Preface.

Coordinate Geometry is the basis of the modern or Analytical method in Mathematical Science. Not only does it afford, in its applications, the best possible exercises in Algebraic reasoning; but, by familiarizing the mind with the ideas of variables and functions, and graphically illustrating them, it is the suitable preparation for the study of the Differential Calculus and the higher branches of Algebra. I have endeavoured to adapt this treatise to the use, both of those who wish to study the Conic Sections by the coordinate method, and of those who intend to pursue the higher branches of Analysis.

The common rectangular equations of the Ellipse, the Parabola and the Hyperbola have been derived from the usual and familiar definitions of these curves; and demonstrations of their properties, founded exclusively upon these equations, occupy the first parts of Chapters V, VI and VII. In this portion of the work, extensive application has been made of the "eccentric angle" in the ellipse, and, in the case of the hyperbola, of a similar auxiliary angle, of which the use is suggested in Salmon's 'Treatise on Conic Sections'. In the Chapter on the Hyperbola, the most useful properties and equations of the three conic sections have been generalized, with especial reference to the manner in which these curves present themselves in Astronomy.

The latter parts of the same Chapters are devoted to more general equations of the curves, and to their discussion by means of the method of combined equations, which is the basis of the "abridged notation." In Chapter VIII, the general equation of the second degree is treated by the method of transformation of coordinates. It has been my object, in this Chapter, to give methods of deriving from an equation in the general form all the circumstances which relate to the position and form of the curve; and also to discuss all the special forms of the equation, not examined in the previous Chapters. If it be desired to abridge the course by leaving out the topics referred to in this paragraph, the student may omit the following portions, which are not necessary to the perusal of the remainder of the work: Chapter III; Chapter IV from Art. 123 to the end ; Chapter V from Art. 151; Chapter VI from Art. 202; Chapter VII from Art. 254; and the whole of Chapter VIII.

In Chapter IX, I have attempted to classify the principles employed in finding the equations of Geometrical Loci, and to explain and illustrate them fully by examples solved in the text, and a carefully graduated series of examples for practice. Chapter X contains a full explanation of the notation employed in Solid Geometry, its application to the plane and the straight line, and a brief notice of the various surfaces of the second degree. It is believed that this Chapter will be found an adequate preparation, in this branch, for the study of standard authors in Analytical Mechanics.

The knowledge of Geometry and Trigonometry essentially prerequisite to the commencement of this study is very slight. I have therefore avoided demonstrations founded upon any geometrical principles or trigonometric formulae except those fundamental ones mentioned in Art. 2.

Nothing tends to impress mathematical principles and methods upon the mind so thoroughly as the solution of numerical examples, with their verifications. Such solutions and verifications will be found in the text of those Articles in which practical methods are explained. Unsolved examples both of a numerical and of an algebraic character are added at the ends of the Articles or sections. To the more difficult examples, especially to those in which algebraic demonstrations are required, are appended hints for their solution.

April, 1869.

Preface.

The difficulties usually encountered on beginning the study of the Differential Calculus, when the fundamental idea employed is that of infinitesimals or that of limits, together with the objectionable use of infinite series involved in Lagrange's method of derived functions, have induced several writers on this subject to return to the employment of Newton's conception of rates or fluxions. The readiness with which this conception is grasped, and the precision it gives to the preliminary definitions, promise an advantage which, however, is in most cases sacrificed by resorting to the use of limits in deducing the formulas for differentiation.

These considerations induced the authors of this work to seek to derive the differentials of the functions by an analytical method founded upon the notion of rates, and entirely independent of the difficult conceptions of infinitesimal increments and of limiting ratios.

The investigation thus initiated resulted in a satisfactory method of obtaining the differentials of the simple functions, which was embodied in a paper communicated to the American Academy of Arts and Sciences, January 14, 1873, by Professor J M Peirce, and published in the Proceedings of the Society. This paper was subsequently rewritten and published as a pamphlet. [On a new Method of obtaining the Differentials of Functions, with especial reference to the Newtonian conception of rates or velocities, by J Minot Rice and W Woolsey Johnson (D Van Nostrand, Publisher, New York, 1875).]

A complete resume of the original paper, by J W L Glaisher, of Trinity College, Cambridge, was published in August, 1874, in the Messenger of Mathematics, Macmillan & Co., London; vol. iv, page 58.

The method alluded to may be briefly described thus: - Denoting an assumed finite interval of time by $dt, dx$ is so defined that $\large\frac{dx}{dt}\normalsize$ is the expression for the rate of $x$ (page 12); and, after establishing a few elementary propositions, which are immediate consequences of the definitions, it is shown (page 17) that, when $y$ is a function of $x, \large\frac{dy}{dx}\normalsize$ has a value independent of the assumed finite value of $dx$. For the mode in which the differentials may be directly deduced, with the aid of this important proposition, the reader is referred to page 23 and to page 39.

It is not the intention of the authors to disparage the use of the limit as an instrument of mathematical research. It is only claimed that the difficulties attending the employment of this notion are so great as to render it desirable to avoid introducing it into the fundamental definitions of a subject necessarily involving many other conceptions new to the student.

The distinction between the view of the differential calculus here presented, and that found in most of the standard works on the subject hitherto published, may be stated thus: - The derivative $\large\frac{dy}{dx}\normalsize$ is usually defined as the limit which the ratio of the finite quantities $\Delta y$ and $\Delta x$ approaches when these quantities are indefinitely diminished: when this definition is employed, it is necessary, before proceeding to kinematical applications, to prove that this limit is the measure of the relative rates of $x$ and $y$. In this work the order is reversed; that is, $dx$ and $dy$ are so defined that their ratio is equal to the ratio of the relative rates of $x$ and $y$, and in Chapter XI, by applying the usual method of evaluating indeterminate forms, it is shown that the limit of $\large\frac{\Delta y}{\Delta x}\normalsize$, when $\Delta x$ is diminished indefinitely, is equal to the ratio $\large\frac{dy}{dx}\normalsize$. Thus the employment of limits is put off until we are prepared to show that the limit has a definite value capable of expression in a language already familiar to the student.

The early introduction of elementary examples of a kinematical character (see pp. 28, 37, and 57; also. Section X, p. 76) which this mode of presenting the subject permits, will be found to serve an important purpose in illustrating the nature and use of the symbols employed.

The method of treating limits employed in Section XXXVIII is a modification of the method given in M J Bertrand's Traité de Calcul différentiel et intégral, pages 41 and 136, edition of 1864. To this valuable work we are also indebted for many other suggestions.

The determination of the maxima and minima values of functions, and the evaluation of indeterminate forms, are so treated as to be independent of Taylor's theorem. In the investigation of these subjects, and in many other cases, we have found it desirable to substitute for the demonstrations commonly given others more in harmony with the conception of rates.

It has been found necessary to devote an unusually large amount of space (from page 230 to page 415) to the geometrical applications of the differential calculus, in consequence of the lack of available text-books on Curve Tracing and on Higher Plane Curves. These pages include Chapter IX, which consists of a brief discussion of the equations and many of the properties of the best-known higher plane curves, and is introduced chiefly for convenience of reference. We trust this chapter will be found a very useful feature of the book.

To facilitate the use of this work as a text-book, it has been divided into short sections, each of which is followed by a copious collection of examples. In the arrangement of these examples the order of subjects in the section has been generally followed, and easy examples usually precede the more difficult ones.

It will not in general be found advisable for the beginner to solve all the examples on reading this work for the first time. They occupy nearly one third of the entire volume, and are intended as a collection from which the instructor may select at his discretion.

Many of these examples were prepared especially for this work. The others were taken chiefly from the collections of Gregory (Walton's edition), Frenet, and Tisserand; and from the treatises of Todhunter, Williamson, and Connel.

J. M. R.
W. W. J.
ANNAPOLIS, MARYLAND,
September 1879.

Preface.

This work, as at present issued, is designed as a shorter course in the Integral Calculus, to accompany the abridged edition of the treatise on the Differential Calculus, by Professor J Minot Rice and the writer. It is intended hereafter to publish a volume commensurate with the full edition of the work above mentioned, of which the present shall form a part, but which shall contain a fuller treatment of many of the subjects here treated, including Definite Integrals, and the Mechanical Applications of the Calculus, as well as Elliptic Integrals. Differential Equations, and the subjects of Probabilities and Averages. The conception of Rates has been employed as the foundation of the definitions, and of the whole subject of the integration of known functions. The connection between integration, as thus defined, and the process of summation, is established in Section VII. Both of these views of an integral - namely, as a quantity generated at a given rate, and as the limit of a sum - have been freely used in expressing geometrical and physical quantities in the integral form.

The treatises of Bertrand, Frenet, Gregory, Todhunter, and Williamsom, have been freely consulted. My thanks are due to Professor Rice for many valuable suggestions in the course of the work, and for performing much the larger share of the work of revising the proof-sheets.

W. W. J.
U. S. NAVAL ACADEMY,
July 1881.

Preface.

This book relates, not to the general theory of curves, but to the definite problem of ascertaining the form of a curve given by its equation in Cartesian coordinates, in such cases as are likely to arise in the actual applications of Analytical Geometry. The methods employed are exclusively algebraic, no knowledge of the Differential Calculus on the part of the reader being assumed.

I have endeavoured to make the treatment of the subject thus restricted complete in all essential points, without exceeding such limits as its importance would seem to justify. This it has seemed to me possible to do by introducing at an early stage the device of the Analytical Triangle, and using it in connection with all the methods of approximation.

In constructing the triangle, which is essentially Newton's parallelogram, I have adopted Cramer's method of representing the possible terms by points, with a distinguishing mark to indicate the actual presence of the term in the equation. These points were regarded by Cramer as marking the centres of the squares in which, in Newton's parallelogram, the values of the terms were to be inscribed; but I have followed the usual practice, first suggested, I believe, by Frost, of regarding them merely as points referred to the sides of the triangle as coordinate axes. It has, however, been thought best to return to Newton's arrangement, in which these analytical axes are in the usual position of coordinate axes, instead of placing the third side of the triangle, like De Gua and Cramer, in a horizontal position.

The third side of the Analytical Triangle bears the same relation to the geometrical conception of the line at infinity that the other sides bear to the coordinate axes. I have aimed to bring out this connection in such a way that the student who desires to take up the general theory of curves may gain a clear view of this conception, and be prepared to pass readily from the Cartesian system of coordinates, in which one of the fundamental lines is the line at infinity, to the generalized system, in which all three fundamental lines are taken at pleasure.

Lists of examples for practice will be found at the end of each section. These examples have been selected from various sources, and classified in accordance with the subjects of the several sections.

W. W. J.
U. S. NAVAL ACADEMY,
November, 1884.

Preface.

The treatment of the subject of Differential Equations here presented will, it is hoped, be found complete in all those portions which bear upon their practical applications, and in the discussion of their theory so far as it can be adequately treated without the use of the complex variable. The topics included and the order pursued are sufficiently indicated by the table of contents. An amount of space somewhat greater than usual has been devoted to the geometrical illustrations which arise when the variables are regarded as the rectangular coordinates of a point. This has been done in the belief that the conceptions peculiar to the subject are more readily grasped when embodied in their geometric representations. In this connection the subject of singular solutions of ordinary differential equations and the conception of the characteristic in partial differential equations may be particularly mentioned. Particular attention has been paid to the development of symbolic methods, especially in connection with the operator $x.\large\frac{d}{dx}\normalsize$, for which, in accordance with recent usage, the symbol $<i><curlytheta> </i>$ has been adopted. Some new applications of this symbol have been made.

The expression "binomial equations" is applied in this work (in a sense introduced by Boole) to those linear equations which are included in the general form $f_{1}(<i><curlytheta></i> )y + x^{s} f_{2}(<i><curlytheta></i> )y = 0$, and which constitute the class of equations best adapted to solution by development in series. In the sections treating of this method a uniform process has been adopted for the secondary or logarithmic solutions which occur in certain cases. The development of the particular integral when the second member is a power of $x$ is also considered. Chapter VIII is devoted to the general solution of the binomial equation in the notation of the hypergeometric series, and Chapter IX to Riccati's, Bessel's and Legendre's equations.

The examples at the ends of the sections have been derived from various sources, and not a few prepared expressly for this work. They are arranged in order of difficulty, and the solutions are given. These have been verified in the proof-sheets, so that it is believed that they will be found free from errors.

The ordinary references in the text are to Rice and Johnson's 'Differential Calculus' and Johnson's 'Integral Calculus', published by John Wiley and Sons uniformly with the present volume.

W. W. J.
U.S. NAVAL ACADEMY,
May, 1889.

Introductory.

Errors of Observation.

1. A quantity of which the magnitude is to be determined is either directly measured, or, as in the more usual case, deduced by calculation from quantities which are directly measured. The result of a direct measurement is called an observation. Observations of the kind here considered are thus of the nature of readings upon some scale, generally attached to an instrument of observation. The least count of the instrument is the smallest difference recognized in the readings of the instrument, so that every observation is recorded as an integral multiple of the least count.

2. Repeated observations of the same quantity, even when made with the same instrument and apparently under the same circumstances, will nevertheless differ materially. An increase in the nicety of the observations, and the precision of the instrument, may decrease the discrepancies in actual magnitude; but at the same time, by diminishing the least count, their numerical measures will generally be increased; so that, with the most refined instruments, the discrepancies may amount to many times the least count. Thus every observation is subject to an error, the error being the difference between the observed value and the true value; an observed value which exceeds the true value is regarded as having a positive error, and one which falls short of it as having a negative error.

3. An error may be regarded as the algebraic sum of a number of elemental errors due to various causes. So far as these causes can be ascertained, their results are not errors at all, in the sense in which the term is here used, and are supposed to have been removed by means of proper corrections. Systematic errors are such as result from unknown causes affecting all the observations alike. These again are not the subjects of the "theory of errors," which is concerned solely with the accidental errors which produce the discrepancies between the observations.

Objects of the Theory.

4. It is obvious that when a set of repeated observations of the same quantity are made, the discrepancies between them enable us to judge of the degree of accuracy we have attained. Speaking in general terms, of two sets of observations, that is the best which exhibits upon the whole the smaller discrepancies. It is obvious also that from a set of observations we shall be able to obtain a result in which we can have greater confidence than in any single observation.

It is one of the objects of the theory of errors to deduce from a number of discordant observations (supposed to be already individually corrected, so far as possible) the best attainable result, together with a measure of its accuracy; that is to say, of the degree of confidence we are entitled to place in it.

5. When a number of unknown quantities are to be determined by means of equations involving observed quantities, the quantities sought are said to be indirectly observed. It is necessary to have as many such observation equations as there are unknown quantities. The case considered is that in which it is impossible to make repeated observations of the individual observed elements of the equations. These may, for example, be altitudes or other astronomical magnitudes which vary with the time, so that the corresponding times are also among the observed quantities. Nevertheless, there is the same advantage in employing a large number of observation equations that there is in the repetition of direct observations upon a single required quantity. If there are $n$ unknown quantities, any group containing $n$ of the equations would determine a set of values for the unknown quantities; but these values would differ from those given by any other group of $n$ of the equations.

We may now state more generally the object of the theory of errors to be, when given more than n observation equations involving $n$ unknown quantities, the equations being somewhat inconsistent, to derive from them the best determination of the values of the several unknown quantities, together with a measure of the degree of accuracy obtained.