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ARITHMETIC introductory to Geometry

2 PLANES, embracing the Eleventh

Uses of the Science

3 Book of Euclid


The Ten Figures, and their local value 6 SPHERICAL GEOMETRY


The Simple Rules of Arithmetic 7-19

The Compound Rules .




26-30 Permanence of Equivalent Forms.. 261

Proportion, and Rule of Three . 31, 33

On the Theory of Indices



36-38 On Impossible Expressions .


Extraction of the Square Root


The Convergent and Divergent Series

Table of Factors


264, 265

The Binomial Theorem 266, 269 et seq.


43 On the Calculation of Logarithms . . 275

Elements, &c., of Euclid

43-48 Calculation of the Arithmetical Values

Propositions of Book I.

48—68 of Quantities expressed by Infinite

Commentaries on, and Exercises , 69, 85 Series


Propositions, &c., of Book II. 87-92 How Logarithms facilitate Calcula-
of Book III. 97–112 tions

276, 277
of Book IV. 116-124 Practical Advantage of Calculating

The Quadrature of the Circle

126 Logarithms to the Base

10, 278

Exercises in the First Four Books of How every number has a Calculable


128 Logarithm


PROPORTION, intended as a Substitute How Value may be derived from the

for Euclid's Fifth Book

129 Logarithmic Series


Propositions, &c., of Book VI. 144-158 The Numerical Value of Logarithms 281

Use of a Table of Logarithms. . 283, 284

ALGEBRA; its general Principles . 161 Method of Finding ihe Characteristic 285

Addition and Subtraction . 166, 168 To Find the Logarithm of a Number

Multiplication and Division 179, 187

not given in the Tables


Exponents, Roots, and Surds 191 Various Problems for Finding Loga-
193 rithmic Numbers



196—201 Method of Finding any Power of a


201-217 Number

289, 291

Ratio and Proportion


Arithmetical and Geometrical Pro- PLANE TRIGONOMETRY



220—224 On representing Lines and Angles by

Extraction of Square Roots 228-232 Numbers


To Extract the Cube Root of Definition of the Science . 292

Compound Quantities.


The Circular Measure of an Angle 293

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Definitions of the Trigonometrical The Distance between two Points . . 369

Sines and Ratios . . . 294, 298 The Mensuration of Areas


The Use of the Negative Sign to

To find the Area of a Rectangle 371

denote Position

298 Of a Trapezoid ..

. 372

Extension of the Definitions of Angles Areas of Trapeziums and Polygons 373

and Trigonometrical Ratios 298, 300 To determine the area of a Circle and

On the Magnitudes of the Trigo-

its Segments

374, 375

nometrical Functions of Angles .

301 Of an Ellipse

. 376

Angles which have the same Sine or Of a Parabola



303 Of a Plane Figure bounded by a Curve 378

The Trigonometrical Functions of dif- Of a Curvilinear Figure


ferent Angles


Of a right Prism, of a Cylinder, and of

Methods of .proving the different For- a Cone.

· 380, 381


305, 308 Of the Surface of a Sphere

. 382

Relation between the Four Funda- The Mensuration of Solids 383, 384

mental Formulas or Expressions 309 The Equality of Parallelepids 385, 386

The Sines and Cosines of Multiples of

Volume of Prism


given Angles

310 Volume of the Pyramid .

388, 389

Ambiguities arising from the use of Volume of a Frustum of a right

Formulas .



Numerical Value of Sines, Cosines, &c. 315 Volume of a Prismoid


Trigonometrical Ratios of Angles 316, 317 | To Find the Solid Content of a Rail-

Formulas connecting inverse Trigono- way Cutting


metrical Ratios

318, 319 Of a Military Earthwork


The Use of Subsidiary Angles 320, 321 Volume of Frustum of a right Cone

Relation between the sides and Angles

Volume of a Sphere

397, 393

of Triangles


Deduced and Derived Formulas 323 SPHERICAL TRIGONOMETRY


Formula for Logarithmic Calculation S25 Sines of the Angles of a Spherical


326, 329 Triangle


On Trigonometrical Series and

Fundamental Formulas



330—530 To prove the Formulas

Method of checking the Calculations . Jil Napier's Analogies


Calculation of Tangents and Cotan- The Solution of Right-angled Spheri-

gents, &c.

352 cal Triangles


Tables of Natural Sines



Tables of Logarithmic Sines .

355 Instruments for Use


Logarithmic Sines of small Angles 357, 359

Delambre's Method



Numerical Solution of Right-angled Properties of the Ellipse



361–366 Construction of an Oval



367 The Parabola


On Heights and Distances

. 367 | The Hyperbola


On determining the Height of



a Tower.

367, 368

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Introductory.-THE present volume is to be devoted to that branch of study implied in the term MATHEMATICS,-a term which comprehends one of the most extensive and important departments of human knowledge. By most people, it is considered also as one of the most difficult departments; and many, with time and talents for the task, are deterred from entering upon a study which would amply repay the expenditure of both, by this mistaken prejudice. Every science, no doubt, has its hard and knotty points; and in no intellectual pursuit can distinction be attained without labour, thought, and perseverance; yet if there be one subject of scientific inquiry which, more than any other, is distinguished by the simplicity, certainty, and obviousness of its fundamental principles,-by the irresistible evidence by which position after position is established, and by the systematic gradations by which layer after layer of the intellectual structure is completed,— that subject is Mathematics.

In other topics of research, there is generally more or less of hypothesis, or conjecture: there are obscure recesses, into which the light of truth and demonstration cannot penetrate, and where fancy and imagination are sometimes permitted to guide our steps. But there are no perplexities of this kind in mathematics,-no ingenious theories to mislead, and no conflicting opinions to bewilder; our progress here is exclusively under the unerring direction of TRUTH herself; and it is her torch alone that lights up the path.

Whatever, therefore, may be the difficulties connected with the study of mathematics, it is plain that they do not arise from our having to grope our way in darkness and uncertainty; the asperities of the road are as clearly revealed before us as the level and


unobstructed track; and all that the earnest student requires, is some friendly hand, to aid him in surmounting these in the earlier stages of his progress.

It is this sort of aid that we here propose to supply. We do not undertake to conduct the scientific inquirer through the entire regions of mathematical research-ours is a far less ambitious aim: we write for the young—for the self-dependent-the solitary-and, perchance, the unfriended student. The office we here take upon ourselves will be performed, if we succeed in the endeavour to assist him. This is the only object at which we now aim, and we think it right thus explicitly to declare it, in order to forewarn those who may desire information on the more recondite researches of science, that the present volume is not intended for them.

It may be proper 'to mention, however, that although we now propose to limit our labours to an: exposition of the elementary principles of mathematical learning, and to economize space as much as possible; yet, within the bounds prescribed, we shall take care that every subject receive a full and fair elucidation, and that it be discussed to an extent amply sufficient for the purposes of general education. We hope, too, by avoiding all attempts at magisterial dignity of style, and addressing our readers in the familiar language of social intercourse, to secure their attention, and win their confidence ; and that we may be fortunate enough, by clearness and simplicity of explanation, to awaken in some a genuine love for science, and a desire to prosecute their researches in writings of wider scope and higher pretensions: we shall endeavour to gratify such desire in a subsequent volume.

We have thought it advisable to commence our work on elementary Mathematics with a preliminary treatise on Arithmetic--the groundwork upon which the entire system, with the exception of pure geometry, ultimately rests. Books on arithmetic, however, are so numerous, and so easily accessible, that we might have been held excused from introducing so hackneyed a subject into a work which, though confessedly of the most elementary character, is, nevertheless, intended to embrace a range of topics beyond the ordinary limits of a school-boy course. But it unfortunately so happens, that books on arithmetic, with few exceptions, are little more than mere depositories of practical rules and mechanical operations; and are, therefore, but ill-suited to prepare the young for that higher kind of exertion -higher, because more intellectual—which science, properly so called, always demands.

A boy who has gone through his “Walkingame,” and who, as matter of course, is then introduced to “ Euclid,” is naturally enough bewildered by the total dissimilarity of the two authors—not from difference of subject, but of manner of exposition: the former has abundantly supplied him with rules, but no reasons; the latter gives him reasons, but no rules : the one has loaded his memory, and employed his fingers; while the other appeals to his judgment, and exercises his understanding.

To make, in this way, the passage from arithmetic to geometry, an abrupt transition from the mechanical to the intellectual, we conceive to be a capital defect in educational training. Arithmetic is as much a science as geometry : there is not a rule in the one, any more than there is a theorem in the other, that is not founded on reason, and demonstrably true. And even viewing arithmetic merely in reference to its practical utility in commercial affairs, to the demands upon it in the counting-house and the shop,—we still contend that its principles should be rationally taught-not authoritatively declared, inasmuch as that which has engaged the understanding, and been received from a conviction of its truth, is more securely retained in the memory than what is committed to it by rote, Rules, unsupported by reasons, are hard to learn, and hard to remember; but, when the


practical precept is associated in the mind with the theoretical principle on which it depends, we learn and remember with ease and satisfaction; for, instead of words, we get knowledge.

These considerations have prevailed with us; and have determined us to render this course of elementary mathematics complete, by commencing at the very foundation. It must be borne in mind, however, that our design is not to exhaust here any subject of which we treat: it is rather to excite an appetite for knowledge, than to satiate it. It will not be expected, therefore, that our treatise on arithmetic is to be co-extensive with what, under that name, is usually put into a schoolboy's hands; the bulk of such books arises, in a great measure, from the system of instruction condemned in the preceding observations ; the object of which system seems to be to inculcate a knack of readily applying rules, by experimenting upon numerous examples, under the guidance of the prescribed directions. This, as the reader has already been made aware, is not our object; we propose to explain principles, and to furnish the reasons that justify the rules; persuaded that, if the former be thoroughly apprehended, there need be but little anxiety felt about the mere verbal memory of the latter.

The subjects to be treated of in the present volume are as follows:-Arithmetic, Geometry, Algebra; Logarithms and Series; Probabilities, and the Principles of Life Assurance; Trigonometry, Conic Sections, Mensuration, Differential Calculus, Integral Calculus; Applications of the last two subjects to Mechanical and Physical inquiries; and a short treatise on the Theory of Equations.

It would extend these introductory remarks far beyond the space that can be allotted to them, to enter into any detailed account here of the several particulars to be introduced under the above-mentioned heads; but we cannot conclude them without a few words more especially addressed to those who have resolved to place themselves under our instructions.

It is a common thing with young students in science to be frequently making inquiries as to the use of what they are learning. “What is the use of this?” is a question put at every turn; generally to the annoyance of the teacher, and often to the discredit of the learner.

The use of any intellectual pursuit-employing the term use in its higher and more honourable signification—is to be realised in the mental satisfaction and the mental elevation it communicates. You do wrong to estimate science solely and exclusively in proportion as it visibly contributes to our animal wants and enjoyments; there is an intellectual pleasure in the very process of acquiring knowledge, while the conscious possession of it raises the human being in the scale of creation, and thus enables him to contemplate its wonders from a more exalted position. It is in this way that knowledge, like virtue, to which indeed it is allied, is said to be its own reward; for the study of science is accompanied with gratifications of the purest and loftiest kind; and is productive of advantages to the student, altogether distinct from the benefits conferred by its applications to the practical purposes of life; it invigorates and enlarges the faculties refines and elevates the desires and adorns and dignifies the entire character, withdrawing our thoughts from what is mean and degrading, and inclining them to the noblest and worthiest of objects to the love and veneration, and therefore to the practice, of TRUTH,

These advantages, though unconnected with outward and tangible results, are surely too precious to be entirely overlooked in any correct estimate of the value of scientific

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