1.1. Advertisement.

The following pages contain a collection of Algebraical Problems designed solely to point out the various methods employed by Analysts in the Solution of Equations. They are arranged in the usual manner: 1. Simple Equations; 2. Pure Quadratics, and others which may be solved without completing the square; and 3. Adfected Quadratics. Many books have been consulted; and, as utility is the sole object of this Publication, wherever a proper Example occurred, it has been taken without hesitation, or altered to suit the purpose. At the head of each Section are given the common Rules; and the whole concludes with a Collection of Problems without Solutions, for the Exercise of the Learner.

2.1. Preface.

The following pages contain the substance of an Elementary Course of Lectures given in St John's College, and originally drawn up for the use of a portion of the students of that society. The want of some treatise, to serve as a Text Book, having been found to discourage the pupil and frequently to cause this branch of philosophy to be neglected, occasioned a recommendation from some friends engaged in the tuition, to commit this to the press. It is, therefore, now submitted, with diffidence, to the judgement of the public; and should it be found to facilitate the progress of the students, for whose use it is principally intended, the purpose of its publication will be answered.

In explaining the principles and elements of any branch of philosophy, originality cannot be expected. To arrange and elucidate the discoveries of others, and in some cases to supply deficiencies, will generally be the aim of those who are engaged in instruction. But to ensure success, it is peculiarly important that the principles should be fixed in the memory by their application to Examples and Problems; and with the hope of effecting this a considerable number have been here introduced.

The term Hydrostatics, in its proper acceptation, signifies that division of the science which treats of the equilibrium of non-elastic fluids; and Hydrodynamics that which relates to their forces and motion: in this respect the terms correspond with those of Statics and Dynamics as applied to solid bodies. But it is not unusual to include the whole doctrine under the general term Hydrodynamics, and to denote the divisions relative to their equilibrium and motion by the terms Hydrostatics and Hydraulics. That part of the science which treats of the mechanical properties of air and the different elastic fluids, is called Pneumatics. These divisions are here comprehended under the general name Hydrostatics; and it is adopted partly because it is the name in common use, and partly because the scientific distinctions are not closely observed in the arrangement, which has been regulated by considerations of convenience, and what experience has taught to be the easiest mode of instruction.

The first Section contains general principles and such propositions as serve to explain the nature of specific gravity. The second treats of the pressure of fluids; and the third of floating bodies and the modes of thence deducing the specific gravities of solids and fluids. The fourth is occupied with the motion of fluids issuing through the orifices of vessels. To this section it was intended to annex the general equations which have been deduced: but the motion of fluids is a subject involved in such difficulties, and so complicated are the formulae, when the necessary considerations are introduced, that in a treatise intended for beginners, it has been deemed expedient to omit them altogether. - In the fifth Section will be found the common principles of resistances; followed in the sixth by the motion of wheels, etc. impelled by water, the screw of Archimedes, and the motion of water in canals. The seventh treats of the nature and properties of elastic fluids; the eighth of the thermometer; and the ninth of the expansion and contraction of fluids and solids, and the corrections thereby afforded to the methods of determining specific gravities. In the tenth are explained the effects of the air's pressure in the case of the barometer, pumps, etc. In the eleventh, the motion of elastic fluids, particularly the air; and in the twelfth, the theory of capillary tubes.

It has not been deemed necessary to enumerate all the authors from whom assistance has been derived, or whose observations have been adopted or altered to suit the plan of the present work. Should it fall into the hands of proficients in the science, they will easily discover them: and to the mere beginner it can be of no service to quote a list of names. But so much has been taken from the Traité d'Hydrodynamique of Bossut, and the Traité de Physique of Biot, that the obligation cannot be passed over without particular acknowledgment.

To the Syndics of the University Press, this opportunity is taken of gratefully acknowledging the liberality with which, from the funds at their disposal, they have defrayed a considerable portion of the expense of printing.

3.1. Preface.

The following pages contain a collection of Problems, which are for the most part an easy application of the Elements of Euclid. They are arranged in what seemed to be the most natural order: The 1st section comprises such as contain the properties of straight lines and angles; the 2nd straight lines and circles: - the 3rd straight lines and triangles; and the 4th parallelograms, squares and polygons. The 5th section contains those which require lines to be drawn in certain directions, but which involve properties of rectangles or squares, or such others as were excluded from the three first. The 6th comprises those by which figures are described, and also inscribed in or circumscribed about each other. The 7th comprehends such as contain the properties of triangles described in or about circles; the 8th those which contain the squares or rectangles of lines connected with circles; and the 9th the construction of triangles. To these is added an Appendix, intended to contain so much of the Elements of Plane Trigonometry, as is necessary for understanding those parts of Natural Philosophy which are the common subjects of Lectures in the University. The Reader who wishes for farther information, is referred to Professor Woodhouse's treatise, or that of Cagnoli, to the latter of which are appended extensive Tables of trigonometrical formulae.

From this performance the only credit expected is that of having endeavoured to place principles in a clear light, and to render a service to the younger students by setting before them a series of Problems, on the solution of which they are recommended to exercise their own ingenuity; for which purpose a table of Contents has been prefixed.

4.1. Preface.

It has been confessed by most of those who have been engaged in the laborious task of tuition, that the principles of any branch of mathematical or philosophical science are more readily comprehended, and more easily imprinted upon the student's memory, when their application to the solution of problems has been pointed out and explained. With the intention of contributing in some measure to this desirable end, the following collection of Problems, which were originally intended to appear in a very different form, are now printed; and it is hoped that, even in this state, they may be found of essential service to such of the younger members of our University, as have not the benefit of a private tutor's superintendence, and access to those sources of assistance which such superintendence usually supplies.

In the following pages, a considerable number of the more easy forms have been placed at the beginning of each section, or subdivision: and the examples have been so disposed, as gradually to lead the student to the solution of such as are more complicated in their form, or involve greater difficulties in the investigation. With the exception of a few, it will be found that no principles are involved, which are not contained in Mr Whewell's Elementary Treatise of Mechanics, whose arrangement however has not been entirely adopted.

Many of the Problems which are here offered have been proposed at public and private examinations in the University, and several may be found scattered elsewhere: but the solutions which have accompanied the latter, have been in many instances tedious and perplexed: they are therefore now proposed to exercise the ingenuity of the Cambridge students; who in general are so trained as to combine quickness of invention with accuracy of investigation, extensiveness of reading with solidity of judgment.

5.1. Preface.

The want of some Collection Of Problems in the several branches of Philosophy, to which the Students at the University might refer, and to the solution of which they might be led to apply the principles taught in the College Lectures, having on many occasions been a subject of regret; - the following Collection, originally intended for the use of a few pupils, is now submitted to the Public, with the hope that it may tend in some measure to remedy the defect complained of, and found to be serviceable to the younger members of the University, for whose use it is more particularly intended. In its formation, advantage has been taken of the questions which during a period of more than fifty years have been proposed at the Examinations: - it has been enlarged by the addition of several which may be found scattered elsewhere, and have been adapted or altered to suit the objects of the present publication: - some have been suggested during a long superintendence of these studies; - and others have been pointed out from various sources, or kindly supplied from the stores of private friends. A small volume of Mechanical Problems having been published a short time ago, it has not been thought necessary to include that branch in the present collection, which is arranged on a similar plan. That the whole series has not been accompanied with Solutions, according to an original plan, is owing to a conviction, - confirmed indeed by the opinion of a very judicious and able Tutor now resident in the University, - that in their present form they may be of greater service to the Students.

6.1. Advertisement.

The following pages contain a collection of Algebraical Problems designed solely to point out the various methods employed by Analysts in the Solution of Equations. They are arranged in the usual manner: 1. Simple Equations; 2. Pure Quadratics, and others which may be solved without completing the square; and 3. Adfected Quadratics. Many books have been consulted; and, as utility is the sole object of this Publication, wherever a proper Example occurred, it has been taken without hesitation, or altered to suit the purpose. At the head of each Section are given the common Rules; and the whole concludes with a Collection of Problems without Solutions, for the Exercise of the Learner.

To the Sixth Edition has been added an Appendix containing a Collection of Problems in Arithmetical, Geometrical, and Harmonic Progressions: and another on the nature of Equations, and the solution of those in higher dimensions.