Wt do the mathematicians mean by considering curves as polygons? Either they are polygons or they are not. If they are, why do they give them the name of curves? Why do not they constantly call them polygons, & treat them as such? If they are not polygons, I think it absurd to use polygons in their stead. Wt is this but to pervert language? to adapt an idea to a name that belongs not to it but to a different idea?
The mathematicians should look to their axiom, Quæ [pg 022] congruunt sunt æqualia. I know not what they mean by bidding me put one triangle on another. The under triangle is no triangle—nothing at all, it not being perceiv'd. I ask, must sight be judge of this congruentia or not? If it must, then all lines seen under the same angle are equal, wch they will not acknowledge. Must the touch be judge? But we cannot touch or feel lines and surfaces, such as triangles, &c., according to the mathematicians themselves. Much less can we touch a line or triangle that's cover'd by another line or triangle.
Do you mean by saying one triangle is equall to another, that they both take up equal spaces? But then the question recurs, what mean you by equal spaces? If you mean spatia congruentia, answer the above difficulty truly.
I can mean (for my part) nothing else by equal triangles than triangles containing equal numbers of points.
I can mean nothing by equal lines but lines wch 'tis indifferent whether of them I take, lines in wch I observe by my senses no difference, & wch therefore have the same name.
Must the imagination be judge in the aforementioned cases? but then imagination cannot go beyond the touch and sight. Say you, pure intellect must be judge. I reply that lines and triangles are not operations of the mind.
If I speak positively and with the air of a mathematician in things of which I am certain, 'tis to avoid disputes, to make men careful to think before they answer, to discuss my arguments before they go to refute them. I would by no means injure truth and certainty by an affected modesty & submission to better judgments. Wt I lay before you are undoubted theorems; not plausible conjectures of my own, nor learned opinions of other men. I pretend not to prove them by figures, analogy, or authority. Let them stand or fall by their own evidence.
N.