Suggested Grade Level	Books
Grade 12	Ray's Analytic Geometry
Grade 12	Ray's Differential and Integral Calculus
Optional / Additional	Ray's Elements of Astronomy
Optional / Additional	Ray's Surveying and Navigation

This is an extraordinary math text beyond what would be found in any high school anywhere. It is 600 pages and is concerned with the geometry of conics and should be studied by the advanced student. Like the Calculus book below, it was printed in extremely low numbers and as such the copy on this CD-ROM is a photocopy of the single copy found at the Library of Congress, which of the hundreds of Ray's Arithmetic books I have handled, is the only one I have ever seen. 

A partial Table of Contents is presented here: in the book it goes on for 29 pages.

CONTENTS. INTRODUCTION: — THE NATURE, DIVISIONS, AND METHOD OF THE SCIENCE. I. DETERMINATE GEOMETRY: Principles of Notation  Examples Principles of Construction  Examples DETERMINATE PROBLEMS: In a given triangle, to inscribe a square In a given triangle, to inscribe a rectangle with sides in given ratio To construct a common tangent to two given circles To construct a rectangle, given area and difference of sides  Examples II. INDETERMINATE GEOMETRY : I°. Development of its Fundamental Principle: The Convention of Co-ordinates Distinction between Variables and Constants; definition of a Function Equations between co-ordinates: their geometric meaning The Locus defined and illustrated II°. Its Method outlined; in what sense it is Analytic: Manner of employing geometric equations to establish properties Special analytic character of the Algebraic Calculus Elements of analysis added by the Convention of Coordinates III°. Its Divisions and Subdivisions: Algebraic and Transcendental Geometry Orders of algebraic loci: Elementary and Higher Geometry Loci in a Plane and in Space: Geometry of Two and of Three Dimensions BOOK FIRST: —PLANE CO-ORDINATE PART I. ON THE REPRESENTATION OF FORM BY ANALYTIC SYMBOLS. CHAPTER FIRST. THE OLDER GEOMETRY: BILINEAR AND POLAR CO-ORDINATES. SECTION I.—THE POINT. BILINEAR OR CARTESIAN SYSTEM OF CO-ORDINATES : Explanation in detail Expressions for Point on cither Axis; — for the Origin POLAR SYSTEM OF CO-ORDINATES : Expression for the Pole; — for Point on Initial Line Distance, in both systems, between any Two Points in a plane Co-ordinates of Point cutting this distance in a given ratio TRANSFORMATION OF CO-ORDINATES: I. To change the Origin, Axes remaining parallel to their first position II. To change the Inclination of the Axes, Origin remaining the same  Particular Cases: —   1. From Rectangular Axes to Oblique   2. From Oblique to Rectangular   3. From Rectangular to Rectangular III. To change System — from Bilinears to Polars, and conversely IV. To change the Origin, and make cither previous Trans formation at the same time GENERAL PRINCIPLES OF INTERPRETATION: I. Any single equation between co-ordinates represents a Locus II. Any two simultaneous equations represent Determinate Points III. Any equation lacking absolute term, represents Locus passing through Origin IV. Transformation of Co-ordinates does not affect Locus,  nor change the Degree of its Equation SPECIAL INTERPRETATION OF EQUATIONS : Tracing their Loci by means of Points Definitions and illustrations Examples: — Equations to some of the Higher Plane Curves SECTION II. — THE BIGHT LINE. A. THE RIGHT LINE UNDER GENERAL, CONDITIONS. I. Geometric Point of View: — Equation to Right Line is always of the First Degree Equation in terms of angle made by Line with axis X, and of its intercept on axis Y Equation in terms of its intercepts on the two axes Equation in terms of its perpendicular from the Origin, and angle of perp'r with axis X Polar equation, deduced geometrically II. Analytic Point of View: — Every equation of First Degree in two variables represents a Right Line. Proof of the theorem by Algebraic Transformation of the general equation of First Degree Proof by means of the Trigonometric Function implied in the equation Proof by Transformation of Co-ordinates Analytic deduction of the Three Forms of the equation Reduction of Ax + By + C = 0 to the form x cos a + y cos B — p = 0 Polar Equation obtained by Transformation of Co-ordinates B. THE RIGHT LINE UNDER SPECIAL CONDITIONS. Equation to Right Line passing through Two Fixed Points Angle between two Right Lines: condition that they shall be parallel or perpendicular Equation to Right Line parallel to given Line ; — perpendicular to given Line Equation to Right Line passing through given Point, and parallel to given Line Equation to any Right Line through a Fixed Point Equation to Right Line through a given Point, and cutting a given line at given angle Equation to Perpendicular through a given Point Length of Perpendicular from (x,y) on x cos a + y cos B - p = 0 ; also on Ax + By + C = 0 Equation to any Right Line through the intersection of two given ones Meaning of equation L + kL' = 0 Equation to Bisector of angle between any two Right Lines Equation to the Right Line situated at Infinity Equations of Condition :  Condition that Three Points shall lie on one Right Line  Condition that Three Right Lines shall meet in One Point  Condition that Movable Right Line shall pass through a Fixed Point C. EXAMPLES ON THE RIGHT LINE. Examples in Notation and Conditions Examples of Rectilinear Loci SECTION III. — PAIRS OF RIGHT LINES: I. Geometric Point of .View:—Equation to a Pair of Right Lines is always of Second Degree. Formation of equations in the type of L M N .... = 0 : their consequent meaning Interpretation of equation LL' = 0 Equation to Pair of Right Lines passing through a Fixed Point Meaning of the equation Ax2 + 2Hxy + By2 = 0 Angle between the Pair Ax2 + 2Hxy + By2 = 0 Condition that they shall cut at right angles Equation to Bisectors of angles between Ax2- + 2Hxy + By2 = 0 Case of Two Imaginary Lines having Real Bisectors of their angles II. Analytic Point of View: — The Equation of Second Degree in two variables,  upon a Determinate Condition, represents Two Right Lines. Proof of the theorem by the mode of forming LL' = 0 Condition on which Ax2 + 2Hxy + By2 + 2Gx + 2Fy + C = 0 represents Two Right Lines SECTION IV.— THE CIRCLE. I. Geometric Point of View : — Equation to Circle is always of Second Degree Equation to the Circle, referred to any Rectangular Axes, deduced from geometric definition Equation to the Circle, referred to Oblique Axes Equation to the Circle, referred to Rectangular Axes with Origin at Center Equation to the Circle, referred to Diameter and Tangent at its extremity Polar Equation to the Circle II. Analytic Point of View: — The Equation of Second Degree in two variables,   upon a Determinate Condition, represents a Circle. Proof of theorem by comparison of the General Equation with that to Circle

 

This is a first-rate calculus textbook. Some of the notation has slightly changed since it was published, but it is hardly worth mentioning. Calculus, after all, was discovered by Isaac Newton several centuries ago, and the basics of it – measuring continuously changing quantities – have not changed.  Calculus is the language Newton invented in order to do physics.  It is 430 pages. 

The Table of Contents is reproduced here for customer interest.

CONTENTS   THE DIFFERENTIAL CALCULUS CHAPTER I. The Method of Limits Fundamental Propositions Magnitudes Considered as Limits Increments and Derivatives Geometrical Illustrations CHAPTER II Functions—classified Derivatives and Differentials Differential Coefficient Differentiation in general CHAPTER III Differentiation of Functions of one Variable CHAPTER IV. Successive Differentiation Notation Leibnitz's Theorem Cauchy's Theorem CHAPTER V. Taylor's Formula Maclaurin's Formula CHAPTER VI. Convergence of Series CHAPTER VII. Estimation of the Values of Functions which assume Indeterminate Forms CHAPTER VIII. Maxima and Minima Rules for determining the same CHAPTER IX. Differentiation of Functions of two or more Variables Successive Differentiation of Functions of two or more Variables Differentiation of Implicit Functions CHAPTER X. Extension of Taylor's Formula to two Variables Extension of Maclaurin's Formula to two Variables Lagrange's Theorem CHAPTER XI. Maxima and Minima of Functions of two Variables CHAPTER XII. Change of Independent Variable Elimination of Constants and Functions CHAPTER XIII. Tangents and Normals to Plane Curves—Linear Coordinates The same when referred to Polar Coordinates CHAPTER XIV. Asymptotes — General Method of Determining Special Method for Homogeneous Equations Circular Asymptotes CHAPTER XV Differentials of Arcs and Plane Areas Angle of Contact Concavity and Convexity CHAPTER XVI. Curvature Radius of Curvature Contact of Curves Evolutes and Involutes CHAPTER XVII. Singular Points of Curves Tracing of Curves from their Equations CHAPTER XVIIL Envelopes of Plane Curves CHAPTER XIX. Tangents and Normals to Curves of Double Curvature Tangents and Normals to Curved Surfaces Differential Equations of Curved Surfaces Envelopes of Curved Surfaces CHAPTER XX. Curvature of Surfaces THE INTEGRAL CALCULUS CHAPTER I. First Principles CHAPTER II. Direct or Simple Integration Integration by Substitution Integration by Parts CHAPTER III. Integration of Rational Fractions CHAPTER IV. Integration of Irrational Functions Binomial Differentials Formulas of Reduction for Algebraic Functions CHAPTER V. Formulas for Logarithmic Functions Formulas for Trigonometric Functions CHAPTER VI. Approximate Integration Definite Integrals Differentiation and Integration under the sign CHAPTER VII. Rectification of Curves Quadrature of Plane Areas CHAPTER VIII. Quadrature of Surfaces of Revolution Volume of Surfaces of Revolution Quadrature of Surfaces in general Volume of Surfaces in general CHAPTER IX. Integration of Functions of two or more Variables CHAPTER X. General Theory of Differential Equations CHAPTER XI. Differential Equations of the First Order and Degree Factors of Integration CHAPTER XII. Integration by Separation of the Variables CHAPTER XIII. Equations of the First Order and Higher Degrees CHAPTER XIV. Singular Solutions of Equations of the First Order CHAPTER XV. Equations of the Second and Higher Orders Integration by Series Linear Equations CHAPTER XVI. Integration of Simultaneous Equations

All right, I will tell you up front, the astronomy book and the book on surveying and navigation are not for everyone. Let me explain why, and why I have chosen to include them.

First, it was my desire to provide the definitive set of Ray's Arithmetic books. I have done that. An entire math curriculum from beginning to end can be found on these CD-ROMs. These two books – Elements of Astronomy and Surveying and Navigation are the final parts of this incredible series. In the interest of completeness alone I would have published them.

Having said that, they are great and useful textbooks in their own right. Let me discuss each one:

Elements of Astronomy. OK – we have learned lots since this book was published. For instance, Pluto had not been discovered. But then, Pluto is no longer a planet anyway. The textbook speculates on the composition of Saturn's rings, and so on. Detractors will find no end of quarrel with the scientific "facts" in the text. Of course, people in 50 years will have the same problems with whatever text is being published now, so this is not a new problem.

What I do like about it is all the information in it that people really care about. This is not a book for planetary scientists, but for people who want to understand the earth and the stars.  It begins with a simple understanding of the way the world works. What a horizon is and why you see the top of ship first as it comes sailing towards you. It then builds on that to latitude and longitude, then measuring a year, then the motion of earth through space, then the motion of the moon and how it affects tides. How you can determine your position by the stars and find the North Star. How to spot the planets in the sky. What the equinox is. How an eclipse happens and how to predict them.

And so on. You get the picture I am trying paint here. All the things people care about – well, what I care about – are perfectly presented here. Completely accurately – they have not changed at all in the last century. The "edge" of pure science – the gases that make up Jupiter – well, this book is all wrong, but I really don't care.

Now don't get me wrong, this book is not fluff. It is a MATH book. It is considered a math book because it does build a lot on the geometry and Trigonometry. If the student wants to "keep up" with the math that is being done in this book, then he needs to have those subjects under his belt. If he contented to just ride along on the conclusions – like me - then I would use this book whenever he showed an interest in the topic.

I recommend this book for the student who is interested in the night sky and how the world works as it spins its way through God's universe. If your child is of a highly scientific bent and wants to understand the pure science aspects of it all, don't use this book.

Now, on to Surveying and Navigation
Again, same story. Nobody navigates this way any more – too much math. So why bother? And few people become surveyors, though that section of the book is still practical and useful today. Interestingly, surveying was the main reason you took Trigonometry in the first place back then. And the US needed lots of surveyors – a frontier to tame, and continent to fill up.

But why would I include this book? First, it is a great Trigonometry book. The first 174 pages don't even really mention surveying – it is Trigonometry textbook. It is a good follow-on to the Geometry and Trigonometry book on this site. After that, the section on surveying contains great "story problems" about the topic, where the student can apply Trigonometry to real-world situations. Then the section on Navigation has more of the same – more practical application of a somewhat abstract math.

I only recommend this book for the student who is interested in higher math and wants to "take it farther." Kind of like extra credit work. Most people – the vast majority – should not use this book, though.

THE science of Elementary Geometry, after remaining nearly stationary for two thousand years, has, for a century past, been making decided progress. This is owing, mainly, to two causes: discoveries in the higher mathematics have thrown new light upon the elements of the science; and the demands of schools, in all enlightened nations, have called out many works by able mathematicians and skillful teachers.

 

Professor Hayward, of Harvard University, as early as 1825, defined parallel lines as lines having the same direction. Euclid's definitions of a straight line, of an angle, and of a plane, were based on the idea of direction, which is, in deed, the essence of form. This thought, employed in all these leading definitions, adds clearness to the science and simplicity to the study. In the present work, it is sought to combine these ideas with the best methods and latest discoveries of the most distinguished writers on Geometry.

 

By careful arrangement of topics, the theory of each class of figures is given in uninterrupted connection. No attempt is made to exclude any method of demonstration, but rather to present examples of all. In explaining the doctrine of limits, the axiom stated by Dr. Whewell is given in the words of that eminent scholar.

 

The books most freely used are, "Cours de Geometrie elementaire, par A. J. H. Vincent et M. Bourdon;" "Geometrie theorique et pratique, etc., par H. Sonnet;" "Die reine elementar-mathematik, von Dr. Martin Ohm;" and "Treatise on Geometry and its application to the Arts, by Rev. D. Lardner."

 

Occasional use has been made of the works of Schlomilch, Briot, Bobillier, Leslie, and Lund; also, of Blanchet's edition of Legendre, Prouhet's edition of Lacroix, and Law's and Lardner's editions of Euclid. The works of Bland, Colenso, and Potts have been used for the selection of exercises.

 

A large portion of the work has been used in class instruction, and the author takes pleasure in acknowledging that many valuable hints have been received from his pupils.

 

The subject is divided into chapters, and the articles are numbered continuously through the entire work. The convenience of this arrangement for purposes of reference has caused it to be adopted at the present day by a large majority of writers upon Geometry, as it had been already by writers on other scientific subjects.

 

If this treatise shall contribute to render the study of Geometry less complex, and more interesting to the student, and hence lead to a wider cultivation and more thorough knowledge of this branch of mathematics, the chief object of the author in its preparation will have been accomplished.

 

ELI T. TAPPAN.

 

MOUNT AUBURN INSTITUTE, March 1864.