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 be supported, and whose superficies they are to touch in as many points as possible, be formed ungeometrically? But I forbear to expatiate on a matter as plain as an axiom of Euclid; trusting, that whoever can perceive the utility of geometrical breeches, will readily agree, à fortiori, to the absolute necessity of geometrical saddles and bridles.

Pursuing my principles, I have demonstrated what is the right line to be drawn by the mathematical rider in every difficult situation. In ascending a horse's back, at what angle to extend the moveable leg, while the fixed one is rested in the stirrup: in leaping, how to regulate the oscillation, or balancing of the body, by attending carefully to that fundamental point which is your centre of motion: in starting, how to dispose of the superfluous momentum, and thereby to preserve in full force the attraction of cohesion between rump and saddle: in rearing, at what angle, formed by the horse's back with the plane of the horizon, it is most advisable to slide down over his tail;