Philosophy Delineated, containing a resolution of divers knotty questions upon suddry philosophical notions

Dr. William Marshall, a doctor at the College of Physicians in London, is not to be confused with William Marshall the engraver. Philosophy Delineated stands out for its abstract engraving, quite in contrast with the earthy content matter. Here's a list of topics to be addressed in this book.

Concerning the Original of Springs, and of their Irregular Ebbings and Flowings.
Of Chymical Multiplications or the increaſing of Quantity of Liquors by Diſtillation.
Of the poſſibility of a Perſons Learning in an hour or two to write his Mind in a Language he is ignorant of.
Concerning Sympathy between ſeveral Parts of Animal bodies,
Freſh Water Springs are Afferted at the Bottom of the Sea.
Of the Sutures in the Skul.
Of the Conſiſtency of Immenſe Volatility with Immenſe Ponderouſneſs
Of the Motion of the Celeſtial Bodies about their particular Axes.
Concerning Angles of Contact.

With such a saucy list of contents, it seems surprising that the half-sheet engraving folded in the back should be purely geometrical. The figures are arranged left-to-right, top-to-bottom As you might expect, these geometrical engravings all belong to the chapter "Concerning Angles of Contact." But the order of the engravings is interesting. These engravings appear in the order that they're introduced, but that is not really the Euclidean order that works from smallest axioms to greatest. Instead, he cross-references between one figure and the other: at times relying on a later proof to further confirm an earlier case, or the reverse.

I'm interested in the relation that William Marshall sees between geometric engravings and the science of optics itself:

It is a ſeeming weighty objection that is urged out of opticks, and the uſage in that ſcience, to demonſtrate in conical figures, the angles of incidence and reflection to be equal, only with reſpect to the right-line tangent, touching the figure in the point of incidence and reflexion, without ſpecial reſpect to the curvature in the conical ſection. But hereto without wrong, either to truth, nature, or that noble ſcience, may be upon good grounds anſwered. 1. That opticks is not pure Geometry, and obſtructed ſtereometry, and mathematicks, wherein quantities, menſurations, and proportions are conſidered merely as in themſelves without relation to matter, and the uſes to which in other faculties they are applicable: but in opticks, is an improvement of what in nature may be obſerved about luminous and viſual beams, and luminous mediums and objects, by mathematical demonſtration and aſſiſtance. Now 2ly. Nature doth not tye it ſelf in its wayes ſo ſtrictly to an indiviſible, abſolute, vertical punctilio, that it cannot reach, without impediments that mathematical exquiſiteness, it will not act at all.

Marshall seems greatly concerned with the translation between pure mathematics and observations about the world. His concern is surprising, given the speed with which he leaps between humors, chemistry, and geology in his first chapter. Nevertheless, Marshall spends the largest, seventh chapter expounding on the principle of mathematical homogeneity and defending a model to compare geometrical objects without recourse to measurement. In this way, Marshall seems to be anticipating Newton's infinitessimal turn.