ABSTRACT
I Have now proposed what I thought to be of Use concerning the Division of the Mathematical Sciences; but have purposely omitted their Distribution into Speculative and Practical, as less just and necessary; since this does not so much intimate Sciences as different Respects of the same Science. For, in my Opinion, every Science is both Speculative and Practical: Speculative, as it speculates, i. e. seeks, investigates and demonstrates Truths (or true Propositions) agreeable to its Object: and Practical, as those Truths when found and demonstrated, may be referred to Use, and reduced into Practice. Ex. gr. The Politician speculates and infers, from well weighed Reasons, what Form of Government, and what Laws will best conduce for the public Welfare: The Moralist considers and pronounces (i e. invents and prescribes some certain Theorem or universal Rule) what is the Duty of a good Man in such and such Circumstances: The Physician examines and determines, from Reason or Experience, the Nature, Virtues, and Temperaments of what they call Simples, whether they be in Faculty Cool or Hot, Dry or Moist, Healthful or Noxious: And so far these Sciences are Speculative. But if any one do use these Conclusions for the directing of his Practice, and attemperate his Actions accordingly: If he manage Affairs of State according to the Prescript of this Political Theorem; if he live according to that Moral Precept; if he take or refuse such a Plant for the fake of his Health; then do these Sciences and Conclusions become Practical And the same happens in our Cafe. The Geometrician demonstrates universally That every Triangle is equal to one half a Parallelogram upon the same Base and of equal Height: That a Cone is equal to one third Part of a Cylinder upon the same Base and of equal Height. Which Propositions and Rules being universally true, the Geometrician finds and approves by Speculation, and they may be always used for determining the Measure of any Triangular Area, or for comparing any Conical Vessel with its circumscribed Cylinder. Both these theorems therefore may be called Practical as they relate to Use and Practice, but they would for that Reason be very unfitly divided into Speculative and Practical. Geometry is one simple Science which may be considered in two Respects, viz. it is stiled Speculative as it is true, and Practical as it is useful. The Absurdity then of this Division is clear. Notwithstanding I shall not very much oppose him who urges that such Geometrical Rules are more useful, which shew the Dimensions of particular Magnitudes, or are accommodated to some certain material Subject; though that perhaps may prejudice the main Science, and it would be better to find them no where, but affixed to their proper Fundamentals in Geometry, along with their Demonstrations, i. e. in their proper Place and native Order: But it ought not therefore to be thought that a Collection of Rules of this Sort do constitute a Part or Species of Geometry distinct from Speculative. Consequently Trigonometry, or the Doctrine concerning the Dimension of Rectangular and Spherical Triangles; Altimetry, which shews the Way of finding Distances; Planimetry, which teaches to describe and measure plain Surfaces; and Stereometry, which compares the Measures of solid Bodies: these are not Parts of Speculative more than Practical Geometry, but belong to the same Science, which in a different Respect may be called, either Speculative or Practical.
