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put into the hands of some young persons, not remarkable for their abilities, and who had hitherto experienced considerable difficulties in understanding the treatises of Walkingame, Bonnycastle, &c. By the use of these copies only, they have, during the above period, made considerable progress, with very little trouble to themselves or others; having each, on an average, made only about five or six applications per month for assistance, beyond what the book supplies. This fact, which from its nature does not depend on single testimony, is highly gratifying to the Author, as it evinces the usefulness of his work, and, he hopes, will operate as a recommendation; at the same time, he wishes to take no improper advantage of the public, but that it may abide a fair trial; and as he would not willingly expose himself to merited ridicule, by becoming his own panegyrist, it remains only for him to adopt the poet's candid request

"Si quid novisti rectius istis,

"Candidus imperti: si non, his utere mecum."

The Author is truly sensible of the honour done him by his pupils and others, whose respectable names compose the list of subscribers: he desires, in particular, to express his gratitude to Dr. Macbride and Richard Berens, Esq. LL. D. for their friendly advice and occasional corrections; for their kindness in undertaking the sole management of getting the work printed, and for other favours.

A BRIEF AND GENERAL ACCOUNT

OF THE

MATHEMATICS.

Pure and Mixed Mathematics defined, and their Nature and Uses explained.

THE term MATHEMATICS, in its original acceptation, means Learning, Science, or Discipline; but in a more restricted and commonly received sense, it is the science of quantity, which treats of magnitudes, considered either as computable or measurable.

The Mathematics comprehends several branches, each of which ranks as a distinct science: these are arranged under two general heads, viz. Pure Mathematics, and Mixed Mathematics.

PURE MATHEMATICS treats of magnitude generally, simply, and abstractedly; it determines the properties and relations of magnitudes and quantities, considered purely as such, and without relation to any material substance whatever. This class comprehends ARITHMETIC, or the science of Numbers; ANALYsis, or the science of general calculation; GEOMETRY, or the science of local extension; and MIXED GEOMETRY, in which Arithmetic and Analysis are combined with pure Geometry.

MIXED MATHEMATICS is pure Mathematics applied to Natural Philosophy or Physics; it combines the properties of Body, Motion, &c. as determined by incontestable experiments, with the doctrine of pure quantity; whence by a methodical and demonstrative chain of reasoning, it deduces conclusions as incontrovertibly evident, as those which Pure Mathematics derives from self-evident principles and definitions.

f The word Mathematics is derived from anos, discipline or science.

The following sciences are comprehended under Mixed Mathematics; viz. 1. MECHANICS, or the science of the equilibrium and motion of solid bodies, treating of the properties and effects of the five mechanical Powers, viz. The Lever, the Axis in Peritrochio, the Pulley, the Inclined Plane, the Wedge, and the Screw; and also of machines of every description compounded of two or more of these. 2. HYDROSTATICS, and HYDRAULICS, comprehending the theory of the nature, gravity, pressure, and equilibrium of fluids; the theory of pumps, syphons, and artificial water-works of every description; to which may be added PNEUMATICS, which treats of the weight, pressure, elasticity, &c. of air and elastic fluids, the air pump, air gun, &c. 3. ASTRONOMY, or the science which treats of the motions, periods, eclipses, distances, magnitudes, and other phenomena of the celestial bodies. 4. OPTICS, or the doctrine of vision, light, and colours, the theory of the eye, the telescope, the microscope, spectacles, and all kinds of reflecting and refracting glasses, to which may be added PERSPECTIVE, or the theory by which visible objects are accurately represented on a plane. 5. AcoUSTICS OF PHONICS, which treats of sound, explaining the nature of the ear, of speaking and hearing trumpets, whispering galleries, &c, including Music, or the science of harmony in sounds.

From various combinations of these branches of mixed Mathematics, we derive a great number of additional branches, as SURVEYING, or the art of dividing, delineating, and measuring land. ARCHITECTURE, civil, military, and naval, or the art of planning and building houses, churches, palaces, castles, ships, &c. PYROTECHNIA, or the art of constructing and managing fireworks, gunnery, &c. NAVIGATION, or the art of conducting a ship at sea from one port to another. CHEMISTRY, or the art of decompounding substances, both solid and fluid, by means of fire. ELECTRICITY, or the investigation of certain powers and their effects, as found in amber, sealing wax, glass, tourmalin, &c. and indeed every part of natural philosophy, and every manual art that can be practised, is connected more or less with mixed Mathematics.

The method by which mathematical truths, which before were doubtful, become evident to the understanding, is called DEMONSTRATION: it is this that peculiarly characterizes accurate knowledge, or true science, and distinguishes it from that spe

cies of knowledge which arises from conjecture, probability, testimony, induction of facts, &c. The latter kinds are called moral evidence, because they are chiefly employed on subjects connected with moral conduct.

On the Difference between Mathematical Demonstration and Moral Evidence.

Demonstration differs from Moral Evidence in the following particulars

1. Demonstration is employed about abstract truths, and the necessary relations of ideas, viz. such as are connected with extension, duration, weight, force, velocity, with whatever else can be accurately expressed by numbers and lines. But Moral Evidence relates to matters of fact, and the constant or variable connections which subsist among things actually existing.

2. They are conducted in a different manner. In demonstration we proceed from known truths to those which were unknown, by steps, each of which is necessarily connected with that which precedes it. In a moral proof, there is no such necessary connection, but it generally consists of a number of independent arguments.

3. In Demonstration it is only necessary to consider one side of the question; for if by Demonstration justly conducted, a proposition be proved true, it is of no consequence what may be urged against it; for whatever is offered as proof on the opposite side, must be a mere fallacy. But in Moral Evidence, there are frequently cogent arguments on both sides of the question; both sides therefore must be carefully examined, and the assent given to that which is supported by the strongest evidence.

4. The contrary to a demonstrated proposition is not only false, but absurd. But the contrary to a proposition established by Moral Evidence, although false, is not necessarily absurd.

8 These particulars were taken for the most part from Gambier's Introduction to the study of Moral Evidence, Chap. 1. A work which I hazard nothing in saying fully merits the encomium which Dr. Johnson has applied to Watts's Improvement of the Mind, viz. "Whoever has the care of instructing others, may be charged with deficience in his duty, if this book is not recommended." While the student is engaged in the study of Euclid's Elements, he should embrace occasional opportunities to read Duncan's or Watts's Logic, and these should be succeeded by Watts's Improvement of the Mind.

5. In demonstration there is a necessary connection between the successive steps, and hence the ideas compared are immediately perceived to agree or disagree. But in Moral Evidence their agreement or disagreement is only presumed; and that on proofs, which are in their nature fallible; the former therefore produces absolute, but the latter can, at the most, produce only moral certainty.

6. As Demonstration is always accompanied with certainty, rules laid down, which are in all cases capable of being demonstrated, will infallibly lead to truth. But in Moral Evidence, no rules can be given, which will direct us how to form an infallible judgment in any particular case.

7. Demonstration terminates in certainty, which is always absolute, and cannot admit of degrees. But the degrees of moral assent may be indefinitely various, from suspicion up to moral certainty.

8. Demonstration requires no accumulation of evidence; for the truth of a proposition as effectually appears from one proof, properly conducted, and as completely commands our assent, as from many. But Moral Evidence admits of, and frequently requires an accumulation of proofs, and each independent argument in favour of the thing to be proved, increases the weight of evidence, but the whole does not compel the

assent.

9. In Demonstration we may reason safely from a conclusion already established; and upon that establish a second, upon these a third, and so on to any length. In Moral Evidence, we can seldom proceed with complete safety beyond the first step s for the second step will be less certain than the first; the third less certain than the second, and so on.

10. All the terms used in a system of Demonstration are previously defined with the greatest accuracy, and are always used in the same sense, so that no dispute can arise, nor any ambiguity have place in their application. But the terms employed in Moral Evidence are not always accurately defined; they are frequently susceptible of very different meanings, and consequently must often lead to uncertainty.

Hence it appears that Demonstration is vastly superior to Moral Evidence. But on the other hand, Moral Evidence is by no means to be lightly esteemed; for although the former is in all cases absolutely conclusive, and the latter not so, yet the

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