Wt need of the utmost accuracy wn the mathematicians own _in rerum natura_ they cannot find anything corresponding wth their nice ideas.
One should endeavour to find a progression by trying wth the scale.
Newton"s fluxions needless. Anything below an M might serve for Leibnitz"s Differential Calculus.
How can they hang together so well, since there are in them (I mean the mathematiques) so many _contradictoriae argutiae_. V. Barrow, Lect.
A man may read a book of Conics with ease, knowing how to try if they are right. He may take "em on the credit of the author.
Where"s the need of certainty in such trifles? The thing that makes it so much esteem"d in them is that we are thought not capable of getting it elsewhere. But we may in ethiques and metaphysiques.
The not leading men into mistakes no argument for the truth of the infinitesimals. They being nothings may perhaps do neither good nor harm, except wn they are taken for something, & then the contradiction begets a contradiction.
a + 500 nothings = a + 50 nothings-an innocent silly truth.
(M417) My doctrine excellently corresponds wth the creation. I suppose no matter, no stars, sun, &c. to have existed before(247).
It seems all circles are not similar figures, there not being the same proportion betwixt all circ.u.mferences & their diameters.
When a small line upon paper represents a mile, the mathematicians do not calculate the 1/10000 of the paper line, they calculate the 1/10000 of the mile. "Tis to this they have regard, "tis of this they think; if they think or have any idea at all. The inch perhaps might represent to their imaginations the mile, but ye 1/10000 of the inch cannot be made to represent anything, it not being imaginable.
But the 1/10000 of a mile being somewhat, they think the 1/10000 inch is somewhat: wn they think of yt they imagine they think on this.
3 faults occur in the arguments of the mathematicians for divisibility _ad infinitum_-
1. They suppose extension to exist without the mind, or not perceived.
2. They suppose that we have an idea of length without breadth(248), or that length without breadth does exist.
3. That unity is divisible _ad infinitum_.
To suppose a M. S. divisible is to say there are distinguishable ideas where there are no distinguishable ideas.
The M. S. is not near so inconceivable as the _signum in magnitudine individuum_.
Mem. To examine the math, about their _point_-what it is-something or nothing; and how it differs from the M. S.
All might be demonstrated by a new method of indivisibles, easier perhaps and juster than that of Cavalierius(249).
(M418) Unperceivable perception a contradiction.
(M419) Proprietates reales rerum omnium in Deo, tam corporum quum spirituum continentur. Clerici, Log. cap. 8.
Let my adversaries answer any one of mine, I"ll yield. If I don"t answer every one of theirs, I"ll yield.
The loss of the excuse(250) may hurt Transubstantiation, but not the Trinity.
We need not strain our imaginations to conceive such little things. Bigger may do as well for infinitesimals, since the integer must be an infinite.
Evident yt wch has an infinite number of parts must be infinite.
Qu. Whether extension be resoluble into points it does not consist of?
Nor can it be objected that we reason about numbers, wch are only words & not ideas(251); for these infinitesimals are words of no use, if not supposed to stand for ideas.
Axiom. No reasoning about things whereof we have no idea. Therefore no reasoning about infinitesimals.
Much less infinitesimals of infinitesimals, &c.
Axiom. No word to be used without an idea.
(M420) Our eyes and senses inform us not of the existence of matter or ideas existing without the mind(252). They are not to be blam"d for the mistake.
I defy any man to a.s.sign a right line equal to a paraboloid, but wn look"d at thro" a microscope they may appear unequall.
(M421) Newton"s harangue amounts to no more than that gravity is proportional to gravity.
One can"t imagine an extended thing without colour. V. Barrow, L. G.
(M422) Men allow colours, sounds, &c.(253) not to exist without the mind, tho" they have no demonstration they do not. Why may they not allow my Principle with a demonstration?
(M423) Qu. Whether I had not better allow colours to exist without the mind; taking the mind for the active thing wch I call "I," "myself"-yt seems to be distinct from the understanding(254)?
(M424) The taking extension to be distinct from all other tangible & visible qualities, & to make an idea by itself, has made men take it to be without the mind.
I see no wit in any of them but Newton. The rest are meer triflers, mere Nihilarians.
The folly of the mathematicians in not judging of sensations by their senses. Reason was given us for n.o.bler uses.
(M425) Keill"s filling the world with a mite(255). This follows from the divisibility of extension _ad infinitum_.
Extension, or length without breadth, seems to be nothing save the number of points that lie betwixt any 2 points(256). It seems to consist in meer proportion-meer reference of the mind.
To what purpose is it to determine the forms of gla.s.ses geometrically?