instruction have proved, as hundreds of individuals would bear testimony, that the theorem here given will save, at the lowest estimate, two-thirds of the labor ordinarily incurred by the rectangular method. A further advantage is that, dispensing with a large and faulty table altogether, it is far more accurate—the computations being executed by aid of the common logarithmic numbers, calculated with greater care and usually extending to six or seven deci. mal places, and the operation being so ordered that, without any additional labor beyond what is absolutely essential to an honest confidence in the result, all gross errors, if any exist, whether of the field or the tables, are detected, and if these have no existence, the smaller and unavoidable ones very much reduced. It is hoped that this book, often requested by my pupils, will prove acceptable to the schools generally. Of the third Part, embracing the mensuration of solids, spherical trigonometry, and navigation, time will permit us to say little more than that, by the method pursued, we have been enabled, within moderate limits, to give a fuller development of these subjects than is usually found in our elementary books. The modification and extension of Napier's Rules demands, how ever, a brief historic notice. I demonstrated and extended these rules by showing : I. When A = 90°, 2. = 90°-Q, B. = 90°- B, C. = 90° - C, sinb = cosac cos B. = tan C. tanc, tana, tanb: sinB = cos A, cosb, = tanc, tan C., since = cosb, cos C. = tan A, tanB.: III. When c= a, or the triangle is isosceles, sinac = tan Atan(1B). sin A= tana, tan(+b), sin(+b) = cosa, cos(B).. sin (1 B) = cos A, cos(16). Having shown the above extension to Mr. Dascum Green, then a pupil, he returned soon after, saying that he had not only verified my forms, but had obtained better ones, and presented the modi. fications of Napier's Rules as I had extended them, substantially as they will be found in the text. This rule, as now extended and modified, possesses a greater simplicity and symmetry, and will enable us, in spherical astronomy, frequently to dispense with complicated figures. I have added a small set of tables, extending to seven decimal places, calculated to answer the wants of the student while pursuing the work, and to make him more ready in using tabular numbers, by compelling him to interpolate by second differences. Afterwards he will find it decidedly to his advantage to possess himself of the tables recently published by Professor Stanley. In conclusion, the advantages which we have endeavored to secure are : 1o. A better connected and more progressive method of geometrizing, calculated to enable the student to go alone. 2o. A fuller, more varied, and available practice, by the introduction of more than four hundred exercises, arithmetical, demonstrative, and algebraical, so chosen as to be serviceable rather than amusing, and so arranged as greatly to aid in the acquisition of the theory. 3o. The bringing together of such a body of geometrical know. ledge, theoretical and practical, as every individual, laying any claim to a respectable education or entering into active life, demands. 4o. The furnishing to those who may wish to proceed on in mathematical learning, of a stepping-stone to higher and more ex. tended works. How well we have accomplished our object it is not, of course, for us to say. We have endeavored to render the work as mechan. ically correct as possible, but, residing at a distance from the place of publication, we can hardly expect that it will be entirely free from typographical imperfections. G. C. W. Genesee Wesleyan Seminary, June, 1848. CONTENTS. SECTION I. Use of the Signs Fractions Simple Equations. 1. Definitions. Mathematics, quantity, proposition. 2. Explanation of the signs to X,:, &c..... 4. Inversions of additions, subtractions, &c... 5. Coëfficients added and subtracted.. 7. Changing the sign of a factor - Powers... 8. Square of polynomial, binomial, residual — Product of sum and dif- 9. Changing the sign of a polynomial... 10. Degree of product, multiplication and division of powers. 11. Multiplying or dividing dividend or divisor..... 12. A fraction an expression of division..... 13. Multiplying or dividing numerator or denominator.. 14. Multiplying a fraction by its denominator.... 15. Fractions multiplied together. ...... 16. Involution and evolution of fractions... 17. Division by a fraction-reciprocal.. 18. Reduction to given denominator... 19. Addition and subtraction of fractions... 20. Fractions cleared of subdenominators.. 21, Scholia - Signs, common factors, &c... 22. Equations Defined --Identical, &c... • SECTION II. Exponents--Proportion-Variation. 1. Reciprocal powers.. 2. Root of a power-extension.. 8. Fractional exponents.. ....... 40 41 41 ........ PAGE 4. All real exponents subject to the same rules.... 5. Ratio defined proportion-homologous and analogous terms--antece- 6. Inversions-compositions---product of extremes and means involution and evolution--equal multiples. .... 7. Inverse or reciprocal proportion.. 8. Continued proportion-geometrical progression .... 9. Variation defined-notation....... 10. Inversion - composition-involution-multiples comparison --com- SECTION III. Analysis of Equations. 1. A single equation resolvable into several distinct equations. Ex- 2. Quadratics-rule for solving-sum and product of roots. 3. Classes of biquadratics solvable as quadratics... 4. Rule for putting problems into equation ..... 5. Discussion of the two values of the unknown quantity.. PLANE GEOMETRY DEPENDING ON THE RIGHT LINE. 63 63 64 64 65 Section I. Comparison of Angles. 1. Definitions. Geometry--solids, surfaces, lines.. 2. Straight line, nature, origin of notion of....... 3. Corollaries-scholiam on parallels... 4. Applications --Straightedge, parallel edges, plane. 5. Sum of adjacent angles constant, method.... 6. Corollaries-right angles, sum of adjacent angles, lines forming one and the same straight line, vertical angles, sum of angles round a point... 7. Applications—Rightangle, Surveyor's Cross... 8. Parallels, method, reversion and superposition .... 9. Corollaries——the converse, conditions determined by the equality of alternate angles, secant perpendicular, lines parallel to the same, angles having parallel sides, angles of parallelogram 10. Application --drawing parallels.. . 11. Sum of external angles of polygon... 12. Corollaries-sum of internal angles of polygon, hexagon, pentagon, quadrilateral, triangle, sum of acute angles of right angled triangle, the external angle formed by producing one side of a triangle. Scholium. 13. Exercises.. 66 67 68 69 69 70 70 71 SECTION II. Equal Polygons-First Relations of Lines and Angles. 1. Polygons, when equal, how proved..... 2. Corollaries-equal triangles, parallelogram divided by a diagonal, distance of parallels, diagonals of parallelogram, isosceles triangle bisected, equilateral triangles. 3. Relation of angles in a triangle of unequal sides.. 4. Corollaries—the converse, a triangle having two equal angles—three. 73 5. Relation of two sides of a triangle to the third...... 6. Consequences--perimeters enveloped and enveloping, the shortest dis- tance from one point to another, shortest distance from a point to a 7. Triangles having two sides of the one equal to two sides of the other, each to each, but the included angles unequal....... 8. Consequences-triangles having their sides severally equal, a quadri- SECTION III. Proportional Lines. 1. Segments of lines intercepted by parallels .... 2. Consequences—similar triangles, &c. .... 3. A right angled triangle divided by a perpendicular.. 4. Consequences relation of the perpendicular to the segments of the hy- pothenuse, &c., square of the hypothenuse, &C..... 5. Relation between the oblique sides of a triangle, the line drawn from the vertex to the base, and the segments of the base~consequences. 82 6. Distance of foot of perpendicular to middle of base. SECTION IV. Comparison of Plane Figures. 1. Rectangles-consequences, measure, right angled triangle........ 2. Trapezoid-consequences, measures, parallelogram, triangle, compar- 3. Triangles having an angle of the one equal to an angle of the other- 4. Exercises... 93 94 BOOK THIRD. PLANE GEOMETRY DEPENDING ON THE CIRCLE, ELLIPSE, HYPERBOLA, AND PARABOLA. i SECTION I. The Circle. 1. Definitions consequence..... 2. Angles at the centre of equal circles-consequences, measures. 101 102 |