imaginary sciences, and yet have no knowledge, or at most a very small one, concerning that which is essential and occurs in reality. One

should please not conclude from this that these glorious studies lead these unhappy ones to such perverse thoughts of their own accord; in many cases, they open the true paths to demonstrate the otherwise entirely unsearchable wisdom of God in the creatures. On the contrary, they are very useful, unless a perverse puffing up, by which one thinks one knows everything, carries these half-philosophers to the misuse of them. This makes them despise others with much conceit, specifically those who have not applied their intellect to the investigation of lines and magnitudes, although they use the same intellect wisely and with much judgment concerning other objects.

§ 11. Thus one finds at present that, in order to make even atheistic writings pass for indisputable truths, their authors have tried to give them the appearance of mathematical demonstrations. A prominent example is seen in the book of B. de Spinoza; which for that reason has also gained so much reputation among many unfortunate people. Since those who do not rightly understand mathematics think, from the outward arrangement of treating things, that what is said therein follows from correct mathematical reasonings.

Perhaps an opportunity will be given later to show more extensively the mistake committed therein, by comparing the demonstrations named there with those of true mathematicians.

To say a word on this in passing:
I. There are two objects with which mathematicians occupy themselves: namely ideas, which they regard only as ideas; and secondly, those which they hold to be ideas of truly existing things.

That is to say, more clearly:
Mathematicians reason either only concerning their ideas, or otherwise concerning things which exist essentially outside their ideas.

II. The first manner is seen in Geometria Speculativa speculative geometry, such as the principles of Euclides, Algebra, etc., where one understands a point as something that has no parts; a line as a length without width; and so forth. Thus one also investigates magnitudes here that have more than three dimensions, etc., all of which everyone knows are only certain ways of our conceptions, without having any essentiality outside of them.

III. The second manner is seen in Aftronomia astronomy, Optica optics, etc., where things that truly exist outside our ideas are investigated.

IV. The ground of the first reasonings, besides the Axiomata axioms, are the Definitien definitions, in which they describe their ideas without concerning themselves whether there is anything in reality that fits them; of which examples were just given. And so it is a truth for them that the three angles in a triangle are equal to two right angles, even if all things were round and not a single triangle essentially existed in the world.