ROBINSON S MATHEMATICAL SERIES. CONIC SECTIONS AND ANALYTICAL GEOMETRY; THEORETICALLY AND PRACTICALLY ILLUSTRATED, BY HORATIO N. ROBINSON, LL.D., LATE PROFESSOR OF MATHEMATICS IN THE U. 8. NAVY, AND AUTHOR OF A FULL OF MATHEMATICS. NEW YOEK: IVISON, PHINNEY & COMPANY, 48 & 50 WALKER STREET. CHICAGO: S. C. GRIGGS & COMPANY, 89 & 41 LAKE STBEET. 1863. R Engineering Library OBINSON S u?LET$, M,ost ERA.CTICAL, and most SCIENTIFIC SERIES of MATHEMATICAL TEXT-BOOKS ever issued in this country. I. Bobinson s Progressive Table Book, $ 12 II. Bobinson s Progressive Primary Arithmetic, - - - - 15 III. Bobinson s Progressive Intellectual Arithmetic, 25 IV. Bobinson s Budiments of Written Arithmetic, 25 V. Bobinson s Progressive Practical Arithmetic, 56 VI. Bobinson s Key to Practical Arithmetic, 50 VII. Bobinson s Progressive Higher Arithmetic, 75 VIII. Bobinson s Key to Higher Arithmetic, 75 IX. Bobinson s New Elementary Algebra, ----- 75 X. Bobinson s Key to Elementary Algebra, 75 XI. Bobinson s University Algebra, 1 25 XII. Bobinson s Key to University Algebra, 1 00 XIII. Bobinson s New University Algebra, - - - - - 1 50 XIV. Bobinson s Key to New University Algebra, - - - 1 25 XV. Bobinson s New Geometry and Trigonometry, - - - 1 50 XVI. Bobinson s Surveying and Navigation, - - - - - 1 50 XVII. Bobinson s Analyt. Geometry and Conic Sections, - - 1 50 XVIII. Bobinson s DifFeren. and Int. Calculus, (in preparation,)- - 1 50 XIX. Bobinson s Elementary Astronomy, 75 XX. Bobinson s University Astronomy, - - - - - -175 XXI. Bobinson s Mathematical Operations, 2 25 XXII. Bobinson s Key to Geometry and Trigonometry, Conic Sections and Analytical Geometry, l 50 Entered, according to Act of Congress, in the year 1860, by HOKATIO N. EOBINSON, LL.D., In the Clerk s Office of the District Court of the United States for the Northern District of New York. PREFACE. In the preparation of the following work the object has been to bring within the compass of one volume of convenient size an ele mentary treatise on both Conic Sections and Analytical Geometry. In the first part, the properties of the curves known as the Conic Sections are demonstrated, principally by geometrical methods ; that is, in the investigations, the curves and parts connected with them are constantly kept before the mind by their graphic representations, and we reason directly upon them. In the purely Analytical Geometry the process is quite different. Here the geometrical magnitudes, themselves, or those having cer tain relations to them, are represented by algebraic symbols, and we seek to express properties and imposed conditions by means of these symbols. The mind is thus relieved, in a great measure, of the ne cessity of holding in view the often-times complex figures required in the intermediate steps of the first method. It is, mainly, at the beginning and end of our investigations that we have to deal with concrete quantity. That is, after we have expressed known and im posed conditions, analytically, our reasoning is independent of the kind of quantity involved, until the conclusion is reached in the form of an algebraic expression, which must then receive its geo metrical interpretation. Much of the value of Analytical Geometry, as a disciplinary study, will be derived from a careful consideration, in each case, of this process of passing from the concrete to the abstract and the 7940O7 (ff) iv PREFACE. converse, and both teacher and student are earnestly recommended to give it a large share of their attention. In both divisions of the work the object has been to present the subjects in the simplest manner possible, and hence, in the first, analytical methods have been employed in several propositions when results could be thereby much more easily obtained; and for the same reason, in the second division, a few of the demonstrations are almost entirely geometrical. The analytical part terminates, with the exception of some exam ples, with the Chapter on Planes. Three others might have been added ; one on the transformation of Co-ordinates in Space, another on Curves in Space, and a third on Surfaces of Revolution and curved surfaces in general : but the work, as it is, covers more ground than is generally gone over in Schools and Colleges, and is sufficiently extensive for the wants of elementary education. Nu merous examples are given under the several divisions in the second part to illustrate and impress the principles. The Author has great pleasure in acknowledging his obligations to Prof. I. F. Quinby, A. M., of the University of Rochester, N. Y., formerly Assistant Prof, of Mathematics in the United States Mili tary Academy, at West Point, for valuable services rendered in the preparation of this treatise, as well as for the contribution to it of much that is valuable both in matter and arrangement. His thor ough scholarship, as well as his long and successful experience as an instructor in the class-room, preeminently qualified him to perform such labor. December, 1861. CONTENTS. CONIC SECTIONS. DEFINITIONS. Conical Surfaces, PAGE 9 Conic Sections, 10 THE ELLIPSE. Definitions and Explanations, 11 Propositions relating to the Ellipse, 13 THE PARABOLA. Definitions and Explanations, 41 Propositions relating to the Parabola, 43 THE HYPERBOLA. Definitions and Explanations, 65 Propositions relating to the Hyperbola, 67 ASYMPTOTES. Definition, 91 Propositions establishing relations between the Hyperbola and its Asymptotes, 91 1* VI CONTENTS. ANALYTICAL GEOMETRY. General Definitions and Remarks, GENERAL PROPERTIES OF GEOMETRICAL MAGNITUDES, CHAP TEE I. OF POSITIONS AND STRAIGHT LINES IN A PLANE AND THE TRANSFORMATION OF CO-ORDINATES. Definitions and Explanations, , 97 Propositions relating to Straight Lines in a Plane, 100 Transformation of Co-ordinates, 119 Polar Co-ordinates, 122 CHAPTER II. THE CIRCLE. LINES OF THE SECOND ORDEB. Propositions relating to the Circle 124 Polar equation of the Circle, 132 Application in the solution of Equations of the second degree, 134 Examples, 139 CHAPTER III. THE ELLIPSE. The description of the Ellipse and Propositions establishing its properties, 140 Example, 167 CHAPTER IV. THE PARABOLA. The description of the Parabola and propositions establishing its properties, 169 Polar equation of the Parabola, 183 Application in the solution of equations of the second degree, 185 Examples 187 CONTENTS. vii CHAPTER Y. THE HYPERBOLA. The Description of the Curve, and Propositions Establishing its Properties, 188 ASYMPTOTES OF THE HYPERBOLA. Definition and Explanation, 201 The Equation of the Hyperbola referred to its Asymptotes, and Properties deduced therefrom, 202 CHAPTER VI. ON THE GEOMETRICAL REPRESENTATION OF EQUATIONS OF THE SECOND DEGREE BE TWEEN TWO VARIABLES. Object of the Discussion, 210 Solution and Discussion of the General Equation, 211 Criteria for the Interpretation of any Equation of the Second Degree between two Variables, 221 APPLICATIONS. First, B*AC<Q, the Ellipse, 222 Second, B*A <7>0, the Hyperbola, 226 Third, * 4^O=0, the Parabola, 231 Examples, 233 CHAPTER VII. On the Intersection of Lines, and the Geometrical Solution of Equations, 237 Remarks on the Interpretation of Equations, 244 viil CONTENTS. CHAPTEE YIII. STRAIGHT LINES IN SPACE. Co-ordinate Planes and Axes, 249 The Equations and Relations of Straight Lines in Space, .... 250 CHAPTEE IX. ON THE EQUATION OF A PLANE. The Equations and Relations of Planes, 258 Examples Relating to Straight Lines in Space and to Planes, . 269 Miscellaneous Examples, 273 CONIC SECTIONS. DEFINITIONS. 1. A Conical Surface, or a Cone is, in its general accept ation, the surface that is generated by the motion of a straight line of indefinite extent, which in its different positions constantly passes through a fixed point and touches a given curve. The moving line is called the generatrix, the curve that it touches the directrix, the fixed point the vertex, and the generatrix in any of its positions an element, of the cone. The generatrix in all its positions extending without limit beyond the vertex on either side, will by its motion generate two similar surfaces separated by the vertex, called the nappes of the cone. 2. The Axis of a cone is the indefinite line passing through the vertex and the center of the directrix. 3. The intersection of the cone by any plane not pass ing through its vertex, that cuts all its elements, may be taken as the directrix ; and when we regard the cone as limited by such intersection, it is called the base of the cone. If the axis is perpendicular to the plane of the base, the cone is said to be right; and if in addition the base is a circle, we have a right cone with a circular base. This is the same as the cone defined in Geometry, (Book VII, Def. 16), and in the following pages it is to be understood that all references are made to it, unless otherwise stated. (9) 10- GONIC SECTIONS. 4. Conic Sections are the figures made by a plane cutting a cone. 5. There are five different figures that can be made by a plane cutting a cone, namely: a triangle, a circle, an ellipse, a parabola, and an hyperbola. HE MARK. The three last mentioned are commonly regarded as embracing the whole of conic sections ; but with equal propriety the triangle and the circle might be admitted into the same family. On the other hand we may examine the properties of the ellipse, the parabola, and the hyperbola, in like manner as we do a triangle or a circle, without any reference whatever to a cone. It is important to study these curves, on account of their exten sive application to astronomy and other sciences. 6. If a plane cut a cone through its vertex, and termin ate in any part of its base, the section will evidently be a triangle. 7. If a plane cut a cone parallel to its base, the section will be a circle. 8. If a plane cut a cone obliquely through all of the elements, the section will represent a curve called an ellipse. 9. If a plane cut a cone parallel to one of its elements, or what is the same thing, if the cutting plane and an element of the cone make equal angles with the base, then the section will represent a parabola. 10. If a plane cut a cone, making a greater angle with the base than the element of the cone makes, then the section is an hyperbola. 11. And if the plane be continued to cut the other nappe of the cone, this latter inter section will be the opposite hyperbola to the former. 12. The Vertices of any section are the points where the cutting plane meets the opposite elements of the cone, or the sides of the vertical triangular section, as A and B. THE ELLIPSE. 11 Hence, the ellipse and the opposite hyperbo las have each two vertices; but the parabola has only one, unless we consider the other as at an infinite distance. 13. The Axis, or Transverse Diameter of a conic section, is the line or distance AB between the vertices. Hence, the axis of a parabola is infinite in length, AB being only a part of it. The properties of the three curves known as the Conic Sections will first be investigated without any reference to the cone whatever ; and afterward it will be shown that these curves are the several intersections of a cone by a plane. THE ELLIPSE. DEFINITIONS. 1. The Ellipse is a plane curve described by the motion of a point subjected to the condition that the sum of its dis tances from two fixed points shall be constantly the same. 2. The two fixed points are called the foci. Thus F, F , we foci. 3. The Center is the point (7, the middle point between the foci. 4. A Diameter is a straight line through the center, and terminated both ways by the curve. 5. The extremities of a diameter are called its vertices. Thus, DD* is a diameter, and JD and D f are its vertices. 6. The Major, or Transverse Axis, is the diameter which passes through the foci. Thus, AA e is the major axis. 7. The Minor, or Conjugate Axis is the diameter at right 12 CONIC SECTIONS. angles to the major axis. Thus, CE is the semi minor axis. 8. The distance between the center and either focus is called the eccentricity when the semi major axis is unity. That is, the eccentricity is the ratio between CA and CF CF; or it is -^7 ; hence, it is always less than unity. The less the eccentricity, the nearer the ellipse approaches the circle. 9. A Tangent is a straight line which meets the curve in one point only; and, being produced, does not cut it. 10. A Normal to a curve at any point is a perpendicular to the tangent at that point. 11. An Ordinate to a Diameter is a straight line drawn from any point of the curve to the diameter, parallel to a tangent passing through one of the vertices of that diame ter. REMARK. A diameter and its ordinate are not at right angles, unless the diameter be either the major or minor axis. 12. The parts into which a diameter is divided by an ordinate, are called abscissas. 13. Two diameters are said to be conjugate, when either is parallel to the tangent lines at the vertices of the other. 14. The Parameter of a diameter is a third proportional to that diameter and its conjugate. 15. The paramater of the major axis is called fhe prin- cipal parameter, or latus rectum ; and, as will be proved, is equal to the double ordinate through the focus. Thus F f G- is one half of the principal parameter. 16. A Sub-tangent is that part of the axis produced, which is included between a tangent and the ordinate, drawn from the point of contact. 17. A Sub-normal is that part of the axis which is includ ed between the normal and the ordinate, drawn from the point of contact. THE ELLIPSE. 13 PROPOSITION I. PROBLEM. To describe an Ellipse. Assume any two points, as F and F and take a thread longer than the distance between these points, A fastening one of its extremities at the point F and the other at the point F . Now if the point of a pencil be placed in the loop and moved entirely around the points F and F* , the thread being constantly kept tense, it will describe a curve as represented in the adjoining figure, and, by definition 1, this curve is an ellipse. PROPOSITION II.-THEOREM. The major axis of an ellipse is equal to the sum of the two lines drawn from any point in the curve to the foci. Suppose the point of a pencil at D to move along in the loop, hold ing the threads F D and FD at A ( equal tension ; when D arrives at A, there will be two lines of threads between F and A. Hence, the entire length of the threads will be measured by F F+%FA. Also, when D arrives at A , the length of the threads is measured by FF + ZF A . Therefore, . FF f +ZFA=FF +2F A i Hence, ... . FA=F A From the expression FF +^FA, take away FA, and add F A , and the sum will not be changed, and we have Therefore, . F D+FD=A A Hence the theorem ; the major axis of an ellipse, etc. 2 14 CONIC SECTIONS. PEOPOSITION III . T H E B E M An ellipse is bisected by either of its axes. Let F, F* be the foci, AA the ma jor and BB f the minor axis of an ellipse ; then will either of these A axis divide the ellipse into equal parts. Take any point, as P in the el lipse, and from this point draw ordinates, one to the ma jor and another to the minor axis, and produce these or dinates, the first to P , the second to P", making the parts produced equal to the ordinates themselves. It is evident that the proposition will be established when we have proved that P and P are points of the curve. First. Fis a point in the perpendicular to PP f at its middle point; therefore FP =FP (Scho. 1, Th. 18, B. 1 Geom.) for the same reason F P f =F f P. Whence, by addition, FP +F f P =FP+F f P. That is, the sum of the distances from P to the foci is equal to the sum of the distances from P to the foci ; but by hypothesis Pis a point of the ellipse; therefore P r is also a point of the ellipse, (Def. 1). Second. The trapezoids P"dCF , PdCF are equal, be cause F C=FC, dP =dP by construction, and the angles at d and C in each are equal, being right angles ; these figures will therefore coincide when applied, and we have P F equal to PF and the angle P F F equal to the angle PFF . Hence the triangles P F F, PFF f are equal hav ing the two sides P F , F Fsmd the included angle P"F F in the one equal, each to each to the two sides PF, FF f and the included angle PFF in the other. Therefore, P F +P"F=PF +F P That is, the sum of the distances from P" to the foci is THE ELLIPSE. 15 equal to the sum of the distances from P to the foci, and since P is a point of the ellipse P" must also be found on the ellipse. Hence the theorem ; an ellipse is bisected, etc. PROPOSITION IV. THEOREM. The distance from either focus of an ellipse to the extremity of the minor axis is equal to the semi-major axis. Let A A 1 be the major axis, J^and F the foci, and CD the semi-minor axis of an ellipse ; then will FD= A | F D be equal to CA. Because F C= CF and CD is at right angles to F F, we have F f D=FD. But, F D+FD=A A Or, 2FD=A A Therefore, FD=A A, or CA. Hence the theorem ; the distance from either focus, etc. SCHOLIUM. The half of the minor axis is a mean proportional between the distance from either focus to the principal vertices. In the right-angled triangle FCD we have But, FD=AC Therefore, ~CD 2 =AG^ FO* = (AC+FC}(A CFC) =AF XAF Or, AF: CD=CD : FA PROPOSITION V. THEOREM Every diameter of an ellipse is bisected at the center. Let D be any point in the curve, and C the center. Draw DC, and produce it. From F draw jF 7 17 parallel; 16 CONIC SECTIONS. to FD; and from F draw FD par allel to F D. The figure DFD F is a parallelogram by construction; and therefore its opposite sides are equal. Hence, the sum of the two sides F D and D Fis equal to F D and DF; therefore, by def inition 1, the point D f is in the ellipse. But the two di agonals of a parallelogram bisect each other ; therefore, DC=CD r , and the diameter DD r is bisected at the center, (7, and DD represents any diameter whatever. Hence the theorem ; every diameter, etc. Cor. The quadrilateral formed by drawing lines from the extremities of a diameter to the foci of an ellipse, is a parallelogram. PROPOSITION YI. THEOREM. A tangent to the ellipse makes equal angles with the two .straight lines drawn from the point of contact to the foci. Let F and F f be the foci and D any point in the curve. Draw F D and FD, and produce F D to //, making DHDF, and draw FH. Bisect FHin T. Draw TD and produce it to t. Now, (by Cor. 2, Th. 18, B. I, Geom.), the angle FDT= the angle HDT, and HTD=its vertical angle F Dl Therefore, FD T=F Dt. It now remains to be shown that Tt meets the curve" only at the point D, and is, therefore, a tangent. If possible, let it meet the curve in some other point, as t, and draw Ft, tH, and F t. (By Scholium 1, Th. 18, B. I, Geom.) Ft=tH. To each of these add F t; Then, F t-}- tH= F t-{- Ft THE ELLIPSE. 17 But F t and tH are, together, greater than F 1 H, because a straight line is the shortest distance between two points ; that is, F t and Ft, the two lines from the foci, are, together, greater than FH, or greater than F D+FD; therefore, the point t is without the ellipse, and t is any point in the line Tt, except D. Therefore, Tt is a tangent, touching the ellipse at D; and it makes equal angles with the lines drawn from the point of contact to the foci. Hence the theorem ; a tangent, etc. Cor. The tangents at the vertices of either axis are perpendicular to that axis ; and, as the ordinates are par allel to the tangents, it follows that all ordinates to either axis must cut that axis at right angles, and be parallel to the other axis. SCHOLIUM 1. From this proposition we derive the following simple rule for drawing a tangent line to an ellipse at any point : Through the given point draw a line bisecting tlie angle included between the line connecting this point with one of the foci and the line produced connecting it with the other focus. SCHOLIUM 2. Any point in the curve maybe considered as a point in a tangent to the curve at that point. It is found by experiment that rays of light, heat and sound are incident upon, and reflected from surfaces under equal angles ; that is, for a ray of either of these principles the angles of incidence and reflection are equal. Therefore, if a reflecting surface be formed by turning an ellipse about its major axis, the light, heat, or sound which proceeds from one of the foci of this surface will be concen trated in the other focus. Whispering galleries are made on this principle, and all theaters and large assembly rooms should more or less approximate this figure. The concentration of the rays of heat from one of these points to the other, is the reason why they are called the foci or burning points. 9* T> 18 CONIC SECTIONS. PEOPOSITION YI I. THEOREM. Tangents to the ellipse, at the vertices of a diameter, are par allel to each other. Let DD be the diameter, and F and F the foci. Draw F D, F D , FD, and FD . Draw the tangents, Tt and Ss, one through the point D, the other through the point D 1 . These tan gents will be parallel. By Cor. Prop. 5, F D FD is a parallelogram, and the angle F D F is equal to its opposite angle, F DF. But the sum of all the angles that can be made on one side of a line is equal to two right angles. Therefore, by leaving out the equal angles which form the opposite an gles of the parallelogram, we have sD f F f +SWF=tDF f + TDF But (by Prop. 6) sD F =SD F-, and also tDF = TDF; therefore, the sum of the two angles in either member of this equation is double either of the angles, and the above equation may be changed to 2SD F=2tDF or SD F^tDF But DF and D F are parallel; therefore SD F and tDF are, in effect, alternate angles, showing that Tt and Ss are parallel. Cor. If tangents be drawn through the vertices of any two conjugate diameters, they will form a parallelogram circumscribing the ellipse. PROPOSITION YIII. THEOREM. If, from the vertex of any diameter of an ellipse, straight lines are drawn through the foci, meeting the conjugate diameter, the part of either line intercepted by the conjugate, is equal to one half the major axis. THE ELLIPSE. 19 Let DD be the diameter, and Tt the tangent. Through the center draw EE f parallel to Tt. Draw F D and DF, and produce DF to K; and from F draw FG parallel to EE f or Tt. 2Tow, by reason of the parallels, we have the following equations among the angles : tDG=DGF\ A] ( tDG=DHK TDF=DFG J But (Prop. 6) WG= TDF; Therefore, DGF=DFG; And, DHK=DKH Hence, the triangles DGF smd DHK are isosceles. Whence, DG=DF, and DH=DK. Because HCia parallel to FG, and F C=CF, therefore, F f H=HG Add, DF=DG and we have F f H+DF=DH But the sum of the lines in both members of this equa tion is F D+DF, which is equal to the major axis of the ellipse; therefore, either member is one half the major axis; that is, DH, and its equal, DK, are each equal to one half the major axis. Hence the theorem ; if from the vertex of any diameter, etc. PROPOSITION IX. THEOREM. Perpendiculars from the foci of an ellipse upon a tangent, meet the tangent in the circumference of a circle whose diame ter is the major axis. Let F , F be the foci, G the center of the ellipse, and D a point through which passes the tangent Tt. Draw F D 20 CONIC SECTIONS. and FD, produce F D to H, mak- - N H ing DH=FD, and produce FD to G, making DG=F D. Then jF .ff and FG are each equal to the major axis, A A. Draw FH meeting the tangent in A T and FG meeting it in t. Draw the dotted lines, CT and Ct. By Prop. 6, the angle .FDT^the angle F*Dt; and since opposite or vertical angles are equal, it follows that the four angles formed by the lines intersecting at D, are all equal. The triangles DF" G and DHF are isosceles by con struction ; and as their vertical angles at D are bisected by the line Tt, therefore F t=tG, FT= TH, and FT and F t are perpendicular to the tangent Tt. Comparing the triangles F GF and F Ct, we find that JFCis equal to the half of F F, and F t, the half of F G; therefore, Ct is the half of FG- ; but A A=FG; hence, Ct=\A A=CA. Comparing the triangles FF H and FCT 9 we find the sides FH and FF f cut proportionally in T and C; therefore, they are equi-angular and similar, and CT is parallel to F H, and equal to one half of it. That is, CT is equal to CA ; and CA, CT, and Ct are all equal ; and hence a circumference described from the center (7, with the radius CA 9 will pass through the points T and L Hence the theorem: perpendiculars from the foci, etc. PROPOSITION X. THEOREM. The product of the perpendiculars from the foci of an dlipse upon a tangent, is equal to the square of one half the minor axis. Produce TC and GF f , and they will meet in the circum ference at S; for FT and F t are both perpendicular to THE ELLIPSE. 21 H the same line Tt, they are there fore parallel ; and the two triangles, GFTand CF S, having a side, FC, of the one, equal to the side, CF , of the other, and their angles equal, each to each, are themselves equal. Therefore, CS=CT, S is in the cir cumference, and SF f FT. "Now, since A A and St are two lines that intersect each other in a circle, therefore (Th. 17, B. 3U, Geom.), SF f x F t=A F xF A; Or, FTxF t=A F xF A. But, by the Scholium to Prop. 4, it is shown that A F xF A= the square of one half the minor axis. Therefore, FTx F 1 1= the square of one half the minor axis. Hence the theorem ; The product of the perpendiculars, etc. Cor. The two triangles, FTD and F tD, are similar, and from them we have TF : F t=FD : DF ; that is, perpendiculars let fall from the foci upon a tangent, are to each other as the distances of the point of contact from the foci. PROPOSITION XI. THEOREM. If a tangent, drawn to an ellipse at any point, be produced until it meets either axis, and from the point of tangency an ordinate be drawn to the same axis, one half of the axis will be a mean proportional between the distances from the center to the intersections of these lines with the axis. Let Tt be a tangent at any point in the ellipse, as P. Draw F P &n&FP,F and F being the foci, and produce ^~^ c G F A F P to Q, making PQ=PF; join T,Q, and draw PG perpendicular to the axis A A . 22 CONIC SECTIONS. The triangles PFT and PTQ are equal, because PT is common, PQ=PF by construction, and the |__ TPF= the angle [_ TPQ (Th. 6). Therefore, TP bisects the angle FTQ, and QT=FT. As the angle at T 7 is bisected by TP, the sides about tliis angle in the triangle F TQ are to each other, as the segments of the third side, (Th. 24, B. H, Geom.) That is, F T : TQ : : F 1 P : PQ Or, F T : FT: : F P : PF From this last proportion we have (Th. 9, B. II, Geom.), F T+FT: F TFT : : F P+PF : F P PF Or, since F T+FT=2CT and F P+PF=*2CA, by substitution we have 2CT: F F : : 2 CM : F PPF (1) Again, because PG is drawn perpendicular to the base of the triangle F PF, the base is to the sum of the two sides, as the difference of the sides is to the difference of the segments of the base, (Prop. 6, PI. Trig.) Whence, F F : F P+PF : : F PPF : 2CG- (2) If we multiply proportions (1) and (2), term by term, omitting in the resulting proportion the factor F F, com mon to the terms of the first couplet, and the factor F P PF, common to the terms of the second couplet, we shall have 2CT-.2CA ::2CA:2CG Or, CT : CA:: CA : CG In like manner it may be proved that Ct : CB :: CB-.Cg Hence the theorem ; If a tangent, drawn to an ellipse, etc. PROPOSITION XII.-THEOREM. The sub-tangent on either axis of an ellipse is equal to the corresponding sub-tangent of the circle described on that axis as a diameter. THE ELLIPSE. Let P be the point of tan- gency of the tangent line Tt to the ellipse, of which A A is the major axis and the center. Draw the ordinate P G to this axis, and produce it to meet A c^~ G~ the circumference of the circle described on AA 1 as a diameter, at J5, and draw EC and BT, T being the inter section of the tangent with the major axis ; then will the line BT be a tangent to the circumference, at the point B. By the preceding theorem we have CT : OA : : GA : CG And since GA= CB, this proportion becomes CT: CB: : CB : CG Hence, the triangles CB T and CBG have the common angle (7, and the sides about this angle proportional ; they are therefore similar (Cor. 2 Th. 17, B. II, Geom.). But CB G- is a right-angled triangle; therefore, CBT is also right-angled, the right angle being at B. Now, since the line BT is perpendicular to the radius CB at its extrem ity, it is tangent to the circumference, and G-T is there fore a common sub-tangent to the ellipse and circle. If a circumference be described on the minor axis as a diameter, it may be proved in like manner that the corresponding sub-tangents of the ellipse and circle are equal. Hence, the theorem ; The sub-tangent on either axis, etc. SCHOLIUM 1. This proposition furnishes another easy rule for drawing a tangent line to an ellipse, at any point. RULE. On the major axis as a diameter, describe a semi-circum ference, and from the given point on the ellipse draw an ordinate to the major axis; draw a tangent to the semi-circumference at the point in which the ordinate produced meets it. The line that con nects the point in which this tangent intersects the major axis with the given point on the ellipse, will be the required tangent. 24 CONIC SECTIONS. SCHOLIUM 2. Because CB T is a right-angled triangle, =G ; \>M.tA G-AG=BG* Therefore, CG GT=A G AG PROPOSITION XIII. THE OEEM. The square of either semi-axis of an ellipse is to the square of the other semi-axis, as the rectangle of any two abscissas of the former axis is to the square of the corresponding ordinate. From any point, as P, of the ellipse of which C is the center, A A the major, and BBf the minor axis, draw the ordinate PG to the major axis; then it is to be proved that ~CA* : ~CB* : : AG GA : PG" Through P draw a tangent line intersecting the axes at Tand t ; then, by Prop. 11, we have CT:: CA:: CA: CG "Whence, CT>CG=CA and by multiplying both members of this equation by CG, it becomes dich may be resolved into the proportion CA 2 :CG 2 ::CT:CG From this we find, (Cor. Th. 8, B. H, Geom.), ZS 2 : Ol 2 W- 2 :: CT: GT (1) Again, drawing the ordinate Pg to the minor axis, we have Ct: CB:: CB: Cg or PG Whence, Ct PG=CB 2 Multiplying both members of this equation by PG, it becomes THE ELLIPSE. 25 a - P 2 =OB 2 -P from which we have the proportion CB 2 : PG* : : Ct : PG By similar triangles we have d : PG:: GT: GT And, since the first couplet in this proportion is the same as the second couplet in the preceding, the terms of the other couplets are proportional. That is, W-.PG 2 :: CT: GT (2) By comparing proportions (1) and ( 2 ), we obtain Cff .PG i .OA 2 : CA 2 CG 2 (3) But CA 2 C(?=(CA+ CG) (CACG)=A G AG-, Whence, by inverting the means in proportion (3) and substituting the values of CA CG , we have finally or, CA 2 : W : : AG AG : PG* By a process in all respects similar to the above, we will find that Hence the theorem ; the square of either semi-axis, etc. SCHOLIUM 1. From the theorem just demonstrated is readily deduced what is called, in Analytical Geometry, the equation of the ellipse referred to its center and axes. If we take any point, as P 9 on the curve, and can find a general relation between A G and PG 9 or between CG- and PG, the equation expressing such relation will be the equation of the curve. Let us .represent CA, one half of the major axis, by A } and CB, one half of the minor axis, by B ; that is, the symbols A and B denote the numerical values of these semi-axes, respectively. Also, denote the CG by x, and PG by y, then A G=A-\-x, and AG=A x- } and by the theorem we have J. : 2 : : (A+x) (Ax) :/ Whence, Or, 3 CONIC SECTIONS. This is the required equation in which the variable quantities, x and y, are called the co-ordinates of the curve, the first, x, being the abscissa j and the second, y, the ordinate; the center C from which these variable distances are estimated, is called the origin of co-ordinates, and the major and minor axes are the axes of co-ordinates. Had we donoted A G by x } without changing y, then we should have AG=2A x, And J. 2 : B* : : (2 A ) x : y> B* Whence, y*=~(2Ax x 2 ), which is the equation of the ellipse JO. when the origin of co-ordinates is on the curve at A . * SCHOLIUM 2. If a circle be described on either axis of an ellipse as a diameter j then any ordinate of the circle to this axis is to the corresponding ordinate of the ellipse, as one half of this axis is to one half of the other axis. Retaining the notation in Scholium 1, and producing the ordinate PG to meet the circumference described on A A as a diameter, at jP, we have, by the theorem, A* : B 2 : : (-4+*) (Ax) : y* But "Whence, Or, That is, (A+x) (Ax) = GP A* :*: A :B: .GP r \y GP : y : : A : B By describing a circle on BB f as a diameter, we may in like manner prove that pg : Pg : : B : A PROPOSITION XIV. -THEOREM. The squares of the ordinate to either axis of an ellipse are to each other, as the rectangles of the corresponding abscissas. B Let AA f be the major, and BB the minor axis of the ellipse, and jF6r, P Gr any two ordinates to the first axis. Denoting CGr by by x, CG by x , PG byy and P 6r by y f , we have, by Scho. 1, THE ELLIPSE. 27 Prop. 13, A*y* + B *x 2 = A 2 JB 2 and A*y f * + B 2 x *=A*B* J3 2 B* Whence, y*=-(A*x*)=(A+x) (Ax) (1) and y****J&^*)-^A+x?) (Ax f ) (2) Dividing equation (1) by equation (2), member by mem ber, and omitting the common factors in the numerator and denominator of the second member of the resulting equation, it becomes y\(A+x] (A-x) y * (A+x )(A x f ) By simply inspecting the figure, we perceive that A+x and A x represent the abscissas of the axis A A , corres ponding to the ordinate y ; and A+x , and A x those corresponding to the ordinate y . By placing the two equations first written above, under the form and proceeding as before, we should find a? (+y)(S-y) in which B+y, B y are the abscessas of the axis corresponding to the ordinate xCG=Pg , and J3 y are those corresponding to the ordinate # = CG f = P g>. Hence the theorem ; the squares of the ordinates, etc. PROPOSITION XV. THEOREM. The parameter of the transverse axis of an ellipse, or, the la- tus rectum, is the double ordinate to this axis through the focus. 28 CONIC SECTIONS. Let F and F r be the foci of an ellipse of which A A and BB f re spectively are the major and mi- nor axes. Through the focus F draw the double ordinate PP . Then will PP f be the parameter of the major axis. We will denote the semi-major axis by -A, the semi- minor axis by JB, the ordinate through the focus by P, and ?ind the distance from the center to the focus by c. The equation of the curve referred to the center and axis, is If in this equation we substitute c for x, y will become P, and we have Transposing the term .B 2 ^ 2 , and factoring the second member of the resulting equation, it becomes A 2 P 2 =B* (A 2 -* 2 ) (l) In the right-angled triangle B CF, since BF=A (Prop. 4) and Bc=B, we have A 2 c 2 =jB 2 . Replacing A 2 c 8 in eq. (1) by its value, that equation be comes A 2 -P 2 = J3 2 -B 2 Or, by taking the square roots of both members, A-P=B-B Whence, A:B::B:P Or, 2A:2B::2B:2P 2P is therefore a third proportional to the major and mi nor axes, and (Def. 14) it is the parameter of the former axis. Hence the theorem ; the parameter, etc. THE ELLIPSE. 29 PROPOSITION XVI. THEOREM. The area of an ellipse is a mean proportional between two circles described, the one on the major, and the other on the mi nor axis as diameters. On the major axis A A of the ellipse represented in the figure, describe a circle, and suppose this axis to be divided into any num ber of equal parts. Through the points of division draw ordinates to the circle, and join the extremities of these consecutive ordinates, and also those of the corresponding ordinates of the ellipse, by straight lines. We shall thus form in the semi-circle a number of trapezoids, and a like number in the semi- ellipse. Let 6r/J, G H be two adjacent ordinates of the circle, and gH g H those of the ellipse answering to them ; and let us denote GH by F, G H by F, gHbyy, g H 1 by y j and the part HH r of the axis by x. The trapezoidal areas, GHH 1 G f , gHH g , are respect ively measured by y+ F y+y r ^ x and^--z (Th. 34, B. I, Geom.) But (Prop. 13, Scho. 2) A:B:: Y:y :: Y>:y Hence (Th. 7, B. II, Geom.) or, A-.Bi: Y+Y-.y+y Y+Y/ x F+F : 2 y+y If the ordinates following F, y f in order, be represented by F", #", etc., we shall also have 3* 30 CONIC SECTIONS. , That is, any trapezoid in the circle will be to the cor responding trapezoid in the ellipse, constantly in the ratio of A to jB; and therefore the sum of the trapezoids in the circle will be to the sum of the trapezoids in the ellipse as A is to B; and this will hold true, however great the number of trapezoids in each. Calling the first sum S, and the second s, we shall then have A:B::S:s But, when the number of equal parts into which the axis AA f is divided, is increased without limit, S becomes the area of the semi-circle and s that of the semi-ellipse. Therefore, A : B : : area semi-circle : area semi-ellipse. Or, A : B : : area circle : area ellipse. By substituting in this last proportion for area circle, its value xA 2 , it becomes A : B : : xA 2 : area ellipse. "Whence area ellipse=7rJ..B, which is a mean proportional between xA 2 and xB 2 . Hence the theorem ; the area of an ellipse, etc. SCHOLIUM. This theorem leads to the following rule in mensu ration for finding the area of an ellipse. ~Ruii E.=Multipli/ the product of the semi-major and semi-minor axes by 3.1416. PROPOSITION XVII. THEOREM. If a cone be cut by a plane making an angle with the base less than that made by an element of the cone, the section is an el lipse. Let VloQ the vertex of a cone, and suppose it to be cut by a plane at right-angles to the plane of the opposite THE ELLIPSE. 31 elements, VN VB, these elements being cut by the first plane at A and B. Then, if the secant plane be not parallel to the base of the cone, the section will be an ellipse, of which AB is the major axis. Through any two points, F and H, on AB y draw the lines KL, MN, parallel to the base of the cone, and through these lines conceive planes to be passed also par allel to this base. The sections of the cone made by these planes will be circles, of which KGL and MIN are the semi-circumferences, passing the first through 6r, and the second through J, the extremities of the perpendiculars to BAj lying in the section made by the oblique plane. The triangles AFL, AHN, are similar ; so also are the triangles BMH, BKF; and from them we derive the fol lowing proportions : AF-.FLr.AHiHN BF:KF::BH:HM By multiplication, AF-BF: FL-KF: : AH-BH: UN- JIM Because KL is a diameter of a circle, and FG an ordi- nate to this diameter, we have and for a like reason, Therefore, AF-BF : FG 2 : : AH-HB : Iff or, AF-BF : AH-HB : : FG 2 : HP This proportion expresses the property of the ellipse proved in (Prop. 14) ; and the section A GIB is, therefore, an ellipse. Hence the theorem ; if a cone be cut, etc. SCHOLIUM. The proportion AF- BF : AH-HB::FG? : Hf would still hold true, were the line AB parallel to the base of the cone, and the section a circle ; the ratios would then become equal 32 CONIC SECTIONS. to unity. The circle may therefore be regarded as a particular case of the ellipse. Jj PROPOSITION XYIII. THEOREM. If, from one of the vertices of each of two conjugate diameters of an ellipse, ordinates be drawn to either axis, the sum of the squares of these ordinates will be equal to the square of the other semi-axis. an ellipse, of which A A is the major and BB r the minor axis ; also let P, P g be any two conjugate diameters. Through the vertices of these diameters draw the tangents to the ellipse and the ordi nates to the axes, as represented in the figure. Then we are to prove that and CB=(PG) 2 +(P G } 2 =(Cgf+(OgJ Now (by Prop. 11) we have GT: CA:: CA : OG, also, Of : CA:: CA: On "Whence, and Therefore, which, resolved into a proportion, gives Of : CT:: OG : Cn (2) By the construction, it is evident that the triangles OPT, CQ t, are similar, as are also the triangles PGT and CQn. THE ELLIPSE. 33 Prom these triangles we derive the proportions Ct : CT: : CQ f : PT CQ! :PT: : On : GT Whence, Ct : CT : : On i GT Comparing the last proportion with proportion (2) above, we have CG: Cn:: Cn: GT Whence, (Crif=CG GT But GT= CTCG; then (Cnf= CG (CTCG), from which we get (Cn) 2 + CG*= CG- CT= CA* (See eq. 1.) Substituting, in this equation, for (<7ft) 2 , its equal CG r > it becomes In a similar manner it may be proved that Hence the theorem ; if from one of the vertices of each, etc. PROPOSITION XIX. THEOEEM. The sum of the squares of any two conjugate diameters of an ellipse is a constant quantity, and equal to the sum of the squares of the axes. The annexed fig ure, being the same as that employed in ^ the preceding prop osition, by that prop osition we have CA=CG + CG and By addition, ~CA 9 CG*+ PG 2 + CG 34 CONIC SECTIONS. But CG and PG are the two sides of the right-angled triangle CPG, and CG and P f G r are the two sides of the right-angled triangle CP f G r ; Therefore, OA. 2 + ~CB* Whence, 4CA 2 +4tCB* The first member of this equation expresses the sum of the squares of the axes, and the second member the sum of the squares of the two conjugate diameters. Hence the theorem ; the sum of the squares of any two, etc. PROPOSITION XX. THEOREM. The parallelogram formed by drawing tangents through the vertices of any two conjugate diameters of an ellipse, is equal to the rectangle of the axes. Employing the figure of the last two propositions, we have, from proposi tion 18, from which, by trans- o position and factoring the second member, we get gG 2 =(CA+CG ) (CACG f )=AG A G f But CA 2 : CB 2 :: A_G -A G^_P f G 2 ; (Prop. 13.) CG 2 : P f G Z CG : P G ^Qn (1) CA : CG (2) (Prop. 11.) CB 2 CB CA Whence, C Or, CA But, CT Multiplying proportions (1) and (2), term by term, omitting, in the first couplet of the resulting proportion, the common factor CA, and in the second couplet the common factor CG, we find CT: CB:: CA: Q f n THE ELLIPSE. 35 Whence, CT- Q f n= CA - CB Or, 4CT-Q n=4i-CB The first member of this equation measures eight times the area of the triangle CQ T, and this triangle is equiva lent to one half of the parallelogram CQ mP, because it has the same base, CQ , as the parallelogram, and its vertex is" in the side opposite the base. This parallelogram is obviously one fourth of that formed by the tangent lines through the vertices of the conjugate diameters; 4CT.Q n therefore, measures the area of this parallelo gram. Also, 4 CA-CB is the measure of the rectangle that would be formed by drawing tangent lines through the vertices of the major and minor axes of the ellipse. Hence, the theorem ; the parallelogram formed, etc. PROPOSITION XXI.-THEOREM. If a normal line be drawn to an ellipse at any point, and] also an ordinate to the major axis from, the same point, then will the square of the semi-major axis be to the square of the semi-minor axis, as the distance from the center to the foot of \ the ordinate is to the sub-normal on the major axis. Let P be the assumed point in the ellipse, and through this point draw the tangent P I 7 , the normal PD, and the ordinate PG, to the major axis ; then C being the center of the ellipse, and A denoting the semi-major, and B the semi-minor axis, it is to be proved that A 2 : B 2 : : CG : DG By (Prop. 13) we have A 2 :B 2 ::A G-AG:TG* (1) and because DPT is a right-angled triangle, and PGr is a 36 CONIC SECTIONS. perpendicular let fall from the vertex of the right-angle upon the hypotenuse, we also have (Th. 25, B. II, Geom.) ~PG?=DG GT But A G- A G= CG- G T (Scho. 2, Prop. 12) Substituting in proportion (1), for the terms of the sec ond couplet, their values, it becomes A 2 : 2 :: CG GT-.DG GT or A 2 : 2 ::CG:DG. Hence the theorem ; if a normal line be drawn, etc. Cor. If CGXj then this theorem will give for the IP A 2 pression. /i mibnormal,D6r, the value x, which is its analytical ex- ST PROPOSITION XXII. THE OEEM. If two tangents be drawn to an ellipse, the one through the vertex of the major axis and the other through the vertex of any other diameter, each meeting the diameter of the other produced, the two tangential triangles thus formed will be equivalent. Let PP f be any diameter of the ellipse whose major axis is AA f . Draw the tangents JJVand PT, the first meeting the diameter produced at and the second the axis pro- duced at T; the triangles CAN and CPT thus formed are equivalent. Draw the ordinate PD; then by similar triangles we have CD: CM:: CP: CN But CD : CA ::CA: CT (Prop. 11) Whence CP: CN: : CA : CT Therefore, CP- CT= CN- CA THE ELLIPSE. 37 Multiplying both members of this equation by sin. C y it becomes CP- CT sin. a= CN- CA sin. or, iCT CP&m.C^CA CNsm.C (1) But CP- sin. C=PD, and CN- sin. C=AN; therefore the first member of equation (1) measures the area of the triangle CPT, and the the second member measures that of the triangle CAN. Hence the theorem ; if two tangents be drawn to an, etc. Cor. 1. Taking the common area CAJEP, from each triangle, and there is left &PEN=&AET. Cor. 2. Taking the common A CDP, from each trian gle, and there is left AP-DT= trapezoidal area PDAN. PROPOSITION XXIII.-THEOREM. The supposition of Proposition 22 being retained, then, if a secant line be drawn parallel to the second tangent, and ordi- nates to the major axis be drawn from the points of intersec tion of the secant with the curve, thus forming two other tri angles, these triangles will be equivalent each to each to the cor- responding trapezoids cut off, by the ordinates, from the trian gle determined by the tangent through the vertex of the major axis. Draw the secant QnS par allel to the tangent PT, and also the ordinates QJR, ng, pro ducing the latter to p. Then A ^ |n //(:S\ ^ ^A "ST is Aj=trapezoid ANVJR, and A%=trapezoid ANpg. V The three triangles, CVE,CPD,CNA are similar, by construction ; therefore, &CNA : AOFD : : CM 2 : : ~CP* "Whence, trapezoid ANPD : &CNA : : ~CA*~C~ff : GZ 2(1) (Th. 8, B. II, Geom.) 4 38 CONIC SECTIONS. In like manner, trapezoid ANVR : &CNA : : CA CR 2 : ~CA 2 (2) Dividing proportion (1) by ( 2 ), term by term, we get trapezoid ANPD m ~CA* trapezoid ANVR ~(JA 2 _ Whence, trapez. ANPD rjrapez. ANVE : : CA 2 UT> 2 : CA Z CR 2 But JPD 2 : ~QR 2 : : A D DA : A R-RA, (Prop. 14) ; and since A D=* CA+ CD, A R= CA+ CR, DA=CACD and RA=CACR, we have ~PD 2 : QlR 2 : : (CA+ CD) (CACD) : (CA+ CR) (CACE):: ~CA 2 Clf : ~CA 2 CIl 2 Therefore, trapezoid ANPD : trapezoid ANVR : :~PD 2 : ~QR 2 , But the trapezoid ANPD=&TPD, (Cor. 2, Prop. 22); whence, ;2 A TPD : trapezoid ANVR ::PD.:: QR (3) and since the triangles TPD and SQR are similar, we have ATPZ) : ASQR : : ~Plf : QR 2 (4) By comparing proportions ( 3 ) and (4) we find A TPD : trapezoid ANVR : : &TPD : &SQR "Whence, trapezoid ANVR=&SQR; and by a similar process we should find that trapezoid ANpg=A.Sng. Hence the theorem ; if a secant line be drawn parallel, etc. Cor. 1. Taking the trapezoid ANpg from the trapezoid- ANVR, and the A% from the &SQR, we have trapezoid gpVR= trapezoid gnQR. Cor. 2. The spaces ANVR, TPVR, and SQR are equiv alent, one to another. Cor. 3. Conceive QR and QS to move parallel to their present positions, until R coincides with C; then QR THE ELLIPSE. becomes the semi-minor axis, the space ANVE the tri angle ANC, and the &QKS equivalent to the ACPI 7 . PROPOSITION XX I Y. -THEOREM. Any diameter of the ellipse bisects all of the chords of the el lipse drawn parallel to the tangent through the vertex of the diameter. * A ST By Cor. 1 to the preceding proposition we have If from each of these equals we subtract the common area gnm VR, there will remain the Aranp, equivalent to the AQw V; and as these triangles are also equi-angular, they are absolutely equal. Therefore, Qmmn. Hence the theorem ; any diameter of the ellipse bisects, etc. REMARK. The property of the ellipse" demonstrated in this proposition is merely a generalization of that previously proved in Prop. 3. PROPOSITION XXV. THEOREM. The square of any semi-diameter of an ellipse is to the square of its semi-conjugate, as the rectangle of any two abscissas of the former diameter is to the square of the corresponding ordi- nate. Let A A be the major axis of the ellipse, CP any semi- diameter and CP its semi- conjugate. Draw the tan- gents TP and AN, the ordi- nate Qm, producing it to meet the axis at S; and P f V, parallel to AN, and in other 40 CONIC SECTIONS. respects make the construction as indicated in the figure. It is then to be proved that OP 2 : OP 2 : : Pm-mP : Qm "Now in the present construction, the triangles CPU and CV R take the place of the triangles SQR and CVR respectively, in Prop. 23 ; and hence by that proposition, the triangles CP V, CAN, and CPT are equivalent one to a,nother. The triangles CPT and CmS are similar ; therefore, "Whence, ACtotf : : CP 2 : Cm AGP7 7 : ^CPTACmS: : CP 2 : CP*Cm Or, A OPT 7 : trapez. mPTS : : ~CP 2 : CP 2 ~Cin <& From the similar triangles, CP r V and mQV, we have A OF V : AwF : : OF 2 : m~Q 2 But area Sm VR+ A CVJR+ Am Q F= area Sm VR+ A <7F--f trapez. mPTS, (Prop. 23.) ; therefore, AwF= trapez. mPTS ; also A<7P ; V =&CPT. Substituting these values in the preceding proportion, it becomes ACP!T : trapez. mPTS : : OP 2 : m 2 ( 2 ) By comparing proportions (1) and (2), we get CP 2 : OP 2 ~Cm : : CP 2 : ^Q 2 Or, CP 2 : CP f2 : : ~CP 2 Cm : m<? "Whence, OP 2 : OF 2 : : (CP+Cm) (CPCm) : ^Q 2 Or, CP 2 : OF 2 : : P m-mP : Hence the theorem ; the square of any semi-diameter , etc. REMARK. The property of the ellipse relating to conjugate diameters, established by this proposition, is but the generalization of that before demonstrated in reference to the axes, in Prop. 13. THE PAEABOLA. 41 THE PARABOLA. DEFINITIONS. 1. The Parabola is a plane curve, generated by the motion of a point subjected to the condition that its distances from a fixed point and a fixed straight line shall be constantly equal. 2. The fixed point is called the focus of the parabola, and the fixed line the directrix. Thus, in the figure, Fis the focus and BB" the directrix of the para bola PFP P", etc. 3. A Diameter of the parabola is a line drawn through any point of the curve, in a direction from the directrix, and at right-angles to it. 4. The Vertex of a diameter is the point of the curve through which the diameter is drawn. 5. The Principal Diameter, or the Axis, of the parabola is the diameter passing through the focus. The vertex of the axis is called the principal vertex, or simply the vertex of the parabola. The vertex of the parabola bisects the perpendicular distance from the focus to the directrix, and all the diam eters of the parabola are parallel lines. 6. An Ordinate to a diameter is a straight line drawn from any point of the curve to the diameter, parallel to the 4* 42 CONIC SECTIONS. tangent line through its vertex. Thus, PD, drawn parallel to the tangent V T, is an ordinate to the diameter VT>. It will he shown that DP=.DGr; and hence PGr is called a double ordinate. 7. An Abscissa is the part of the diam eter hetween the vertex and an ordinate. Thus, V f D is the ahscissa corresponding to the ordinate PD. 8. The Parameter of any diameter of the parahola is one of the extremes of a proportion, of which any ordi nate to the diameter is the mean, and the corresponding abscissa the other extreme. 9. The parameter of the axis of the parahola is called the principal parameter, or simply the parameter of the parabola. It will be shown to be equal to the double ordinate to the axis through the focus. Thus, BB f , .the chord drawn through the focus at right-angles to the axis, is the parameter of the parabola. The principal parameter is sometimes called the latus- rectum. 10. A Sub-tangent, on any diameter, is the distance from the point of intersection of a tangent line with the diameter produced to the foot of that ordinate to this diameter that is drawn from the point of contact. 11. A Sub-normal, on any diameter, is the part of the diameter intercepted be tween the normal to the curve, at any point, and the ordinate from the same point to the diameter. Thus, in the figure, V N being any diameter, PT a tangent, and PN a normal at the point P, and PQ an ordinate to the diameter; then TQ is" a sub-tangent and QN& sub-normal on this diameter. THE PARABOLA. 43 "When the terms, sub-tangent and sub-normal, are used without reference to the diameter on which they are ta ken, the axis will always be understood. PROPOSITION I.-PROBLEM. To describe a parabola mechanically. Let CD be the given line, and F the given point. Take a square, as DBG, and to one side of it, GB, attach a thread, B and let the thread be of the same length 31 as the side GB of the square. Fasten one c end of the thread at the point G, the other end at F. Put the other side of the square against the given line, CD, and with the point of a pencil, in the thread, bring the thread up to the side of the square. Slide the side BD of the square along the line CD, and at the same time keep the thread close against the other side, permitting the thread to slide round the point of the pencil. As the side BD of the square is moved along the line CD, the pencil will describe the curve represented as passing through the points Fand P. For <7P+P^=the length of the thread, and GP+PB=ihe length of the thread. By subtraction, PFPB=0, or PF=*PB. This result is true at any and every position of the point P; that is, it is true for every point on the curve corresponding to definition 1. Hence, FV= VH. If the square be turned over and moved in the opposite direction, the other part of the parabola, on the other side of the line FH, may be described. Cor. It is obvious that chords of the curve which are perpendicular to the axis, are bisected by it. 44 CONIC SECTIONS. PROPOSITION II. THEOREM. Any point within the parabola, or on the concave side of the curve, is nearer to the focus than to the directrix; and any point without the parabola, or on the convex side of the curve, is nearer to the directrix than to the focus. Let jPbe the focus and HB f the directrix B of a parabola. First. Take A, any point within the curve. From A draw AFio the focus, and AB per- B pendicular to the directrix; then will AF be less than AB. Since A is within the curve, and B is without it, the line AB must cut the curve at some point, as P. Draw PF. By the definition of the parabola, PB= PF; adding PA to each member of this equation, we have PB+PA=BA=PA+PF But PA and PF being two sides of the triangle APF, are together greater than the third side AF; therefore their equal, BA, is greater than AF. Second. Now let us take any point, as A , without the curve, and from this point draw A F to the focus, and A B r perpendicular to the directrix. Because A is without the curve and F is within it, AF must cut the curve at some point, as P. From this point let fall the perpendicular, BP, upon the directrix, and draw A B. As before, PB=PF; adding A P to each member of this equation, and we have A P+PB=A P+PF=A F. But A P and PB being two sides of the triangle A PB, are together greater than the third side, A B ; therefore their equal, A F, is greater than A B. Now A B, the hy potenuse of the right-angled triangle A BB is greater than either side; hence, A B is greater than A B ; much more then is A F greater than A B . Hence the theorem; any point within the parabola, etc. THE PAEABOLA. 45 Cor. Conversely: If the distance of any point from the directrix is less than the distance from the same point to the fo cus, such point is without the parabola; and, if the distance from any point to the directrix is greater than the distance from the same point to the focus, such point is within the parabola. First. Let A be a point so taken that A B <A F. Now A is not a point on the curve, since the distances A B and A f F are unequal; and A r is not within the curve, for in that case A B would be greater than A F according to the proposition, which is contrary to the hy pothesis. Therefore A being neither on nor within the parabola, must be without it. Second. Let A be a point so taken that AB>AF. Then, as before, A is not on the curve, since AF and AB are unequal ; and A is not without the curve, for in that case AB would be less than AF, which is contrary to the hypothesis. Therefore, since A is neither on nor without the parabola, it must be within it. PROPOSITION III. THEOREM. If a line be drawn from the focus of a parabola to any point of the directrix, the perpendicular that bisects this line will be a tangent to the curve. Let F be the focus, and HD the di rectrix of a parabola. Assume any point whatever, as B, in B the directrix, and join this point to the focus by the line BF; then will tA, the U F v F perpendicular to BF through its middle point t, be a tan gent to the parabola. Through B draw BL perpendicu lar to the directrix, and join P, its intersection with tP, to the focus. Then, since P is a point in the perpendic ular to BF at its middle point, it is equally distant from the extremities of BF; that is, PB=PF. P is there- 46 CONIC SECTIONS. fore a point in the parabola, (Def. 1). Hence, the line tP meets the curve at the point P. "We will now prove that all other points in the line tP are without the parahola. Take A 9 any point except P in the line tP, and draw AF, AB; also draw AD perpen dicular to the directrix. AF is equal to AB, because A is a point in the perpendicular to BF at its middle point; but AB, the hypotenuse of the right-angled triangle ABD, is greater than the side AD; therefore AD is less than AF, and the point A is without the parabola. (Cor., Prop. 2). The line tA and the parabola have then no point in common except the point P. This line is there fore tangent to the parabola. SCHOLIUM 1. The triangles BPt and FPt are equal; therefore the angles FPt and BPt are equal. Hence, to draw a tangent to the parabola at a given point, we have the following RULE. From the given point draw a line to the focus, and an other perpendicular to the directrix, and through the given point draw a line bisecting the angle formed by these two lines. The bi secting line will be the required tangent. SCHOLIUM 2. Just at the point Pthe tangent and the curve co incide with each other ; and the same is true at every point of the curve. Now, because the angles BPt and FPt are equal, and the angles BPt and LPA are vertical, it follows that the angles LPA and FPt are equal. Hence it follows, from the law of re flection, that if rays of light parallel to the axis VF be incident upon the curve, they will all be reflected to the focus F. If there fore a reflecting surface were formed, by turning a parabola about its axis, all the rays of light that meet it parallel with the axis, will be reflected to the focus ; and for this reason many attempts have been made to form perfect parabolic mirrors for reflecting telescopes. If a light be placed at the focus of such a mirror, it will reflect all its rays in one direction ; hence, in certain situations, parabolic mirrors have been made for lighthouses, for the purpose of throwing all the light seaward. Cor. 1. The angle BPF continually increases, as the THE PABABOLA. 47 pencil P moves toward "F, and at V it becomes equal to two right angles ; and the tangent at V is perpendicular to the axis, which is called the vertical tangent. Cor. 2. The vertical tangent bisects all the lines drawn from the focus of a parabola to the directrix. Let Vt be the vertical tangent ; then because the two right-angled triangles FVt and FHB are similar, and VF= VH, we have Ft=tB. PROPOSITION I V. THEOREM. The distance from the focus of a parabola to the point of contact of any tangent line to the curve, is equal to the dis tance from the focus to the intersection of the tangent with the axis. Through the point P of the parabola of which F is the focus and BH the directrix, draw the tangent line PT, meeting the axis produced at the point f k v ir i T; then will FP be equal to FT Draw PB perpendicular to the directrix, and join F,B. The angles BPT and TPF are equal, (Scho. 1, Prop. 3) ; and since PB is parallel to TG, the alternate angles BP T, and PTC are also equal. Hence the angle TPF is equal to the angle PTF, and the triangle PFT is isosceles; therefore FP=FT. Hence the theorem ; the distance from the focus to, etc. ! SCHOLIUM. To draw a tangent line to a parabola at a given point, we have the following RULE. Produce the axis, and lay off on it from the focus a dis tance equal to the distance from the focus to the point of contact. The line drawn through the point thus determined and the given point will be the required tangent. 48 CONIC SECTIONS. PROPOSITION V. THEOREM. The perpendicular distance from the focus of a parabola to any tangent to the curve, is a mean proportional between the distance from the focus to the vertex and the distance from the focus to the point of contact. In the figure of tlie preceding proposi- tion draw in addition the vertical tangent Vt; then we are to prove that Ft 2 = VF-FP. Because TtF and VFt are f H y g D c similar right-angled triangles, we have TF :Ft::Ft: VF. But TF=PF, (Prop. 4) ; therefore, PF : Ft : : Ft : VF Whence, Wf^PF. VF Hence, the theorem ; the perpendicular distance from^etc. PROPOSITION YI. THEOREM. The sub-tangent on the axis of the parabola is bisected at the vertex. In the figure which is constructed as in the two preceding propositions, draw in addition the ordinate PD, from the point of contact to the axis ; then we T H v F D are to prove that TD is hisected at the vertex V. The two right-angled triangles TFt and tFP have the side Ft common, and the angle FTt equal to the angle FPt ; hence the remaining angles are equal, and the tri angles themselves are equal; therefore tT=tP. From the similar triangles TDP, TVt, we have the proportion Tt: tP: : TV: VD But tT=tP; whence TV= VD Hence the theorem ; the sub-tangent on the axis, etc. THE PARABOLA. 49 Cor. Since TV=TD, it follows that Vt^PD. That is, The part of the vertical tangent included between the vertex and any tangent line to the parabola, is equal to one half of the ordinate to the axis from the point of contact PROPOSITION VII. THEOREM. The sub-normal is equal to twice the distance from the focus to the vertex of the parabola* In the figure (which is the same as that B of the last three propositions), PC is a normal to the parabola at the point (7, . and DC is the sub-normal ; it is to be T H v F D proved that DC=2FV. Because BH and PD are parallel lines included be tween the parallel lines BP and HD, they are equal. BF and PC are also parallel, since each is perpendicular to the tangent PT ; hence BF=PC, and also the two tri angles HBF and DPC are equal. Therefore HF=DC; but HF=2FV; whence DC=2FV. Hence the theorem ; the sub-normal is equal to twice, etc. SCHOLIUM. This proposition suggests another easy process for constructing a tangent to a parabola at a given point. RULE. Draw an ordinate to the axis from a given point, and from the foot of this ordinate lay off on the axis, in the opposite direction of the vertex, twice the distance from the focus to the vertex. Through the point thus determined and the given point draw a line, and it will be the required tangent. PROPOSITION YII I. THEOREM. Any ordinate to the axis of a parabola is a mean proportion al between the corresponding sub-tangent and sub-normal. 5 D 50 CONIC SECTIONS Assume any point, as P, in the parabo la of which. F is the focus and HB the directrix. Through this point draw the tangent PT, the normal P(7, and the or- T H v F D c" dinate PD to the axis. Then in reference to the point P, TD is the sub-tangent, and 1) the sub-normal on the axis ; and we are to prove that TD : PD : : PD : DC The triangle TFC is right-angled at P, and PD is a perpendicular let fall from the vertex of this angle upon the hypotenuse. Therefore, PD is a mean proportional between the segments of the hypotenuse, (Th. 25, B. II, Geom.) Hence the theorem ; any ordinate to the axis, etc. SCHOLIUM 1. For a given parabola, the fourth term of the pro portion, TD : PD : : PD : DC, is a constant quantity, and equal to twice the distance from the focus to the vertex, (Prop. 7). By placing the product of the means of this proportion equal to the product of the extremes, we have PZ> 2 ~ TD-DC=TD-2DC, which may be again resolved into the proportion \TD\PV\ :PD:2DO Or, VD:PD : :PD:2DG But VD is the abscissa, and PD is the ordinate of the point P ; hence (Def. 8) 2DC is the parameter of the parabola, and is equal to four times the distance from the focus to the vertex, or to twice the distance from the focus to the directrix. SCHOLIUM 2. If we designate the ordinate PD by y, the abscissa VD by X, and the parameter by 2p, the above proportion becomes x^ : y : : y : 2p Whence, y =2px. This equation expresses the general relation between the abscissa and ordinate of any point of the curve, and is called, in Analytical Geometry, the equation of the" parabola referred to its principal ver tex as an origin. Cor. The sub-normal in the parabola is equal to one-half of the parameter. p THE PARABOLA. 51 PROPOSITION IX. THEOREM. The parameter, or latus rectum, of the parabola is equal to twice that ordinate to the axis which passes through the focus. Let F be the focus, and BB the direc trix of a parabola ; and through the focus B draw a perpendicular to the axis intersecting R the curve at P and P r . From P and P let fall the perpendiculars P_B, P B , on the direc- trix. Then will ZPF be equal to 2FH, or to the parameter of the parabola. By the definition of the parabola, PFPB; and be cause PP and BB are parallel, and the parallels PB and FH are included between them, we have PB=FH. Hence PF=FH, or 2PF=2FH= the parameter. Scho. 1, Prob. 8. Cor. Since the axis bisects those chords of the parabola which are perpendicular to it, FP=FP ] . That is, FP ; therefore PP =2FH. That is, The parameter of the parabola is equal to the double ordi nate through the focus. PROPOSITION X. THEOREM. The squares of any two ordinates to the axis of a parabola are to each other as their corresponding abscissas. Let y and y r denote the ordinates, and x and x f the abscissas of any two points of the parabola; then, by Scho. 2, Prop. 8, we have the two following equations : y 2 =2px and y 2 =2px Dividing the first of these equations by the second, member by member, we have Whence y* : y 2 : : x : x r Hence the theorem ; the squares of any two ordinates, etc. 52 CONIC SECTIONS. PEOPOSITION XI. TIIEOREM. If a perpendicular be drawn from the focus of a parabola to any tangent line to the curve, the intersection of the perpen dicular with the tangent will be on the vertical tangent. Let F be the focus, and BH the di- B| rectrix of the parabola, and PT a tan gent to the curve at the point P. From jFdraw FB perpendicular to the tangent, T H v r D c~ intersecting it at t, and the directrix at J3. We will now prove that the point t is also the intersection of the ver tical tangent with the tangent PT. Because the triangle TFP is isosceles, the perpendicu lar Ft bisects the base PT; therefore tP=tT. Again, since Vt and DP are both perpendicular to the axis, they are parallel, and the vertical tangent divides the sides of the triangle TDP proportionally. Hence, TV: VD:: Tt : tP; but TV= YD (Prop. 6) therefore, Tt=tP. That is, the tangent PT is bisected by both the perpen dicular let fall upon it from the focus, and the vertical tangent. Therefore the tangent PT, the vertical tangent and the perpendicular FB, meet in the common point t. Hence the theorem ; if a perpendicular be drawn, etc. PROPOSITION XI I. THEOREM. The parameter of the parabo j a is to the sum of any two or- dinates to the axis, as the difference of those ordlnates is to the difference of the corresponding abscissas. Take any two points, as P and Q, in the parabola repre sented in the following figure, and through these points draw the double or dinates Pp and Qq. VD and VE are the corresponding abscissas. Draw PS and pt parallel to the axis. Then, since THE PARABOLA. 53 PD=Dp and QE=Eq, we have QE+PD C^ = Qt, equal to the sum of the two ordinates ; and QEPD= QS, equal to their differ ence; also VEVD=DE, equal to the v( difference of the corresponding abscissas. We are now to prove that pv 2p : Qt : : QS : DE *^^ in which 2p denotes the parameter of the parabola. Because PD and QE are ordinates to the axis, we have (Scho. 2, Prop. 8) PD*=2p VD (1) and QE=2p-VE (2) Whence QE* Pff==2p (VE VD)=2p-DE (3) But QE 2 PTf= ( QE+ PD} ( QEPD)= Qt- QS, therefore Qt-QS=2p-J)E (4) Whence 2p : Qt : : QS : DE Hence the theorem ; the parameter of the parabola, etc. Cor. By dividing eq. (4) by eq. (2), member by member, we obtain Qt-QS_DE Whence VE : DE : : QE* : Qt- QS PROPOSITION XIII. THEOREM. If a tangent line be drawn to a parabola at any point, and from any point of the tangent a line be dravm parallel to the axis terminating in the double ordinate from the point of contact, this line will be cut by the curve into parts having to each other the same ratio as the segments into which it divides the double ordinate. 54 CONIC SECTIONS. Take any point as P in the parabo la represented in the figure, and of which VD is the axis, and through this point draw the tangent PTto the curve, and the double ordinate PQ to the axis. Assume a point in the tan gent at pleasure, as JL, and through it PJ draw AC parallel to the axis, cutting// the curve at B and the double ordinate at C. Then we arc to prove that AB:BC::PC: CQ By similar triangles we have PC: CA :: PD: DT; but DT=2DV(Prop. 6) therefore PC: CA : : PD : 2DV (1) But D V: PD : : PD : 2p (Scho. 2, Prop. 8) or 2DF: PD : : 2PD : 2p. Inverting terms, PD : 2DV: : 2p : 2PD=PQ (2) By comparing proportions (1) and (2), we get PC: CA::2p: PQ But 2p : CQ:: PC: BC (Prop. 12) Multiplying the last two proportions, term by term, we have 2p-PC: CA-CQ : : 2p-PC: BC PQ The first and third terms of this proportion are equal ; therefore the second and fourth are also equal. Hence we have the proportion CA: BC:: PQ : CQ Whence by division, CABC :BC:: PQCQ : CQ or AB:BC: : PC: CQ If we take any other point, H, on the tangent, and through it draw the line HL parallel to the axis, inter secting the curve at K and the ordinate at L 9 we will have, in like manner, HK: KL: : PL: LQ Hence the theorem ; if a tangent be drawn, etc. THE PARABOLA. 55 PROPOSITION XIV . T H E O R E M . If any two points be taken on a tangent line to a parabola, and through these points lines parallel to the axis be drawn to meet the curve, such lines will be to each other as the squares of the distances of the points from the point of contact. The figure and construction being the same as in the foregoing proposi tion, we are to prove that AB : HK : : PA 2 : PH 2 We have AB : BC : : PC : CQ (1) (Prop. 13.) Multiplying the terms of the second PJ couplet of this proportion by PC, it/ becomes AB: BC::~PC 2 : POCQ_ (2) But, (Cor. Prop. 12) VD:BC:: ~P1? : PC- CQ (3) Dividing proportion ( 2 ) by proportion (3), term by term, we have AB ,..l 2 .i YD PI? Whence, AB :VZ>:: ~PC* : PI? (4) From the similar triangles, APC and TPD, we get the proportion * ** rt . A n . o St?\ PA*:PT ::PC :PD By comparing proportions W and (5) we find AB : YD : : ~FA In like manner we can prove that HK : VD:: PH 2 : Dividing proportion (6) by proportion (7), term by term, we have ^:l::S:l MR PH* Whence, AB : HK : : PA : Pjf Hence the theorem ; if any two points be taken, etc. 56 CONIC SECTIONS. APPLICATION. Conceive PH to be the direction in which a body thrown from the surface of the earth, would move, if it were undis turbed by the resistance of the air and by the force of gravity. It would then move along the line PH, passing over equal spaces in equal times. When a body falls under the action of gravity, one of the laws of its motion .is, that the spaces are proportional to the squares of the times of descent ; hence, if we suppose gravity to act upon the body in the direction AC, the lines AJ3, TV, HK, etc., must be to each other as the squares PA , PT , PH , etc. ; that is, the real path of a projectile in vacuo, possesses the property of the parabola that has been demonstrated in this proposition. In other words, The path of a projectile, undisturbed by the resistance of the air y is a parabola, more or less curved, depending upon the direction and intensity of the projectile force. PEOPOSITION XY. THEOBEM. The abscissas of any diameter of the parabola are to each other as the squares of their corresponding ordinates. . Let P be any point on a parabola, PL a tangent line, and PF a diame ter through this point. From the points B, V,K, etc., assumed at pleas ure on the curve, draw ordinates and parallels to the diameter, forming the quadrilaterals PCBA, PD VT, etc. !N"ow, since the ordinates to any di ameter of the parabola are parallel to the tangent line through the vertex of that diameter, these quadrilaterals are parallelograms and their opposite sides are equal. But, by the preceding proposition, we have AB : TV: HK, etc., : :~PA 2 : ~PT 2 : PH\ etc. or PCiPD: PE, etc., : : ~C 2 : ~Vff : KE*> etc. THE PARABOLA. 57 By definition 6, PC is the ordlnate and BC the abscis sa of the point B, and so on. Hence the theorem ; the abscissas of any diameter, etc. PROPOSITION XY I. THEOREM. If a secant line be drawn -parallel to any tangent line to the \parabola, and ordinates to the axis be drawn from the point of contact and the two intersections of the secant with the curve, these three ordinates will be in arithmetical progression. Let CT be the tangent line to the parabola, and EH the parallel secant. Draw the ordinates EG, CD, and HI, to the axis VI, and through E draw EK parallel to VI. We are now to prove that The similar triangles, HKE and. CDT, give the pro- p jrti n HK : KE:: CD: DT=2VD and, by proposition 12, we have 2p:KL: : HK : KE. Therefore 2p : KL : : CD : 2 VD, (1) and from the equation, y*=2px, we get, by making y= CD and x= VD, 2p: 2OD: : CD: 2FZ> (2) By dividing proportion (1) by (2), term by term, we shall have KL "Whence KL=2CD But KL=HI+KI=HI+EG; therefore HI+ EG= 2 CD Hence the theorem; if a secant line be drawn, etc. 58 CONIC SECTIONS. SCHOLIUM 1. If we draw CM parallel, and MN perpendicular to VI, then 2CD=2MN=EG-}-HI; and since MNis parallel to each of the lines EG and HI, the point M bisects the line EH. That is, the diameter through G bisects its ordinate EH) and as HE is any ordinate to this diameter, it follows that A diameter of the parabola divides into equal parts all chords of the curve parallel to the tangent through the vertex of the diameter. SCHOLIUM 2. Hence, as the abscissas of any diameter of the parabola and their ordinates have the same relations as those of the axis, namely; that the ordinates are bisected by the diameter, and their squares are proportional to the abscissas ; so all the other prop erties of this curve, before demonstrated in reference to the abscis sas and ordinates of the axis, will likewise hold good in reference to the abscissas and ordinates of any diameter. PROPOSITION XVII . T H E E E M . The square of an ordinate to any diameter of the parabola is equal to four times the product of the corresponding abscissa and the distance from the vertex of that diameter to the focus. Let "FJTbe th.e axis of aparaola, and through any point, as P, of the curve, draw the tangent P T, and the diameter PW; also draw the secant Qq, parallel PT, and pro duce the ordinate QN, and the di ameter P W, to meet at D. From the focus let fall the perpendicular FY upon the tangent, and draw FP and VY. We are now to prove that Because FYis perpendicular to PT, Qv parallel to PT and DQ parallel to each of the lines PM and VY, the triangles DQv, PMT, TFFand TFFare all similar. Whence Qv : QI> : : TF : TP (1) But ~TF*=PF* and TF= PF- VF. (Prop. 5) THE PARABOLA. 59 Substituting these values in proportion (1) and dividing the third and fourth terms of the result by PF, it becomes ~~Qvi~Qtf : :PF: VF (2) Again, from the triangles QDv and PM T we get QD :Dv ::PM: MT=2VM : : PM 2 : 2PM- VM But (Scho. 2, Prop. 8) PM*=VF VM Whence QD : Dv : : 4 VF- VM : 2PM- VM; : : 4 F.F : 2PM therefore 2PM >=4 VF-Dv (3) By subtracting the equation QN Z =4t VF- F^Vfrom the equation PM 2 =4 VF- VM, member from member, we have 4: VF- ( VM VN) Whence ^ (PM+QN] (PMQN)=(PM+QN) DQ=VF-DP (4) Subtracting eq. (4) from eq. (3), member from member, we obtain (PMQN] i>=4 VF (DvDP)= VF-Pv and because PMQNDQ, this last equation becomes Substituting this value of JDQ 2 in proportion (2), we have ? : 4VF- Pv : : PF : VF or ^v 2 : 4Pv: : PF : 1 Whence ~Qv=PF - Pv Hence the theorem ; the square of an ordinate, etc. Cor. If, in the course of this demonstration, we had used the triangle vdq in the place of vDQ, to which it is similar, we would have found that qv 2 =4PF Pv^ whence Qv=qv. And since the same may be proved for any ordi- nate, it follows that 60 CONIC SECTIONS. All the ordinates of the parabola to any of its diameters are bisected by that diameter. SCHOLIUM. The parameter of any diameter of the parabola has been defined (Def. 8) to be one of the extremes of a proportion, of which any ordinate to the diameter is the mean and the corresponding abscissa the other extreme. Now, we have just shown that Qv =qv =^PF Pv. Whence, Pv : Qv : : Qv : 4PF. 4PF, which remains constant for the same diameter, is therefore the parameter of the diameter PW. And as the same may be shown for any other diameter, we conclude that The parameter of any diameter of the parabola is equal to four times the distance from the vertex of that diameter to the focus. PKOPOSITION XYIII. THEOREM. The parameter of any diameter of the parabola is equal to the double ordinate to this diameter that passes through the focus. Through any point, as P, of the pa- / Jfy rabola of which F is the focus and V the vertex, draw the diameter PW, the tangent P T, and, through the focus the double ordinate BD, to the diameter. It is now to be proved that 4PF, or the parameter to this diameter, is equal to BD. Because PW is parallel to TX, and BD to TP, TPvF is a parallelogram, and Pv TF. But PF=FT (Prop. 4), hence Pv=PF. By the preceding proposition, Bv=4tPF-Pv =4PF-PF Whence, Bv=2PF ; therefore, 2Bv=J3D=PF i Hence the theorem ; the parameter of any diameter, etc. PROPOSITION XIX.-THEOREM. The area of the portion of the parabola included between the curve, the ordinate from any of its points to the axis, and THE PARABOLA. 61 the corresponding abscissa, is equivalent to two thirds of the rectangle contained by the abscissa and ordinate. Let VD be the axis of a parabo la, and VIP any portion of the curve. Draw the extreme ordinate PI) to the axis, and complete the rectangle VAPD ; then will the area included between the curve VIP, the ordinate PD, and the abscissa FD, be equiva lent to two thirds of the rectangle VAPD. Take any point J, between P and the vertex, and draw PI, producing it to meet the axis produced at E. Now, from the similar triangles, PQI and PDEy we get the proportion PQ: QI: : PD : DE: Whence PQ - DE= QI PD= GD - PD. (l) If we suppose the point I to approach P, the secant line PJE will, at the same time, approach the tangent PT; and finally, when I comes indefinitely near to P, the secant will sensibly coincide with the tangent PT, and DE may then be replaced by DT=2DV=2PA. Under this sup position, eq. (1) becomes 2PQ - PA=PD - GD. That is, when the rectangles GDPH and. APQ C become indefinitely small, we shall have Eect. GDPH=2HQct. APQC. We will call Kect. GDPH the interior rectangle, and Eect. APQC the exterior rectangle. If another point be taken very near to J, and between it and the vertex, and with reference to it the interior and exterior rectangles be constructed as before, we should again have the interior equivalent to twice the exterior rectangle. L et us conceive this process to be continued until all possible interior and exterior rectangles are constructed ; then would we have Sum interior rectangles=2 sum exterior rectangles. 62 CONIC SECTIONS. But, under the supposition that these rectangles are in definitely small, the sum of the interior rectangles be comes the interior curvilinear area, and the sum of the exterior rectangles the exterior curvilinear area, and the two sums make up the rectangle APD F. Therefore, if this rectangle were divided into three equal parts, the in terior area would contain two of these parts. Hence the theorem ; the area of the portion of the, etc. PROPOSITION XX. THEOREM. If a parabola be revolved on its axis, the solid generated will be equivalent to one half of its circumscribing cylinder. Conceive the parabola in the fig- ure, which is constructed as in the last proposition, to revolve on its axis VD. We are then to find the measure of the volume generated. T E V^G ~i> The rectangle ID will generate a cylinder having D Q for the radi us of its base, and DGr for its axis; and the rectangle AI will generate a cylindical band, whose length is CT, and thickness PQ. The solidity of the cylinder =nDQ*^D& The solidity of the band =7r(PZ) 2 ~DQ*) F= x[PD*(PDPQy] 7<3W[2PD - PQ P 2 ] VG E~ow, under the supposition that the point 1 is indefi nitely near to P, DQ may be replaced by PD, VG by FD, - and PQ 2 may be neglected as insensible in comparison with 2PD-PQ. These conditions being introduced in the above expressions for the solidities of the cylinder and band, they become The solidity of the cylinder =7rPZ) 2 DG The solidity of the band = 2nPD -PQ-VD THE PARABOLA. 63 "Whence, sol. of cylinder : sol. of band : : ~TI? -DG: 2PD- PQ VD (1) But, when /and P are sensibly the same point, PQ : GD : : PD : 2 VD therefore, The terms in the last couplet of proportion (1) are there fore equal, and we have sol. of cylinder : sol. of band : : 1 : 1 or sol. of cylinder=sol. of band. In the same manner we may prove that any other inte rior cylinder is equivalent to the corresponding exterior band. Hence the sum of all the possible interior solids is equivalent to the sum of the exterior solids. But the two sums make up the cylinder generated by the rectan gle VDPA; therefore either sum is equivalent to one half of the cylinder. Hence the theorem ; if a parabola be revolved, etc. REMARK. The body generated by the revolution of a parabola about its axis is called a Paraboloid of Revolution. PROPOSITION XXI. THEOREM. If a cone be cut by a plane parallel to one of its elements, the section will be a parabola. Let M VN be a section of a cone by a plane passing through its axis, and in this section draw AH parallel to the element VM. K Through AH conceive a plane to be passed perpendicular to the plane M VN; then will M" ]t -^|-- the section DA GI of the cone made by this last plane, be a parabola. In the plane MVN, draw MN and KL perpendicular to the axis of the cone, and through them, pass planes perpendicular to this axis. The sections of the cone, by these planes, will be circles, 64 CONIC SECTIONS. of which MN and KL, respectively, are the diameters. Through the points F and H, in which AH meets KL and MN, draw in the section DA GI the lines FG and HI, perpendicular to AH. Because the planes DAI and MVN are at right angles to each other, FG is perpendic ular to KL, and HI is perpendicular to MN. Now, from the similar triangles AFL, AHN, we have AF:AH::FL:HN (1) By reason of the parallels, KF=MH; multiplying the first term of the second couplet of proportion (1) by KF, and the second term by MH, it becomes AF: AH: : FL-KF: HN MH (2) But FG- is an or din ate of the circle of which KL is the diameter, and HI an ordinate of the circle of which JOT is the diameter: therefore FL-KF=FG\ and JZ2V-JfJJ=S? (Cor., Th. IT, B. Ill, Geom.) Substituting, for the terms of the second couplet, in pro portion (2), these values, it becomes A F : AH : : FG 2 : ~H1 2 This proportion expresses the property that was dem onstrated in proposition 15 to belong to the parabola. Hence the theorem ; if a cone be cut by a plane, etc. Cor. From the proportion, AF: AH: : FG 2 : HI 2 we , . get j- 1 = -TTT , that is, -r-=p or -jjj- which is a third proportional to any abscissa and the corresponding ordi nate of the section, is constant, and (by Def. 8) is the para meter of the section. THE HYPERBOLA. 65 THE HYPERBOLA. DEFINITIONS. 1. The Hyperbola is a plane curve, generated by the motion of a point subjected to the condition that the difference of its distances from two fixed points shall be constantly equal to a given line. REMARK 1. The distance between the foci is also supposed to be known, and the given line must be less than the distance between the fixed points } that is, less than the distance between the foci. REMARK 2. The ellipse is a curve confined by two fixed points called the foci ; and the sum of two lines drawn from any point in the curve is constantly equal to a given line. In the hyperbola, the difference of two lines drawn from any point in the curve, to the fixed points, is equal to the given line. The ellipse is but a single curve, and the foci are within it ; but it will be shown in the course of our investigation, that The hyperbola consists of two equal and opposite branches, and the least distance between them is the given line. 2. The Center of the hyperbola is the middle point of the straight line joining the foci. 3. The Eccentricity of the hyperbola is the distance from the center to either focus. 4. A Diameter of the hyperbola is a straight line pass ing through the center, and terminating in the opposite branches of the curve. The extremities of a diameter called its vertices. 6* E 66 CONIC SECTIONS. 5. The Major, or Transverse Axis, of the hyperbola is the diameter that, produced, passes through the foci. 6. The Minor, or Conjugate Axis, of the hyperbola bisects the major axis at right-angles; and its half is a mean proportional between the distances from either focus to the vertices of the major axis. 7. An Ordinate to a diameter of the hyperbola is a straight line, drawn from any point of the curve to meet the diameter produced, and is parallel to the tangent at the vertex of the diameter. 8. An Abscissa is the part of the diameter produced that is included between its vertex and the ordinate. 9. Conjugate Hyperbolas are two hyperbolas so related that the major and minor axes of the one are, respectively, the minor and major axes of the other. 10. Two diameters of the hyperbola are conjugate, when either is parallel to the tangent lines drawn through the vertices of the other. The conjugate to a diameter of one hyperbola will ter minate in the branches of the conjugate hyperbola. 11. The Parameter of any diameter of the hyperbola is a third proportional to that diameter and its conjugate. 12. The parameter of the major axis of the hyperbola is called the principal parameter, the latus-rectum, or simply the parameter ; and it will be proved to be equal to the chord of the hyperbola through the focus and at right- angles to the major axis. EXPLANATORY REMARKS. Thus, let FT be two fixed points. Draw a line between them, and bisect it in C. Take GAj CA , each equal to one half the given line, and CA may be any distance less than CF; A A is the given line, and is called the major axis of the hyperbola. Now, let us suppose the curve already found and represented by ADP. Take any point, as P, and join P, F and P } F r ; then ; by Def. 1, the difference between PF THE HYPEEBOLA. 67 and PF must be equal to the given line A A ; and conversely, if PF PF=A A, then P is a point in the curve. By taking any point, P, in the curve, and joining P, F and P } F a triangle PFF is always formed, having F F for its base, and A A for the difference of the sides ; and these are all the conditions necessary to define the curve. As a triangle can be formed directly opposite PF F, which shall be in all respects exactly equal to it, the two triangles having F F for a common side ; the difference of the other two sides of this opposite triangle will be equal to A A, and correspond with the con dition of the curve. Hence, a curve can be formed about the focus F f } exactly similar and equal to the curve about the focus F. We perceive, then, that the hyperbola is composed of two equal curves called branches, the one on the right of the cen ter and curving around the right-hand focus, and the other on the left of the center and curving around the left-hand focus. In like manner, by making CB equal to a mean proportional between FA and FA , and constructing above and below the center the branches of the hyperbola *of which BB =ZCB is the major, and A A the minor axis, we have the hyperbola which is conjugate to the first. PP is a diameter of the hyperbola, PT a tangent line through the vertex of the diameter, and QQ , parallel to PT and terminating in the branches of the conjugate hyperbola, is conjugate to the diameter PP . IID is the ordinate from the point H to the diameter CP, and PD is the corresponding abscissa. PROPOSITION I. -PROBLEM. To describe an hyperbola mechanically. Take a ruler, F H, and fasten one end at the point JP, on which the ruler may turn as a hinge. At the other end, at tach a thread, the length of which is less than that of the 68 CONIC SECTIONS. ruler by the given line A A. Fas ten the other end of the thread at F. "With the pencil, P, press the thread against the ruler, and keep it at equal tension between the points H and F. Let the ruler turn on the point F , keeping the pencil close / \ to the ruler and letting the thread slide round the pen cil ; the pencil will thus describe a curve on the paper. If the ruler be changed, and made to revolve about the other focus as a fixed point, the opposite branch of the curve can be described. In all positions of P, except when at A or A , PF r and PF will be two sides of a triangle, and the difference of these two sides is constantly equal to the difference be tween the ruler and the thread ; but that difference was made equal to the given line A A ; hence, by Definition 1, the curve thus described must be an hyperbola. Cor. From any point, as P, of the hyperbola, draw the ordinate PD to the major axis, and produce this ordinate to P , making DP equal to PD; and draw FP, FP , F P and F P . Then, because F D is a perpendicular to PP at its middle point, we have FP=FP , and F*P=* F P ; whence F PFP=F P FP , and P is a point of the hyper bola. Therefore, PP is a chord of the hyperbola at right angles to the major axis, and is bisected by this axis ; and as the same may be proved for any other chord drawn at right angles to the major axis, we conclude that All chords of the hyperbola which are drawn at right angles to the major axis are bisected by that axis. It may be proved, in like manner, that All chords of the hyperbola which are drawn at right angles to the conjugate axis are bisected by that axis. THE HYPERBOLA. 69 PROPOSITION II. THEOEEM. If a point be taken within either branch of the hyperbola, or on the concave side of the curve, the difference of its distances from the foci will be greater than the major axis; and if a point be taken without both branches, or on the convex side of both curves, the difference of its distances from the foci will be less than the major axis. Let A A be the major axis, and F and F f the foci of an hyperbola. Within the branch APX take any point, Q, and draw FQ and F Q; then we are to prove F A 7 A F First. That F QFQ is greater than A A . Since Q is within the branch APX, the line F Q must cut the curve at some point, as P. Draw PF and FQ. By the definition of the hyperbola, FT PF= AA . Adding PQ+PF to both members of this equation, it becomes F PPF+PQ+PF=AA + PQ+PF or, F Q=AA + PQ+PF. But PQ and PF being two sides of the triangle FPQ, are together greater than the third side FQ. Therefore F Q>AA + FQ; and, by taking FQ from both members of this inequality, we have F Q-FQ>AA . Second. Take any point, q, without both branches of the hyperbola, and join this point to either focus, as F. Then since q is without the branch APF, the line qF must cut the curve at some point, P. Draw qF, qF , and PF . Because P is a point on the curve, we have PF f PF =AA . Adding Pq+PFto the members of this equa tion it becomes PF r PF+ Pq+ PF= A A + PF+ Pq or, PF +Pq=AA +PF+Pq=AA +qF. 70 CONIC SECTIONS. But PF f and P/, being two sides of the triangle F Pq, are together greater than the third side qF . "Whence qF <AA r -\-qF; and by taking qF from both members of this inequality, we have qF qF<A A . Hence the theorem ; if a point be taken, etc. Cor. Conversely : If the difference of the distances from any point to the foci of an hyperbola be greater than the major axis, the point will be within one of the branches of the curve ; and if this difference be less than the major axis, the point will be without both branches. For, let the point Q be so taken that F Q FQ>AA f ; then the point Q cannot be on the curve ; for in that case we should have F Q FQ= A A f . And it cannot be with out both branches of the curve, for then we should have F Q FQ<AA , from what is proved above. But it is contrary to the hypothesis that F Q FQ is either equal to or less than A A ; hence the point Q must be within one of the branches of the hyperbola. In like manner we prove that, if the point q be so cho sen that qF f qF<AA , this point must be without both branches of the hyperbola. PEOPOSITION III. THEOKEM.- A tangent to the hyperbola bisects the angle contained by lines drawn from the point of contact to the foci. Let F , F be the fgci, and -P any point on the curve; draw PF , PF and bisect the angle F PF\>j the line TT ; this line will be a tangent at P. If TT be a tangent at P, ev- F^A 7 c TA"F ery other point on this line will be without the curve. THE HYPERBOLA. 71 Take PG=PF and draw GF; TT bisects GF, and any point in the line TT is at equal distances from F and G (Scho. 1, Th. 18, B. I, Geom). By the definition of the curve, F f Gr=A A the given line. ~Now take any other point than P in T T , as E, and draw EF , EF and EG. Because JEFia equal to EG we have EF EF=EF EG. ^uiEF EG, is less than F G, because the differ ence of any two sides of a triangle is less than the third side. That is, EF EF is less than A A; consequent ly the point E is without the curve (Prop. 2), and as E is any point on the line TT , except P, therefore, the line TT , which bisects the angle at P, is a tangent to the curve at that point. Hence the theorem ; a tangent to the hyperbola, etc. SCHOLIUM. It should be observed that by joining the variable point, P, in the curve, to the two invariable points, F and F, we form a triangle; and that the tangent to the curve at the point P, bisects the angle of that triangle at P. But when any angle of a triangle is bisected, the bisecting line cuts the base into segments proportional to the other sides. (Th. 24, B. II, Geom). Therefore, F P : PF=F T : TF Kepresent P Pby r and PF by r; then r : r=F f T : TF But as / must be greater than r by a given quantity, a, therefore, r+a : r=F T : T F Or, 1+^ : \=F T : TF Let it be observed that a is a constant quantity, and r a variable one which can increase without limit; and when r is immensely great in respect to a, the fraction - is extremely minute, and the first term r of the above proportion would not in any practical sense differ from the second; therefore, in that case, the third term would not essen- 72 CONIC SECTIONS. tially differ from the fourth; that is, F T does not essentially differ from FT when r, or the distance of P from F is immensely great. Hence, the tangent at any point P, of the hyperbola, can never cross the line FF at its middle point, but it may approach within the least imaginable distance to that point. If, however, we conceive the point P to be removed to an infinite distance on the curve, the tangent at that point would cut AA at its middle point C } and the tangent itself is then called an asymptote. PKOPOSITION I Y. THEOREM. Every diameter of the hyperbola is bisected at the center. Let F and F be the foci, and A A 1 the major axis of an hyperbo la. Take any point, as P, in one of the branches of the curve ; draw PF and PF , and complete the parallelogram PFP F . We will now prove that P is a point in the opposite branch of the hyperbola, and thai PP passes through, and is bisected at, the center, C. Because PFP F is a parallelogram, the opposite sides are equal; therefore F PPF=FP P F ; but since J is, by hypothesis, a point of the hyperbola, F P PF AA ; hence FP P f F = AA , and P is also a point of the hyperbola. Again, the diagonals, F F, P P of the parallelogram, mutually bisect each other ; hence C is the middle point of the line joining the foci, and (Def. 2) is the center of the hyperbola. P P is therefore a diameter, and is bi sected at the center, C. Hence, the theorem ; every diameter of the hyperbola, etc. PROPOSITION Y. THEOREM. Tangents to the hyperbola at the vertices of a diameter are parallel to each other. - . THE HYPERBOLA. 73 At the extremities of the diam eter, PP , of the hyperbola repre sented in the figure, draw the tan gents TT and VV. We are now to prove that these tangents are parallel. By proposition (Prop. 3) TT bisects the angle FPF > and V V also bisects the angle F P F. But these angles being the opposite angles of the parallelogram FPF P , are equal; therefore the [__T PF=ihe [__PT / F=ihe [_ VP F- But the LJs PT F, VP F, formed by the line FP meet ing the tangents, are opposite exterior and interior an gles. The tangents are therefore parallel (Cor. 1, Th. 7, B. I, Geom). Hence the theorem ; tangents to the hyperbola, etc. PROPOSITION Y I . THEOREM. The perpendiculars ktfall front the foci of an hyperbola on any tangent line to the curve, intersect the tangent on the circum ference of the circle described on the major axis as a diameter. In the hyperbola of which A A is the major axis, F and F the foci, and C the center, take any point in one of the branches, as P, and through it draw the tan gent line TH . From the foci let fall on the tangent the perpendic ulars FH, F H , draw PF and PF , and produce FH to intersect PF in 6r. We are now to prove that H and H f are in the circumference of a circle of which AA f is the diameter. Draw CH 9 producing it to meet F H in Q. Then, because Pil is a tangent to the curve, it bisects the angle FPF ; therefore the right-angled triangles, FPH and 74 CONIC SECTIONS. HPG, being mutually equiangular, and having the side PH common, are equal. "Whence, FIIHG and PF= PG. But, by the definition of the hyperbola, F f PPF =AA f ; }QncQF f PPG=F f G=AA f . Since CH bisects the sides F F and FG of the triangle FGF , we have F F:FC::F f G: CH but F F=2FC; therefore F f G=2CH=AA If then with C as a center and CA as a radius, a cir cumference be described, it will pass through the point H. Again ; the triangles FHC and F r CQ are in all respects equal ; hence CQ= CH, and Q is alstf a point in the cir cumference of the circle of which A A is the diameter. Therefore, the right-angled triangle QH H, having for its hypotenuse a diameter HQ of this circle, must have the vertex, H of its right angle at some point in the cir cumference. Hence the theorem; t\e perpendiculars let fall, etc. PROPOSITION VII. THEOEEM. The product of the perpendiculars let fall from the foci of an hyperbola upon a tangent to the curve at any point, is equal to the square of the semi-minor axis. Resuming the figure of the pre ceding proposition ; then, since the semi-minor axis, which we will represent by 13, is a mean propor tional between the distances from either focus to the extremities of the major axis, we are to prove that jB 2 = FA x FA =FHx F H By the construction, the triangles FHC and CQF f are equal; therefore FH=F Q (1) THE HYPERBOLA. 75 Multiplying both members of eq. (1) by F H it be comes FH - F H =F> Q F H (2) Again, it was proved in the last proposition that the points H, H f and Q were in the circumference of the cir cle described on A A as a diameter; therefore F H and F A are secants to this circumference, and we have F Q : F A : : F A :^ H f (Cor., Th. 18, B. EJ, Geom). Whence, F Q - F H =F A - F A (3) But F A ^FA, F A=FA , and F Q=-FH. Making these substitutions in eq. ( 3 ) it becomes FH - F H =FA - FA =B 2 . Hence the theorem : the product of the perpendiculars, etc. Cor. 1. The triangles PFH, PF H are similar ; therefore, PF : PF f : : FH : F H That is : The distances- from any point on the hyperbola to the foci, are, to each other, as the perpendiculars let fall from the foci upon the tangent at that point. Cor. 2. From the proportion in corrollary 1, we get __ PF-F H =,2 PF-F H -FH FH= -- pF , ; whence FH =- But by the proposition, F H FH=& ; _ 2 ^2 . pF 2 . pF therefore, FH = pF , ^^CA+PF^ tecause F AA =2CA, and PG=PF. In like manner it may be proved that BZ PF _B\2 CA+PF) PF PF PROPOSITION YI II. THEOREM. If a tangent be drawn to the hyperbola at any point, and al so an ordinate to the major axis from the point of contact, then will the semi-major axis be a mean proportional between the 76 CONIC SECTIONS. distance from the center to the foot of the ordinate, and the dis tance from the center to the intersection of the tangent with this axis. Let A A be the major axis, F$ f the foci and C the center of the hyperbola. Through any point, as P, taken on one of the branch es, draw the tangent PT intersect ing the axis at T; also draw PF, PF f to the foci, and the ordinate PM to the axis. "We are now to prove that CT : CA. :: CA: CM. Because P T bisects the vertical angle of the triangle FPF (Prop. 3), it divides the base into segments pro portional to the adjacent sides (Th. 24, B. II, Geom.) Therefore, F T: TF : : F P: PF. Whence, F TTF:F / T+ TF: : F PPF: F P+PF That is, 2CT : F F : : AA =2CA : F P+PF Or, by inverting the means, 2 CT : 2 CA : : F F : F P+PF (D Now, making MF"=MF, and drawing PF", we have, from the triangle F PF", F F" : F P+PF" : : F PPF" : F MMF" (Prop 6, PL Trig.) But, because the triangle FPF " is isosceles, and PM is a perpendicular from the vertical angle upon the base, therefore the preceding proportion becomes 2CM: F P+PF: : 2CA : F F or, 2 CM : 2 CA : : F P+PF : F F (2) Multiplying proportions (1) and (2), term by term, ob serving that the terms of the second couplet of the result ing proportion are equal, we have THE HYPERBOLA. 77 Whence, CT-CM=CA; which, resolved into a proportion, becomes CT:CA: : CA : CM. Hence the theorem ; if a tangent be drawn, etc. SCHOLIUM. The property of the hyperbola demonstrated in this proposition is not restricted to the major axis, but also holds true in reference to the minor axis. The. tangent intersects the minor axis at the point t, and PG is an ordinate to this axis from the point of contact. Now, the simi lar triangles tCT, TIIF, give the proportion Ct : FH:\ CT : TH (1) and from the similar triangles PMT, TF IT, we also have PM: F U r.MT: ITT (2) Multiplying proportions (1) and (2), term by term, we get Ct-PM : FH-F H : : CT-MT : TH-H T (3) But FH-F H =B i (Prop. 7). Moreover, drawing the ordinate TV, and the radius CVoi the circle, and the line VJ\i, we have by the proposition CT: CA .-.CA : CM or, CT: CV::CV: CM Therefore, the triangles VCT and MCV, having the angle C common and the sides about this angle proportional, are similar (Cor. 2, Th. 17, B. II, Geom.) ; and because the first is right-angled, the second is also right-angled, the right angle being at F; hence VT*=CT-MT(Th. 25, B. II, Geom). Also, A A and HH r are two chords of a circle intersecting each other at T; hence IIT-TH =AT TA = VT* (Th. 17, B. Ill, Geom). Substituting for the terms of proportion (3) these several values, it becomes Ct-PM: B*:: VT* : FT ::1 : 1 Whence, Ct-PM=JP Therefore, Ct : B : : B : PM= CG 7* 78 CONIC SECTIONS. Cor. It has been proved that the triangle CVM is right- angled at V; therefore, VM is a tangent at the point V to the circumference on A A as a diameter, and Of is its sub-tangent. But TMia also the sub-tangent on the ma jor axis of the hyperbola answering to the tangent PT; hence If a tangent be drawn to the hyperbola at any point, and through the point in which the tangent intersects the major axis an ordinate be drawn to the circle of which this axis is a diam eter, the sub-tangent on the major axis corresponding to the tan gent through the extremity of this ordinate will be the same as that of the tangent to the hyperbola. PROPOSITION IX. THEOREM. In any hyperbola the square of the semi-major axis is to the square of the semi-minor axis, as the rectangle of the dis tances from the foot of any ordinate to the major axis, to the vertices of this axis, is to the square of the ordinate. Resuming the figure to Propo- sition 8, the construction of which needs no further explanation, we are to prove that ~CA 2 rCif : : A M-AM: PM\ assuming CB to represent the semi-minor axis. From the similar triangles PMT, TIIF and TH F , we derive the proportions PM-.FH: : MT : TH PM: F H : : MT : TH r Whence pjf 2 . F H-F H f : : MT 2 : TH- TR (1) But FII-F H is equal to the square of the semi-minor axis (Prop. 7); and because the chords, HH f and AA , of the circle intersect each other at T, we have THE HYPERBOLA. 79 (Th. IT, B. in, Geom.) These values of tlie consequents of proportion (1) be ing substituted, it becomes PM 2 : "SO 2 : : MT* : W (2) The triangles CVTand TVM are similar, and give the proportion : VT* :: VM 2 : OF 2 =aT (3) Comparing proportions (2) and (3), we find that 2 PM : C : : VM :OA (4) Because M Vis a tangent and MA f a secant to the cir cle A VA H , we have VM*= A M - AM (Th. 18, B. Ill, Geom.) Placing this value of VM 2 in proportion (4) and invert ing the means of the resulting proportion, it becomes PM 2 : A M- AM: : ~BC* :~CA* or, "CT : ~BG 2 : : A M- AM: PM* Hence the theorem ; in any hyperbola the square of the, etc. Cor. Proportion (4) above may be put under the form ~CA 2 : ~BC 2 : : VM* : PM 2 (a) and from the right-angled triangle CVM we have from which, because CV= CA, we get Also, the right-angled triangles CVM, VTMare similar; therefore, CM:VM: : VM : MT - Whence VM 2 = CM- MT. Now, if in proportion (a) we place for VM these val ues, successively, we shall have the two proportions C : : : CM-MT: PM (b) and "CT : ~BC? : : CM -^OA 2 : PM 2 (c) 80 CONIC SECTIONS, SCHOLIUM 1. Let us denote CA by a, CB by b, CMbj x, and PM byy; then A M=x-}~ a and AM=x-a. Because CM* OA* =(CM+CA) (CMCA)=AM t - Disproportion (c), by substitu tion, now becomes a * : tf : : (^-fa) (x a) : y*. (a ) Whence aY=Vx*a?V or, ay 6V= a 2 6 9 . This equation is called, in analytical geometry, Ae equation of the hyperbola referred to its center and axes, in which x, the distance from the center to the foot of any ordinate to the major axis, is called the abscissa. The equation a 2 ^* & 2 x 2 = a 2 & 2 therefore ex presses the relation between the abscissa and ordinate of any point of the curve. SCHOLIUM 2. Let y denote tlfe ordinate and x the abscissa of a second point of the hyperbola; then we shall have a a : V : : (x +a) (x a) : y * Comparing this proportion with proportion (a ), scholium 1, we find f :y":i (x-f-a) (x a) : (x +a) (x a} That is : In any hyperbola the squares of any two ordinates to the major axis are to each other, as the rectangles of the corresponding distances from the feet of these ordinates to the vertices of the axis. A similar property was proved for the ellipse and the parabola. PROPOSITION X. THEOREM. The parameter of the major axis, or the latus-rectum, of the hyperbola is equal to the double ordinate to this axis through the focus. Through the focus F of the hyperbo- la, of which AA is the major and BB f the minor axis, draw the chord PP f at right angles to the major axis; then de- noting the parameter by P, we are to prove that AA : BE : : BB : PP =P (Def. 11.) THE HYPERBOLA. 81 By definition 6, BC =FA FA, and by proposition 9 we have ~AC 2 : ~BC 2 : : FA FA : PF=(JPP ) 2 (Cor. Prop. 1.) "Whence ~AC* : ~BG Z : : ~B(f : (^PPJ Therefore AC .BG: :JBC: \PP (Th. 10, B. II, Geom.) Multiplying all the terms of this last proportion by 2, it becomes or, AA f : BB : : BB : PP f Hence the theorem ; the parameter of the major axis, etc. PROPOSITION XI. THEOREM. If from the vertices of any two conjugate diameters of the hyperbola ordinates be drawn to either axis, the difference of the squares of these ordinates will be equal to the square of one half the other axis. Let AA , BB f be the axes, and PP , QQ f any two conjugate diam eters of the conjugate hyperbolas represented in the figure. Then, drawing the ordinates QV, PM, to the major axes, and the ordinates PS=MC, QD= VC, to the minor axis, it is to be proved that and that ~GB*==~QV 2 -. Draw the tangents PT and Qt, the first intersecting the major axis at Tand the minor axis at T r , and the second intersecting the minor axis at t and the major axis at t. ISTow, by proposition 8, we have, with reference to the tangent PT, CT: CA::CA: CM, 82 CONIC SECTIONS. and by the scholium to the same proposition, we also have, with ference to the tangent Qt to the conjugate hyperbola, Ct: CA =CA: : CA : CV The first proportion gives CA = CT- CM, and the sec ond ~CA 2 = Ct CV, Whence CT- CM= Ct CV, which, in the form of a proportion, becomes CM i CV:: Ct : CT (l) From the similar triangles tCQ, CTP, we get Ct:CT::QC:PT (2) and from the triangles CQV, TPM QC: PT: : CV : TM (3) Comparing proportions (1), (2) and (3), it is seen that CM i CV: : CV: TM "Whence CV 2 = CM- TM; but TM= CMCT; Therefore ~CV**= CM 2 CT- CM. And because CT - CM= CA (Prop. 8), we have or, ~CA 2 =CM 2 ~CV 2 Again we have CT : CB:: CB: PM (Scho., Prop. 8) and Ct : CB : : CB : CD=QV (Prop. 8) Whence CT - PM= Ct - Q V, which, resolved into a proportion, becomes PM: QV: : Ct : CT f (4) From the similar triangles, T f CP, Ct Q, we get Ct : CT f : : t Q: CP (5) And from the triangles t DQ, CPM, we also get t Q: CP: : t D : PM (6) From proportions (4), (5) and (6) we deduce THE HYPERBOLA. 83 PMi Whence PM == Q V- t D; but t D= -Ct r therefore, PM 2 =QV 2 Ct f - QV=~QV* ^ And because Ct f OD= CB 2 (Prop. 8) we " ave or CB 2 =QV 2 PM 2 Hence the theorem ; from the vertices of any two, etc. Cor. By corollary to proposition 9 we have ~CA 2 :~CB 2 : : O 2 ~OZ 2 : PM* In like manner, in reference to the conjugate hyperbo la, we shall have :CA : ij Ctf: CV* or, By composition, -^C : :~CA : : ~QV 2 : :~CA 2 i~CA 2 +CV* But by this proposition we have ~CA 2 = CM 2 CV 3 ; hence 7?A* therefore ~GB 2 : ~QV 2 : : "OT : CM Whence CB: QV:: CA: CM or, CAiCBn CM-. QV CM* PROPOSITION XII. THEOREM. The difference of the squares of any two conjugate diameters of an hyperbola is constantly equal to the difference of the squares of the axes. In the figure, which is the same as that of the preceding proposi tion, PP f and QQ f are any two con jugate diameters (Def. 10). It is to be proved that PP 2 ~QQ 2 =AA f2 SB? By proposition 11 we have 84 CONIC SECTIONS. and CA 2 = CM 2 ~CB = QV 2 PM therefore CA 2 CB 2 = CM*+ PM 2 ( CV 2 + Q V 2 } Multiplying each member of this equation by 4, observ ing that 4iCA 2 AA &c., it becomes AA 2 BB 2 =PP 2 QQ* 2 Hence the theorem ; the difference of the squares, etc. PROPOSITION XIII. THEOREM. The parallelogram formed by drawing tangent lines through the vertices of any two conjugate diameters of the hyperbola is equivalent to the rectangle contained by the axes. Let LMNO be a parallelogram formed by drawing tangent lines through the vertices of the two con jugate diameters PP f , QQ f of the conjugate hyperbolas represented in the figure. It is to be proved that area LMNO=AA xBB f . We have CA : CB : : CS : QV (1) (Cor. Prop 11.) Also, CT: CA : : CA : CS (2) (Prop. 8.) Multiplying proportions (1) and (2), term by term, omit ting in the first couplet of the result the common factor CA, and in the second the common factor CS, we find CT: CB:: CA: QV Whence CT - Q V= CA - CB But CT- QV measures twice the area of the triangle CQT, and this triangle is equivalent to the half of the parallelogram QCPL, because they have the common base QC and are between the same parallels QC, LT (Th. 30, B. I, Geom.) THE HYPERBOLA. 85 Now the parallelogram QCPL is one-fourth of the par allelogram LMNO, and CA CB measures one fourth of the rectangle contained "by the axes ; therefore the paral lelogram and rectangle are equivalent. Hence the theorem ; the parallelogram formed, etc. PROPOSITION XIV. THEOREM. If a tangent to the hyperbola be drawnthrough the vertex of the transverse axis, and an ordinate to any diameter be drawn from the, same point, the semi-diameter will be a mean proportional between the distances, on the diameter, from the center to the tan gent, and from the center to the ordinate. Let CA be the semi-major axis and CP any semi-diameter of the hyper- bola. Draw the tangents At, PT, the ordinate AH to the diameter, and the ordinate PMto the major axis. It is T AV M now to be proved that~OP 2 = Ci - CH. "We have CT : CA : : CA : CM, (Prop. 8) also CAiCti: CM: OP from the similar A s CAt, CMP Multiplying these proportions term by term, omit ting in the result the common factor in the first couplet, and also that in the second, we find CT:Ct::CA:CP (1) Again we have CP .CT: : CH: CA from the similar A s CPT, CHA. Proceeding with these last proportions as with those above, we find CP^Ct:: CH: CP Whence, CP 2 =Ct-CH. Hence the theorem ; if a tangent to the hyperbola, etc. Cor. 1. From proportion (1) we get CT- CP= Ct - CA; but the triangles CTP, CAt, having a common angle, C, are 86 CONIC SECTIONS. to each other as the rectangles of the sides about this an gle (Th. 23, B. II, Geom.) Therefore ACTP=ACLi. Cor. 2. If from the equivalent areas ACTP, A(M we take the common area C TVt there will remain A TA Y== Cor. 3. If we add to each of the triangles TAV, tVP, the trapezoid VAMP, we shall have area &TMP= area tAMP. PROPOSITION XY.-THEOREM. If through any point of an hyperbola there be drawn a tan gent, and an ordinate to any diameter, the semi-diameter will be a mean proportional between the distances on the diameter from the center to the tangent, and from the center to the ordi nate. Take any point as D on the hy perbola of which CA is the semi- major axis, and through this point B draw the tangent DT and the semi- diameter CD, also take any other point, as P, on the curve, and draw c the tangent Pi, the ordinate PIfto the diameter through D, and the ordinates PQ and DG to the axis. The semi-diameter CD and the tangent Pi intersect each other at t . "We will now prove that Let CB represent the semi-conjugate axis, then by co rollary to proposition 9 (proportion (b)) we have ~3Z 2 : "CB 2 :: CG TG and -CA 2 : "OS 2 : : CQ-tQ : Whence C& TG : CQ : tQ : D(? : ~PQ* but ~D(? : ~PQ 2 : :~TG? : ~LQ\ from the similar A s TGD, LQP-, THE HYPERBOLA. 37 therefore CG TG : CQ-tQ : : TG : LQ (1) Drawing Dm parallel to Pt we have te similar A s j tQP which give the proportion Da : PQ : : Gm : Qt. (2) The A s TGD, LQP also give DG : PQ : : TG : LQ (3) From proportions (2) and (3) we get TG:LQ::Gm:Qt (4) .Multiplying proportions (1) and (4) term by term, there results, CG ~TG 2 : CQ-tQ LQ: : ~TG 2 Gm : ~LQ 2 -Qt Dividing the first and third terms of this proportion by !T6r 2 and the second and fourth terms by Qt LQ it be comes CG : CQ : : Gm : LQ or CG : Gm:: CQ: LQ (5) Whence CG : CGGm : : CQ : CQLQ That is CG : Cm : : CQ : CL (6) Again CT-CG= CA*= CQ - Ct, (Prop. 8.) therefore CG^: Ct : : CQ : CT The antecedents in this last proportion and in propor tion (6) are the same, the consequents are therefore pro portional, and we have Ct: CT:: Cm : CL We have also, Cm : CD : : Ct : Ct from the similar A s CmD, Ctt r And CT: CD:: CL : Off from the similar A s CTD CLH By the multiplication of the last three proportions term by term we find Ct Cm-CT:~CD 2 CT^: Cm-Ct-CL: CL Ct -CH Whence CT:~CD 2 -CT:: CL : CL Ct -CII -or l:~CD 2 : : I : Ct f -CII therefore ~CI?= Ct - CII 88 CO NIC SECTIONS. Hence the theorem ; if through any point of an, etc. REMARK. The property of the hyperbola just established is the generalization of that demonstrated in the preceding proposition. PROPOSITION XVI. THEOREM. The square of any semi-diameter of the hyperbola is to the square of its semi-conjugate as the rectangle of the distances from the foot of any ordinate to the first diameter, to the ver tices of that diameter, is to the square of the ordinate. Let PP f and QQ be any two conjugate diameters of the conju gate hyperbolas represented in the figure. Through any point as Cr draw the tangent G-T inter secting the first diameter at T and the second at T f , and from the same point draw the ordinates GrH, GrK, to these diameters. "We will now prove that, ~CP 2 : ~C : : PH-P H* _ By the preceding proposition we have CP = CT- CH and multiplying each member of this equation by CH it ~ becomes CP 2 - CH= CT- GH Whence CPj^Cff* 1 1 CT : Offfrom which by division we get CP 2 : CH 2 CP 2 : : CT : CHCT=TH, (1) Again we havel7 2 = CT C2"(Prop. 15) and multi plying each member of this equation by CK it becomes Whence CQ : CK : : CT : CK=GH (2) The similar A s TCT , THGr give the proportion CT : aH: : CT: TH (3) Comparing proportions (2) and (3) we obtain CQ : CK* ::CT: Til (4) THE HTPERBOLA. 89 And by comparing proportions (1) and (4) we obtain ~CQ* : ~CK 2 : ~CP 2 : CH 2 CP 2 or CP 2 : ~CQ 2 : CH 2 CP 2 : CK 2 =GH 2 But because CP=CP and ~CH 2 ~CP 2 =(CHCP) (CH+CP) = PH- (CH+CP) the last proportion above becomes ~CP* : ~CQ 2 : : PH-P H: GH* Hence the theorem ; The square of any semi-diameter , etc. REMARK. The property of the hyperbola with reference to any two conjugate diameters just demonstrated is the same as that with reference to the axes established in proposition 9. Cor. If the ordinate GH be produced to intersect the curve at G f and the above construction and demonstra tion be supposed made for the point G f instead of 6r, we should finally get the same proportion as before, except the fourth term, which would be G H ; therefore, G H GH. Hence we conclude that Any diameter of the hyperbola bisects all the chords drawn parallel to a tangent line through the vertex of that diameter. PROPOSITION XYII . T H E R E M . The squares of the ordinates to any diameter of the hyper bola are to one another as the rectangles of the corresponding distances from the feet of these ordinates to the vertices of the diameter. Resuming the figure to the proposition which precedes and drawing any other ordinate gh to the diameter PP , it is to be proved that ~GH 2 :~gh 2 : . PH P H : Ph-P f h By the foregoing proposition we have two proportions following, viz : ~CP 2 :~CQ 2 :: PH-P H^ CP 2 : CQ 2 :: Ph -P h : gh 8* 90 CONIC SECTIONS. Since the ratio CP 2 : CQ 2 is common to these pro portions the remaining terms are proportional. That is GH 2 i^h 2 i-.PH -P f H : Ph -P h Hence the theorem The squares of the ordinates, etc. PROPOSITION XYIII.-THEOKEM. If a cone be cut by a plane making an angle with its base greater than that made by an element of the cone, the section will be an hyperbola. Let the A s MVN, BVR\)Q the sections of two opposite cones by a plane through the common axis, and PH a line in this section not pass ing through the vertex, and making with MN the \_BHN> the [_BMN. Through this line pass a plane at right angles to the first plane, mak- ing in the lower cone the section IGAG f I f ; then will this section be one of the branches of an hyperbola. Let KL and MNloe the diameters of two circular sec tions made by planes at right ^ angles to the axis of the cone, and at F and JT, the intersections of these lines with BH, erect the perpendiculars FG, HI to the plane MVN. FG is the intersection of the plane of the section IGA G f l with the plane of the circle of which KL is the diameter and is a common ordinate of the section and oi the circle ; so likewise is HI a common ordinate of the section and of the circle of which MN is the diameter. Now by the similar A s AFL, AHN, and BFK, BHM we have AFiAHi-.FL .HN (1) and BF:BH::FKiHM (2) Multiplying proportions (1) and (2), term by term, we get THE HYPERBOLA. 91 AF -BF : AH-BH : : FL-FK : HN -HM (3) But because LGK and NIM are semi-circles, FGr 2 = FL-FK and ~ILI*=HN HM. Substituting these values for the terms of the last couplet of proportion (3) it be comes AF-BF: AH-BH-. : W : ~HI* If we denote any two ordinates of the corresponding section of the opposite cone by ^ and hi we should have in like manner Af-Bf : Ah -Bh : : (fg) 2 : (hi) 2 If, therefore, AB be taken as a diameter of the curves cut out of the opposite cones by a plane through AH, at right angles to the plane VMN 9 we have proved that these curves possess the property which was demonstra ted in the preceding proposition to belong to the hyper bola. Hence the theorem ; if a curve be cut by a plane, etc. ASYMPTOTES. DEFINITION. An Asymptote to a curve is a straight line which continually approaches the curve without ever meeting it, or, which meets it only at an infinite distance. We shall for the present assume, what will be after wards proved, that the diagonals of the rectangle con structed by drawing tangent lines through the vertices of the axis of the hyperbola possess the property of asymp totes, and they are therefore called the asymptotes of the hyperbola. PROPOSITION XIX. THEOREM. If an ordinate to the transverse axis of an hyperbola be produced to meet the asymptotes, the rectangle of the segments into which it is divided by either of its intersections with the curve willbe equivalentto the square of the semi-conjugate axis. 92 CONIC SECTIONS. Let CA, CB be the semi-axes and Ct, Ct the asymptotes of an hyperbola. Through any point, as P, of the curve, B draw the ordinate PQ to the major axis and produce it to meet the asymptotes at n c and n . By the enunciation we are re quired to prove that CB 2 =Pn Pn By Cor. proposition 9 we have ~CA 2 : "OB 2 : :~C 3 ~CZ a : ~PQ 2 (1) And from the similar triangles CAB , CQn $, 2 C : =C : :~CQ 2 : ~fyf (2) Comparing proportions (1) and (2) we find ~CQ 2 j -CQ^-CA 2 : ~Q^ 2 : ~PQ 2 which gives by division CA * . C * . : - or ~SI 2 : Qn 2 ~PQ 2 : :~CQ 2 : Qn* (3) From proportions (2) and (3) we get In this proportion the antecedents are the same the consequents are therefore equal ; that is ~ ~ (QnPQ)=Pn-Pri Hence the theorem ; if an ordinate to the major axis, etc. Cor. Let us take another point p in the curve and from it draw the ordinate pQ f to the major axis ; then, as be fore, we shall have CB 2 <= pt -pt f ; t and t r being the in tersections of the ordinate, produced, with the asymptotes. Whence Pn Pn =pt -pt r , which in the form of a pro portion becomes Pn : Pt : : pt : Pn PROPOSITION XX . T H E R E M . The parallelograms formed by drawing through the different points of the hyperbola lines parallel to and meeting the asymp totes are equivalent one to another, and any one is equivalent to one half of the rectangle contained by the semi-axes. IV P THE HYPERBOLA. Let CA, CB be the semi-axes and Cn, Cn the asymptotes of an hyperbola. From any point, as P, of the curve draw the ordi- J nate PQ to the transverse axis, producing it to meet the asymptotes at n, n f , and through P and the vertex A draw parallels to the b asymptotes, forming the parallelograms PmCl, AECD. This last is a rhombus because its adjacent sides CE, CD are equal, being the semi-diagonals of equal rectangles. It will now be proved that Area PmCi = area AECD=\ Eect. AB BC. By the proposition which precedes we have ~CI?=Pn - Pn (1) And from the similar triangles AB E, Pnm, and the similar triangles ADb , Pin we also have AE : AB f =CB : : mP : Pn AD:Ab f =CBi : Pt : Pn Multiplying these proportions, term by term, we find AE AD : ~CB 2 : : mP Pt : Pn - Pn By equation (1) the consequents of this proportion are equal, therefore the antecedents are also equal. That is, AE AD=mP Pt If the first member of this equation be multiplied by sin. [_DAE, and the second member by the sine of the equal \_mPt it becomes AE- AD sin. DAE=mP Pt sin mPt But AE -AD sin D AE measures the area of the rhom bus AECD and mP Pt sin. mPt measures the area of the parallelogram PmCt; therefore the parallelogram and the rhombus are equivalent. Moreover, because the A s AEC, ADC are equal, and the A s AEC, AEB f are equivalent, it follows that the rhombus AECD is equiva- 94 CONIC SECTIONS. lent to the &AB C, or, to one half of the rectangle con tained by the semi-axes. Hence the theorem; the parallelograms formed, etc. Cor. 1. If from the rhombus AECD and the parallel ogram PmCtihe common part be taken, there will remain the parallelogram AKtD, equivalent to the parallelogram PmEK, and if to each of these the curvilinear area AKP be added, we shall have Area APmE= area APtD. Had we proceeded in the same way with the parallelo gram PmCt and any parallelogram other than AECD we should have had a like result ; therefore If from any two points in the hyperbola parallels be drawn to each asymptote, the area bounded by the parallels to one asymptote, the other asymptote, and the curve will be equivalent to the other area like bounded. SCHOLIUM. If the product AE-AD, which is a constant quan tity be denoted by a, the distance Cm by or, and the distance mp= Ct by y, then, by this proposition, we shall have the equation xy=a, which, in analytical geometry, is called the equation of the hyperbola referred to its center and asymptotes. Cor. 2. In the equation xy=a,y expresses the distance of any point of the curve from the asymptote on which x is estimated. From this equation we get y=-. Now 2 it is evident that as x increases y decreases, and finally when x becomes infinite, y becomes zero. That is, the asymptote continually approaches the hyperbola without ever meeting it, or without meeting it within a finite dis tance. "We were, therefore, justified in assuming that the diagonals of the rectangle formed by the tangents through the vertices of the axes were asymptotes to the hyperbola. ANALYTICAL GEOMETRY. (95) ANALYTICAL GEOMETRY. GENERAL DEFINITIONS AND REMARKS. Analytical Geometry, as the terms imply, proposes to in vestigate geometrical truths by means of analysis. In it the magnitudes under consideration are represent by sim- bolg, such as letters, terms, simple or combined, and equa tions ; and problems are then solved and the properties and relations of magnitude established by processes pure ly algebraic. A single letter, without an exponent, will aJwjjjs be un derstood as denoting the length of a line ; and in general, any expression of the first degree denotes the length of a line and is, for thin reason, said to be linear ; so likewise, an equation all of whose terms are of the first degree is call ed a linear equation. An expression of the second degree will represent the meas ure of a surface, and an expression of the third degree will represent the measure of a volume. "When a term is of a higher degree than the third, a sufficient number of its literal factors, to reduce it to this degree, must be regarded as numerical or abstract. The subject of Analytical Geometry naturally resolves itself into two parts. First. That which relates to the solution of determinate problems; that is, problems in which it is required to de termine certain unknown magnitudes from the relations which they bear to others that are known. In this case we must be able to express the relations between the known and unknown magnitudes by independent equa tions equal in number to the required magnitudes. (96) GENERAL PROPERTIES. 97 After having obtained, by a solution of the equations of the problem, the algebraic expressions for the quanti ties sought, it may be necessary, or, at least desirable, to construct their values, by which we mean, to draw a geo metrical figure in which the parts represent the given and determined magnitudes, and have to each other the rela tions imposed by the conditions of the problem. This is called the construction of the expression. This branch of analytical geometry, which may be termed Determinate Geometry ! , being of the least impor tance, relatively, will be omitted, after this reference, in the present treatise, and we shall pass at once to division. Second. That which has for its object to discover and discuss the general properties of geometrical magnitudes. In this the magnitudes are represented by equations ex pressing relations between constant quantities, and, either two or three indeterminate or variable quantities, and for this reason it is sometimes called Indeterminate Geometry. GENERAL PROPERTIES OF GEOMETRICAL MAGNITUDES, CHAPTER I. OF POSITIONS AND STRAIGHT LINES IN A PLANE, AND THE TRANSFORMATION OF CO-ORDINATES. DEFINITIONS. 1. Co-ordinate Axes are two straight lines drawn in a plane through any assumed point and making with each other any given angle. One of these lines is the axis of . abscissas or the axis of X; the other is the axis of ordinates, or the axis of Y, and their intersection is the origin of co ordinates. 2. Abscissas are distances estimated from the axis of Y on lines parallel to the axis of X ; ordinates are distances 9 98 ANALYTICAL GEOMETRY. estimated from the axis of X on lines parallel to the axis of Y. 3. The abscissa and ordinate of a point together are called the co-ordinates of the point. 4. The co-ordinate axes are said to be rectangular when they are at right angles to each other, otherwise they are oblique. 5. The two different directions in which distances may be estimated from either t axis, on lines parallel to the other, are distinguished by the signs plus and minus. 6. Abscissas are designated by the letter x and ordi- nates by the letter y, and when unaccented they are called general co-ordinates, because they refer to no particular one of the points under consideration. "When particular points are to be considered the co-ordinates of one are denoted by x and y ; of another by x" and y", etc., which are read x prime, y prime, x second, y second, etc. ILLUSTKATIONS. Through any point A draw the lines XX , YY f making with each other any given angle. Call XX the axis of abscissas and YY r the axis X- of ordinates. A is the origin of co-or dinates, or zero point. The four angu lar spaces into which the plane is divi ded are named, respectively,^^/, second, third, and. fourth angles. YAX is the first angle, YAX is the second angle, YAX 1 is the third angle, and Y AX is the fourth angle. Take any point, as P, in the first angle, and from it draw Pp parallel to the axis of Y and Pp r parallel to the axis of X, the first meeting the axis of X at p, and the second the axis of Fat p f ; then p P=Ap is the abscissa, and pP=Ap r is the ordinate of the point P. Now produce Pp r to P making p P f =p P, and from P draw a parallel to the axis of Y meeting the axis of X at p"; then the point P is in the second angle, and p P GENERAL PROPERTIES. 99 *= Ap" is its abscissa, and p"P f =Ap r is the ordinate. By like constructions we determine the position of the point P in the third angle, and that of the point P " in the fourth angle. It is evident that the abscissas of these four points are numerically equal, as are likewise their ordinates ; but if we have reference to the algebraic signs of the co-ordi nates, each point will be assigned to its appropriate angle and will be completely distinguished from the others. Abscissas estimated to the right of the axis of Y are posi tive and those estimated to the left are negative. Ordinates estimated from the axis of JT upwards are positive, those estimated downwards are negative. We shall therefore have for points In the 1st angle, x positive, y positive. " " 2d " x negative, y positive. " " 3d " x negative y negative. " " 4th " x positive y negative. From what precedes we see that the position of a point in the plane of the co-ordinate axis is fully determined by its co-ordinates. To construct this position we lay off on the axis of X the given abscissa, to the right, or to the left of the origin, according to the sign ; also lay off on the axis of Y the given ordinate, upwards from the origin if the sign be plus, downwards if it be minus. The lines drawn through the points thus found, parallel to the co ordinate axes, will intersect at the required point and fix its position. As rectangular co-ordinates are more readily appre hended than oblique, and as discussions and algebraic expressions are generally less complicated where refer ences are made to the former, than when made to the latter, rectangular co-ordinates will be habitually em ployed in the following pages. When we have occasion to use others it will be so stated. 100 ANALYTICAL GEGMETKY, PROPOSITION I- To find the equation of a straight line, Let XX , YT be two rectangu- Y iar co-ordinate axes. A being the origin draw any line as L f L through this point, and designate the natu ral tangent of the angle LAX by a. Then take any distance on AX as AP, and represent it by x, and the perpendicular distance PMy. Then by trigonometry we have" Ead : tan. MAP :;AP: PM or 1 : a : : x : y Whence y~ax (l) Now this equation is general ; that is, it applies to any point M on the line AL, because we can make x greater or less, and PM will be greater or less in like proportion and M will move along on the line AL as we move P on the line AX. Because the point M will continue on the line AL through all changes of x and y, we say that yax is the equation of the line AL. Now let us diminish x to 0, and the equation .reduces to ?/=0 at the same time, which brings M to the point A. Let x pass the line YY , then AP becomes #, and the corresponding value of y will be P M 1 , and,being be low the line X X, will, therefore,be minus. Therefore y=ax. is the general equation of the line L _Z7, extending indefi nitely in either direction. If the tangent a becomes less, the line will incline more towards the line X X. When a=0 the line will coincide with Xlt>. Fow let AP*" fce +&W& a become a, then P" M"> will correspond to y, x& beepmes minm y, because it is STRAIGHT LiJN ES. 101 below the axis &X . Or, algebraically y= ax, indica ting some point JM. " below the horizontal axis. It is, therefore, obvious that yax may represent any line, as LL , passing through A from the list into the %d quadrant, and that y= ax may be made to represent any line, as L*I/", passing through A from the 2d into the 4th quadrant. Therefore y=&x may be wade to represent any straight line passing tJirouyh the zero point. In case we have a and x, that is, both a and x mi nus at the same time, their product will be -+(ix, showing that y must be ^>fe? by the rules of algebra. As an exercise, Set the learner examine these lines and see whether they ^correspond to the equation. When we have a we must draw the line from A to the right and below AX ; then XAL " is the angle whose natural tangent is a. But the opposite angle X^AU is the same in value. When we have x we must take the distance as AP to the left of the axis YY 7 , and the corresponding line P M" is above XX , and therefore plus, as it ought to be. But the equation of a straight line passing through the zero point is not sufficiently general for practical application ; we will therefore suppose a line to pass in any direction across the axis YY , cutting it at the distance AB or AD (6) or b distance above or below the zero point A, and find its equation. Through the zero point A draw a line, AN, parallel to ML. Take any point on the line AX and through P draw 9* 102 ANALYTICAL GEOMETRY. PM parallel to A Y, then ABMNv?\\\ be a parallelogram. Put AP=x. PM=y. The tangent of the angle NAP=a. Then will NP=ax. To each of these equals add NM=b, then we shall have y=ax+b for the relation between the values of x and y correspond ing to the point M, and as M is any variable point on the line ML corresponding to the variations of x, this equa tion is said to be the equation of the line ML. "When b is minus the line is then QL , and cuts the axis YY in 1), a point as far below A as B is above A. Hence we perceive that the equation may represent the equation of any line in the plane YAX. If we give to a, x, and b, their proper signs, in each case of application we may write y=ax+b for the equation of any straight line in a plane. Cor. Since the equation yax+b truly expresses the relation between the co-ordinates of any point of the line, it follows that if the co-ordinates x and y 1 of any partic ular point of the line be substituted for the variables x and y the equation must hold true ; but if the co-ordinates x" and y", of any point out of the line be substituted for the variables, the equation cannot be true. "What appears in the particular case of a straight line are general principles which we thus enunciate, viz : 1st. If the co-ordinates of a particular point, in any line whatever, be substituted for the variables in the equation of the line, the equation must be satisfied; but if the co-ordinates of a point out the Ine, be substituted for the variables in its equa tion, the equation cannot be satisfied. 2d. If the co-ordinates of any point be substituted for the va riables in the equation of a line, and the equation be satisfied, the STRAIGHT LINES. 103 point must be on the line ; but if the equation be not satisfied by the substitution, the point cannot be on the line. These are principles of the highest importance in ana lytical geometry, and should be thoroughly committed and fully understood by the student. SCHOLIUM. Instead of rectangular, let us as- Y sume the oblique co-ordinate axes AX and A Y t making with each other an angle denoted by m. Through the origin draw the line AP making with the axis of x the angle PAD=n ; then the angle PAjy=m n. Take any point as P in the line and from it draw PD* and PD parallel, respectively, to the axes of X and Y. From the triangle APD we have (Prop. 4, Sec. 1, Plane Trig.) PI): AD:: Sin. PAD=Sm. PAD 1 or y : x::S m. n : Sin. (m n.) Whence y= sin * n sm.m n But - is constant for the same line and may be repre- sin. (ra n sented by a. Therefore, for any straight line passing through the origin of a system of oblique co-ordinate axes we have, as before, the equation yax. And if we denote by b the distance from the origin to the point at which a parallel line cuts the axis of Y above or below the origin we shall also have for the equation of this line in which it must be remembered that a denotes the sine of the angle that the line makes with axis of x divided by the sine of the angle it makes with the axis of Y. To fix in the minds of learners a complete comprehension of the equation of a straight line, we give the following practical EXAMPLES. 1. Draw the line whose equation is y=2x-}-3. (1) Then draw the line represented by y= x-j-2 (2) and determine where these two lines intersect. 104 ANALYTICAL GEOMETKY. 1 2 Draw FF and XX at right angles, and taking any convenient unit of meas ure lay it off on each of the axes from the origin in both positive and negative directions a sufficient number of times. Equation (1) is true for all values of x and y. It is true then when x=Q. But when x=0 the point on the line must be on the axis FF. Whenaj=0. y=3. Y This shows that the line sought for must cut FF at the point +3. The equation is equally true when ?/=0. But when y=0, the corresponding point on the line sought must be on the axis XX , and on the same supposition the equation becomes That is, midway between 1 and 2 is another point in the line which is represented by y 2ar-f-3, but two points in any right line must define the line; therefore ML is the line sought. Taking equation (2) and making x=Q will givey=2, and making ?/ will give x=2; therefore MQ must be the line whose equation is y x-\-2, and these two lines with the axis XX form the tri angle LMQj whose base is 3| and altitude about 2J. But let the equations decide, (not about,) but exactly the posi tion of the point M of intersection. This point being in both lines, the co-ordinates x and y corres ponding to this point are the same, therefore we may subtract one equation from the other, and the result will be a true equation, 3s+l=0. Or x= J. Eliminating x from the two equations we find y=2^. 2. For another example we nequire the projection of the line repre sented by the equation *==-- * 2 420 Making e=0, then y 2. Making y=0, then x= 840. Using the last figure, we perceive that the line sought for must STRAIGHT LINES. 105 pass through S two units below the zero point, and it must take such a direction S V as to meet the axis XX! at the distance of 840 units to the left of zero. Hence its absolute projection is practi cally impossible. 8. Construct the line whose equation is y=2x-{-5. 4. Construct the line whose equation is y 3# 3. 5. Construct the line represented by 2y 3a?-j-5. 6. Construct the line represented by y4x 3. The lines represented by equations 4 and 6 will intersect the axis of Y at the same point. Why ? 7. Construct the line whose equation is y=r2#-}-3. 8. Construct the line whose equation is y=. 2x 3. The last two lines intercept a portion of the axis of Y which is the base of an isosceles triangle of which the two lines are the sides. What are the base and perpendicular, and where the vertex of the triangle 1 ANS. The base is 6, the perpendicular 1, vertex on the axis of X. Construct the lines represented by the following equations. 9. 3a?+5y 15=0 11. *+y+2=0 12. a?+y+3=0 13. 2x- y+4=0 PROPOSITION II To find the distance between two given points in the plane of the co-ordinate axis. Also, to find the angle made by the line joining the two given points, and the axis of X. Let the two given points be P and , and because the point P is said to be given, we know the two distances AN=x f , NP=y r . And because the point Q is given we know the two distances. AM=x" and MQ=y". P N ^ Q R M: 106 ANALYTICAL GEOMETRY. Then, AM AN=NM=PR=x" x ; and MQMR=QR=y"y . In the right angled triangle PRQ we have (PJR) 2 +(Q) 2 =(PQ) 2 . But D=PQ. That is D 2 =(x" x } 2 +(y" yj, Or D=^(x"x f+(y"yJ Thus we discover that the distance between any two given points is equal to the square root of the sum of the squares of the differences of their abscissas and ordinates. If one of these points be the origin or zero point, then r =0 and ?/ =0, and we have a result obviously true. To find the angle between PQ and AX. PR is drawn parallel to A X, therefore the angle sought is the same in value as the angle QPR. Designate the tangent of this angle by a, then by trigo nometry we have PR-. RQ:: radius : tan. QPR. That is, x"x r : y"y f : : 1 : a. fy f Whence a== x" _ x f In case #"=?/ , PQ will coincide with PR, and be paral lel to A X, and the tangent of the angle will then be 0, and this is shown by the equation, for then x" x In case x"=x f , then PQ will coincide with RQ and be parallel to A Y, and tangent a will be infinite, and this too the equation shows, for if we make x"=x r or x"x f =0, the equation will become y"y f aV. __ ^_=oo STRAIGHT LINES. 107 PROPOSITION III. To find the equation of a line drawn through any given point. Let P be the given point : The equation must be in the form y=ax+b (1) Because the line must pass through the given point whose co-ordinates are x and ?/ , we must have y =ax r + b. (2) Subtracting equation (2) from equation (1) member from member, we have y y r =a(x x f ) ( 3 ) for the equation sought. In this equation a is indeterminate, as it ought to be, because an infinite number of straight lines can be drawn through the point P. "We may give to y r and x their numerical values, and give any value whatever to a, then we can construct a particular line that will run through the given point P. Suppose # =2, 2/ =3, and make a=4. Then the equation will become y _3=4(z 2). Or y4,x 5. This equation is obviously that of a straight line, hence equation (3) is of the required form. PROPOSITION IY. To find the equation of a line which passes through two given points. Let AX and A Y be the co-ordinate axes, and P and Q the given points. Denote the co-ordinates of P by x , y r and of Q by x", y". The required equation must be of the form 108 ANALYTICAL GEOMETKY. VVe will now determine such vrlues for a and b as will cause the Ime represented by this equation to pass through the given points. As the line is to pass through the point P, the co-ordinates x , y f of this point when substituted for the variables x, y must satisfy the equation, and we shall have y f =ax f -\-b Q N M: (2) And because the line is to pass through the point , whose co-ordinates are x",y" we will also have y"=ax"+b (3) Subtracting eq. (2) from eq. (3) member from member, we get "Whence a= y "~ y (4) x u x From eqs. (1) and (2) we obtain in like manner y y =a(x x f ] (5) Substituting for a in eq. (5) its value in eq. (4) we find yy = y -^-.(xx } ( 6 ) x x for the equation sought. If we subtract eq. (3) from eq. (1) member from mem ber, and substitute for a in the resulting equation its value in eq. (4) we find yy"-- for the required equation. By simply clearing eqs. (6) and (7) of fractions and re ducing, it may be shown that they are in fact but different forms of the same equation. To prove that either of these equations is that of a line passing through the points P and , we have but to sub- STRAIGHT LINES. 109 stitute in it, for x and y, the co-ordinates of these points. It will be found that when these substitutions are made for either point, the equation will be satisfied. We will illustrate the use of these equations by the fol lowing EXAMPLES. 1. The co-ordinates of P are x =3, 2/ =4, and of , *=-i, y=3. "What is the equation of the line that passes through these points ? Here And the equation y y r =& ~~^ -(x x ) becomes x x By substituting in the equation y ?/" 7~ 7 (# x") U " *C we gety 3=j(x+l) ory=Jx-f 3J, the same as that above. 2. Find the equation of the straight line that is deter mined by the points whose co-ordinates are x f 4, y 1 and z"=4i, V= V Ans. y= ^x Ijf. 3. The co-ordinates of one point are z =6, 2/ =5, and of another they are x"= 3, y"=S. What is the equation of the straight line that passes through these points ? Ans. PROPOSITION V. To find the equation of a straight line which shall pass through a given point and make,with a given line, a given angle. The equation of the given line must be in the form v=ax+6. (1) 10 110 ANALYTICAL GEOMETEY. Because the other line must pass through a given point its equation must be (Prop. III.) y y =a (x x f ). ( 2 ) "We have now to determine the value of a . "When a and a are equal, the two lines must be paral lel, and the inclination of the two lines will be greater or less according to the relative values of a and a . Let PQ be the given line, making with the axis of JTan angle whose tangent is a and Pit the other line which shall pass through the given point P and make with P, a given an- Q gle QPR. The tangent of the / angle PPJTis equal to a . Because PRX=PQR+QPR. QPR=PRZPQR Tan. P#=tan. (PRXPQR.) As the angle QPR is supposed to be known or given, we may designate its tangent by m, and m is a known quantity. Now by trigonometry we have m=tan. (PRXPQR}=~- f . (3) Whence a 1 ma This value of a put in eq. (2) gives for the equation sought. Cor. 1. "When the given inclination is 90, m its tan gent is infinite, and then a f _. "We decide this in the a following manner. An infinite quantity cannot be increased or diminished STRAIGHT LINES. Ill relatively, by the addition or subtraction of finite quanti ties, therefore, on that supposition, 1 ma ma a APPLICATION. To make sure that we comprehend this proposition and its resulting equation, we give the fol lowing example : The equation of a given line is y=2x+6. Draw another line that will in tersect this at an angle of 45 and pass through a given point P, whose co-ordinates are Draw the line MN correspond ing to the equation y=2x+6. Lo cate the point P from its given co~ ordinates. Because the angle of intersection is to be 45, w=l, and a=2. Substituting these values in eq. (4) we have Or y= 3z+12J. Constructing the line MJR corresponding to this equa tion, we perceive it must pass through P and make the angle NMR 45, as was required. The teacher can propose any number of like examples. Cor. Equation (3) gives the tangent of the angle of the inclination of any two lines which make with the axis of X angles whose tangents are a and a . That is, we have in general terms a f a I+aa f In case the two lines are parallel m=0. "Whence a =a, an obvious result. 112 ANALYTICAL GEOMETRY. In case the two lines are perpendicular to eacli other, m must be infinite, and therefore we must put to correspond with this hypothesis, and this gives ~-i a a result found in Cor. 1. To show the practical value of this equation we require the angle of inclination of the two lines represented by the equations y=%x 6, and y= x+2. Here a=3 and a f = 1. Whence --* This is the natural tangent of the angle sought, and if we have not a table of natural tangents at hand, we will take the log. of the number and add 10 to the index, then we shall have in the present example 10.301030 for the log. tangent which corresponds to 63 26 6" nearly. The sign of the tangent determines the direction in which the angles are estimated. 2. What is the inclination of the two lines whose equa tion are and 3y=__2-f 6 ? Ans. The tangent of their inclination is 4f Log. 4.75 plus 10=10.676694. The inclination of the lines is therefore 78 6 5". 3. Find the equation of a line which will make an an gle of 56 with the line whose equation is As the required line is to pass through no particular point, but is merely to make a given angle with the known line, we may assume it to pass through the origin of co-ordinates. Its equation will then be of the form STRAIGHT LINES. 113 y=a r x. We must now determine such a value for a f that the two lines will make with each other an angle of 56. Represent the tangent of the given angle hy t; then "by corollary (2) 1+fa Tn the tables we find that log. tangent of 56 to be 10. 171013, from which subtracting 10 to reduce it to the log. of the natural tangent and we have 0.171013 for the log. of /. The number corresponding to this is 1.483. Whence a/ ~i-== 1.483 From which we find a = 1,473 nearly and the equa tion of the line making with the given line, an angle of 56 is therefore y= 1.473z. PROPOSITION VI. To find the co-ordinates which will locate the point of inter section of two straight lines given by their equations. We have already done this in a particular example in Prop. I, and now we propose to deduce general expressions ibr the same thing. Let y=ax+b be the first line. And y=a x-\-b f be the second line. For their point of intersection y and x in one equation will become the same as in the other. Therefore we may subtract one equation from the other, and the result will be a true equation. For the sake of perspicuity, let x l and y l represent the co-ordinates of the common point, then by subtraction (aa f )x 1 Whence x .= fc^l (o-O 10* 114 ANALYTICAL GEOMETRY. EXAMPLE. At what point will the lines represented by the two equations y= 2x+l and y=5x+~LQ intersect each other. Here a= 2, a =5, 6=1, =10. Whence 3=$, y= -34. If we take another line not parallel to either of these, the three will form a triangle. Then if we locate the three points of intersection and join them, we shall have the triangle. PROPOSITION VII. To draw a perpendicular from a given point to a given straight line and to find its length. Let y=ax+b be the equation of the given straight line, and x , y the co-ordinates of the given point. The equation of the line which passes through the giv en point must take the form yy =a r (xx f ). (Prop. 3.) And as this must be perpendicular to the given line, we must have a = -. Therefore the equations for the two lines must be y=ax-\-b for the given line; (1) and y y = _(# x ); CL 1 fx f \ Or y -x+ ( - +y ) for the perpendicular line (2) a \a / Let x l andf/j represent the co-ordinates of the point of intersection of these two lines. Then by Prop. 6, - STRAIGHT LINES. 115 +pW) a \a 1 , -a ,.- Or we may conceive x and y to represent the co-ordin ates of the point of intersection, and eliminating y from eqs. (1) and (2) we shall find x as above, and afterwards we can eliminate y. Now the length of the perpendicular is represented by Whence v C*, x^+(y l y ?=I>. (Prop. H.) 1 (b+ax y y + ( a*+l / - perpendicular. If we put u=b+ax r y 9 the quantities under the radi cal will become u "Whence the perpendicular EXAMPLES. 1. The equation of a given line is y=3x 10, and the co-ordinates of a given point are x =4 and 2/ =5. What is the length of the perpendicular from this given point to the given straight line ? Ans. y^N/90. 2. The equation of a line is 7/= 5x 15, and the co ordinates of a given point are x / =4 and y =5. What is the length of the perpendicular from the given point to the straight line ? Ans. 7.84+ . 116 ANALYTICAL GEOMETBY. PROPOSITION VIII. To find the equation of a straight line which will bisect the angle contained by two other straight lines. Let y=ax+b (1) and y=a x+V (2) be the equations of two straight lines which intersect ; the co-ordinates of the point of intersection are fb b f \ du ab f /T\ TTT *i= 7 2/i= (Prop. YI. \a a I a a "We now require a third line which shall pass through the same point of intersection and form such an angle with the axis of X (the tangent of which may be repre sented by m) that this line will bisect the angle included between the other two lines. "Whence by (Prop. Y.) the equation of the line sought must be y y = m ( X j) (3) in which we are to find the value of m. Let PN represent the line cor- responding to equation (1) PM the line whose equation is (2), and PR the line required. Now the position or inclination of PN to AX depends entirely on the value of a, and the inclination of PM depends on a and both are A independent of the position of the point P. Now RPN^RPX NPX* and M Whence by the application of a well known equation in plane trigonometry, (Equation (29), p. 253 Plane Trig.) we have tan. RPN=t&n. ( TfPX r NPZ r }= m ~~ a I+am And tan. MPR=tsm. (MP3 7 of m 1 1-fa w STRAIGHT LIKES. 117 But by hypothesis these two angles RPN and MPR are to be equal to each other. Therefore "Whence m z +m=l. * (4) a -k-a This equation will give two values of m; one will cor respond to the line PR, and the other to a line at right angles to PR. If the proper value rn be taken from this equation and put in eq. (3), we shall have the equation required. Practically we had better let the equations stand as they are, and substitute the values of #, a x, and y, cor responding to any particular case. To illustrate the foregoing proposition we propose the following EXAMPLES, Two lines intersect each other : is the equation of one line. 0-) Is tlmt of the other line, (2) Find the equation of the line which bisects tjie apgle contained by these two lines : Here a= 2, a =4, 6=5, & =6. Whence x l = i, and y l = ** Thus (3) becomes And eq. (4) becomes "Whence m=0,1097 or w= 9.1097. y _y = (Or y V = 118 ANALYTICAL GEOMETRY. Equation (4) is that of the line required ; (3) that of the line at right angles to the line required. All will be ob vious if we construct the lines represented by the eqs. (1), (2), (3), and (4). For another example, find the equation of a line which bisects the angle contained by the two lines whose equa tions are Here a=l, a = 20. Whence (4) becomes m 2 ffm=l. Therefore m= 0.385, or +2.6. NOTE. Two straight lines whose equations are y=ax-{- b and y a-^-l) will always intersect at a point (unless a a ) and with the axis of Fform a triangle. The area of such triangle is expressed by From the given equations we find the co-ordinates of the intersection of the lines to be For the line bisecting the angle included between the given lines we have either y 2 3 4 T 2 = 0.385(3+ J?) a) or, y-W=2.6(*+if) (2) By transposition and reduction (l) becomes y= 0.385Z+11.75 (3) and (2) becomes #=2.6z+12.76 (4) The lines represented by eqs. (3) and () are at right an gles to each other. The latter line bisects the angle in cluded between the given lines, and the former the adja cent or supplemental angle. 3. From the intersection of two lines whose equations are STRAIGHT LINES. 119 4 (1) and 2?/=3a;+4 (2) A third line is drawn making, with the axis of -J, an angle of 30. Find the intersection of the given lines and the equation of the third line. ( The co-ordinates of the points of intersection Ans.-l are #,= T \, ^ 1 =ff, and the required equation I is?/ fj=0,5773(z+A). 4. Two lines are represented by the equations and "What kind of a triangle do these lines form with the intercepted portion of the axis of F, and what are its sides and its area ? ( The triangle is isosceles ; its base on the axis Ans. < of F is 2, the other sides are each 1.201+, and Mts area 0. 66+. 5. Two lines are given by the equations 2?/+3j:r= 2J and 2y |x=4 Required the equation of the line drawn from the point whose co-ordinates are ic"=3, y^O to the intersection of the given lines and the distance between these two points. , f The equation sought is y= 0.717^+2.1523 and I the distance is ^(1.8) 2 +(2.52) 2 . TRANSFORMATION OF CO-ORDINATES. It is often desirable to change the reference of points from one system of co-ordinate axes to another differing from the first either in respect to the origin or the direc tion of the axes, or both. The operation by which this is done is called the transformation of co-ordinates. The 120 ANALYTICAL GEOMETRY. "V" system of co-ordinate axes from which we pass is the prim itive system and that to which we pass is the new system. Let J. JTand A Y be the primi tive axes. Take any point, as A , the co-ordinates of which referred to AX and A Y are x=a, y=b and through it draw the new axes A X, and A 1 Y 1 parallel to the primative axes. Then denoting the co-ordinates of any point, as M, referred to the primitive axes by x and y, and the co ordinates of the same point referred to the new axes by x and y , it is apparent that A TV I X Y A By giving to a and b suitable signs and values we may place the new origin at any point in the plane of the prim itive axes and the above formulas are those for passing from one system of axes to a system of parallel axes hav ing a different origin. The formulas for the transformation of co-ordinates must express the values of the primitive co-ordinates of points in terms of the new co-ordinates and those quanti ties which fix the position of the new in respect to the primitive axes. PROPOSITION IX. To find the formulas for passing from a system of rectangu lar to a system of oblique co-ordinates from a different origin. Let AX, A Fbe the primitive axes and A X, A Y the new axes. Through any point as M draw MP parallel to A Y and MP perpendicular to AX. Then A P f is the new abscissa, P M the new ordinate of the point M, and AP and PM are respectively the primitive abscissa and ordinate of the same point. STRAIGHT LINES. 121 Let AB=a, BA =b, AP=x, y PM=y,A f P =x , P M=y f the an gle X A X v =m, and the angle Y f A f A"=n. ~Now by trigonome try we have A f K=x Qos.m,KP =LH=x sin. m P H=KL=y f cos. n. "A And MH=y f sin, n. Whence x=a+x cos.m+y coa.n,yb-}-x sin. w-f ?/ sin.w, the formulas required. SCHOLIUM. In case the two systems have the same origin, we merely suppress a and b, and then the formulas required are x=x cos. cos. i. y=x sin. m-{-y f sin. n. PROPOSITION X. To fold the formulas for passing from a system of oblique co ordinates to a system of rectangular co-ordinates, the origin be ing the same. Take the formulas of the last problem xx* cos. m+y r cos. ft, y=x f sin. m-\-y sin. ft. We now regard the oblique as the primitive axes, and require the corresponding values on the rectangular axes. That is, we require the values of x f and y . If we multi ply the first by sin. ft, and the second by cos. ft, and sub tract their products, y will be eliminated, and if x be eliminated in a similar manner, we shall obtain f __x sin. ft y cos. ft ,_y cos. m x sin m sin. (ft m) sin. (ft m) SCHOLIUM. If the zero point be changed at the same time in reference to the oblique system, we shall have x sin. n y cos. n ,_ i \_J/ cos.ra x sin. m x =a+ sn. w m} We will close this subject by the following 11 122 ANALYTICAL GEOMETRT. EXAMPLE. The equation of a line referred to rectangular co-ordi nates is y=a x+b f . Change it to a system of oblique co-ordinates having the same zero point. Substituting for x and y their values as above, we have x f sin. m+y sin. n=a (xcoa.m-)-y f cos. n]+b r . Eeducing ,__( cos. m sin. m)x , b sin. n a cos. m sin. n a 1 cos. m POLAR CO-ORDINATES. There are other methods by which the relative posi tions of points in a plane may be analytically established than that of referring them to two rectilinear axes inter secting each other under a given angle. For example, suppose the line AB to revolve in a plane about the point A. If the angle that this line makes with a fixed line passing through A be known, and also the length of AB, it is evident that the extremity B of this line will be determined, and that there A! X is no point whatever in the plane the position of which may not be assigned by giving to AB and the angle which it makes with the fixed line appropriate values. The variable distance AB is called the radius vector, the angle that it makes with the fixed line the variable angle and the point A about which the radius vector turns, the pole. The radius vector and the variable angle together consti tute a system of polar co-ordinates. STRAIGHT LINES. 123 Denote variable angle BAD by v, the radius vector by r and by x and y, the co-ordinates of B referred to the rectangular axes A X, A Y; then by trigonometry we have xr cos. v and y=r sin. v. Now from the first of these we have r= (v may re- cos, v volve all the way round the pole), and as x and cos. v are both positive and both negative at the same time, that is, both positive in the first and fourth quadrants, and both negative in the second and third quadrants, therefore r will always be positive. Consequently, should a negative radius appear in any equation, we must infer that some incompatible conditions have been admitted into the equation. PROPOSITION XI. To find the formulas for changing the reference of points from a system of rectangular co-ordinate axes to a system of polar co-ordinates. Let A X, A Y be the co- y ordmate axes, A. the pole AB the radius vector of any point, and AD parallel to A X the fixed line from which the va riable angle is estimated. De note the co-ordinates A E, AJEof the pole by a and b and A. the radius vector AB by r. D EC X Draw B C perpendicular to A X; then is A 1 C=x the abscissa, and BC=y the ordi- nate of the point B. From the figure we have A r C=A E+EC=A f E+AF=A E+AB cos.v and BC=BF+FC=BF+AE=AE+AB sin. v 124 Whence ANALYTICAL GEOMETKY. x=a-\-r cos. v y=b-\-r sin. v. SCHOLIUM. If instead of estimating the variable angle from the line AD, which is parallel to the axis A X, we estimate it from the line AH which makes with the axis the given angle HAD=m we shall have x=a-{-r cos. (v-\-m) y=b-\-r sin. (x-\-ni) CHAPTER II. THE CIRCLE. LINES OF THE SECOND ORDER. Straight lines can be represented by equations of the first degree, and they are therefore called lines of the first order. The circumference of a circle, and all the conic sections, are lines of the second order, because the equa tions which represent them are of the second degree. PEOPOSITION I. To find the equation of a circle. Let the origin be the center of the circle. Draw AM to any point in the circumference, and let fall MP perpendicular to the axis of X. Put AP=x, PM=y and AMR. Then the right angled triangle APM gives and this is the equation of the circle when the zero point is the center. THE CIECLE. 125 When y=0, x z =R*, or x=R, that is, P is at X or J/. When =0, y 2 =R 2 , or dby=R, showing that Jf on the circumference is then at Y or Y". When x is positive, then P is on the right of the axis of Y, and when negative, P is on the left of that axis, or between A and A 1 . When we make radius unity, as we often do in trigo nometry, then x*+y 2 =l, and then giving to x or y any value plus or minus within the limit of unity, the equation will give us the corresponding value of the other letter. In trigonometry y is called the sine of the arc XM, and x its cosine. Hence in trigonometry we have sin. 2 +cos. 2 =l. Now if we remove the origin to A and call the distance A f P=x,ihen AP=x R, and the triangle APM gives (z~ RJ+y^R*. Whence y ? =2Rxx*. This is the equation of the circle, when the origin is on the circumference. When x=Q,y=Q at the same time. When x is greater than 2jR, y becomes imaginary, showing that such an hy pothesis is inconsistent with the existence of a point in the cir cumference of the circle. There is still a more general equation of the circle when the zero point is neither at the center nor in the circumference. The figure will fully illustrate. Let AB=c, BC=b. Put AP Y =x, or AP =x, and PM or P f M "=y, CM, CM , &c. each=J2. In the circle we observe four equal right angled- triangles. The numerical expression is the same for each. Signs only indi cate positions. 11* 126 ANALYTICAL GEOMETRY. Now in case CDM is the triangle we fix upon, We put AP=x, then BP= CD=(xc), PM=y, Ml}=yCB=(yb}. Whence (x c) 2 +(y b) 2 =R 2 (1) In case CDM r is the triangle, we put AP=x and PM r =y> Then ( X c) 2 +(by) 2 =R 2 (2) In case CD M" is the triangle, we put AP =x, P M" =!/ Then ( C xf+(y b) 2 =S z (3) If CD M" is the triangle, we put FM"=y. Then (cx) 2 +by) 2 =R 2 (4) Equations (1), (2), (3) ? and (4), are in all respects numer ically the same, for (c x) 2 (x c) 2 , and (b -y) 2 =(y b) 2 . Hence we may take equation (1) to represent the general equation of the circle referred to rectangular co-ordinates. The equation (xc) 2 +(yb) 2 =R 2 (1) includes all the others by attributing proper values and signs to c and b. If we suppose both c and b equal 0, it transfers the zero point to the center of the circle, and the equation becomes x 2 +y 2 =2^ To find where the circle cuts the axis of X we must makey=0. This reduces the general equation (1) to (xc) 2 +b 2 =R 2 . Or ( X c) 2 =R 2 b 2 . ~Now if b is numerically greater than J, the first mem ber being a square, (and therefore positive,) must be equal to a negative quantity, which is impossible, showing that in that case the circle does not meet or cut the axis of JT, and this is obvious from the figure. In case 6=jR, then (x c) 2 =0, or x=c, showing that the THE CIRCLE 127 circle would then touch the axis of X. If we make #=0, eq. (1) becomes Or This equation shows that if c is greater than R, the circle does not cut the axis of F, and this is also obvious from the figure. If c be less than R, the second member is positive in value, and showing that if* the circumference cut the axis at all, it must be in two points, as at Jf", M" . PROPOSITION II. The supplementary chords in the circle are perpendicular to each other, DEFINITION. Two lines drawn, one through each ex tremity of any diameter of a curve, and which intersect the curve in the same point, are called supplementary chords. That is, the chord of an arc, and the chord of its sup plement. In common geometry this proposition is enunciated thus: All angles in a semi-circk are right angles. The equation of a straight line which will pass through the given point B, must be of the form (Prop. HI. Chap. I.) y-y =a(xx ). (1) The equation of a straight line which will pass through the given point JT, must be of the form yy r =a (xx ). (2) 128 ANALYTICAL GEOMETRY. At the point B, y =0, and z = E, or Therefore eq. (1) becomes y=a(x+K). (3) And for like reason eq. ( 2 ) becomes y=a (x ll\ (4) For the point in which these lines intersect x and y in eq. (8) are the same as x and y in eq. (4) ; hence, these equations may be multiplied together under this sup position, and the result will be a true equation. That is, . ?/3=aa (z 2 jR 2 ). (5) But as the point of intersection must be on the curve, by hypothesis, therefore, x and y must conform to the fol lowing equation : y*+x*=R*. Or y*= l(x* R*}. (6) Whence aa = 1 , oraa +l-j-O. This last equation shows that the two lines are perpen dicular to each other, as proved by (Cor. 2, Prop. 5., Chap. 1.) Because a and a f are indeterminate, we conclude that an infinite number of supplemental chords may be drawn in the semi-circle, which is obviously true. PROPOSTION III. To find the equation of a line tangent to the circumference of a circle at a given point. Let C be the center of the cir cle, P the point of tangency, and Q a point assumed at pleasure in the circumference. Denote the co-ordinates of P by z , y , and those of Q 9 by of , y", f~ c I \ The equation of a line passing through two points whose co-or- THE CIRCLE. 129 dinates are x , y and x", y" is of the form (Prop. 4, Chap. 1). y x-. (1) JO """"* We are to introduce in this equation, first, the condi tion that the points P and Q are in the circumference of the circle, which will make the line a secant line, and then the further condition that the point Q shall coincide with the point P, which will cause the secant line to be come the required tangent line. Because the points P and Q are in the circumference of the circle, we have x n+ y n=%* and x" 2 +y" 2 =IP Whence by subtraction and factoring, (x +x") (x x")+(y +y") (y y")=0 (2) from which we find y y" x +x" This value of ^ ^ substituted in equation (1) gives us X X for the equation of the secant line, "Now, if we suppose this line to turn about the point P until Q unites with P, we shall have x"=x r and y"=y f , and the secant line will become a tangent to the circum ference at the point P. Under this supposition eq. (3) becomes y-y =- x - r (x-x 1 ), (4) y x in which _ is the value of the tangent of the angle which the tangent line makes with axis of X. I 130 ANALYTICAL GEOMETRY. By clearing this equation of fractions, and substituting for x f2 +y f2 its value, JR\ we have finally lor the equation of the tangent line, yy +xx f =R 2 . (5) This is the general equation of a tangent line ; ,2/ , are the co-ordinates of the tangent point, and #, y, the co-ordinates of any other point in the line. SCHOLIUM 1. For the point in which the tangent line cuts the axis of X, we make^ =:0, then Q For the point in which it meets the axis of J", we make x =Q, and SCHOLIUM 2. A line is said to be normal to a curve when it is perpendicular to the tangent line at the point of contact. Join A, Pj and if APT is a right angle, then A P is a normal, and AB, a portion of the axis of X under it, is called the sub normal. The line BT under the tangent is called the subtangent. Let us now discover whether APT is or is not a right angle. Put a = the tangent of the angle PAT, then by trigonometry But Whence a=- aa= Or Eq. (6) 1 Therefore AP is at right angles to PT. (Prop. 5. Chap. 1.) That is, a tangent line to the circumference of a circle at any point is perpendicular to the radius drawn to that point. SCHOLIUM 3. Admitting the principle, which is a well-known truth of elementary geometry, demonstrated in the preceding scho lium, we would not, in getting the equation of a tangent line to the THE CIRCLE. 131 circle, draw a line cutting the curve in two points, but would draw the tangent line PT at once, and admit that the angle APT was a right angle. Then it is clear that the angle APB= the angle PTB. Now to find the equation of the line, we let x and y r represent the co-ordinates " A - of the point P, and x and y the general co-ordinates of the line, and a the tangent of its angle with the axis of X, then (by Prop III, Chap. I,) we have Now the triangle APB gives us the following expression for the tangent of the angle APB, or its equal PTB, This value of a put in the preceding equation, will give us y -y=- x - t (x -x). y Or y tyy ^x t+xx . Whence xx ^R*^ same as before. PROPOSITION IY. To find the equation of a line tangent to the circumference of a circle, which shall pass through a given point without the circle. Let H (see last figure to the preceding proposition) be the given point, and x" and y" its co-ordinates, and x and y the co-ordinates of the point of tangency P. The equation of the line passing through the two points H and P must be of the form yy"=a(xx") (!) in which a= ^ &.. x x" Since PH is supposed to be tangent at the point P, 132 ANALYTICAL GEOMETRY. " and x and y are the co-ordinates of this point, equation (6) Prop. 3, gives us . Placing this value of a in equation (1) we have for the equation sought. This equation combined with which fixes the point P on the circumference will deter mine the values of x and ?/ , and as there will be two real values for each, it shows that two tangents can be drawn from H, or from any point without the circle, which is obviously true. SCHOLIUM. We can find the value of the tangent PT by means of the similar triangles ABP, PBT, which give x : R : : y r : PT. x More general and elegant formulas, applicable to all the conic sections, will be found in the calculus for the normals, subnormals, tangents and subtangents OF THE POLAR EQUATION OF THE CIRCLE. The polar equation of a curve is the equation of the curve expressed in terms of polar co-ordinates. The variable distance from the pole to any point in the curve is called the radius vector, and the angle which the radius vector makes with a given straight line is called the vari able angle. THE CIRCLE 188 PROPOSITION T. To find the polar equation of the circle. When the center is the pole or the fixed point, the equa tion is and the radius vector It is then constant. 2s~ow let P be the pole, and the co-ordinates of that point referred to the center and rectangular axes be a and 6. Make PJf=r, and MPJP=v the variable angle; AN =x and NM=y. Then (Prop. 11, Chap. 1.) we have xa-\-r cos. v, and y=b+r sin v. These values of x and y substituted in eq. (1), (ob serving that cos. 2 y+sin. 2 tf=l,) will give ?^+2(a cos. v+b sin. v)r+a?-)-b 2 j which is the polar equation sought. SCHOLIUM 1. P may be at any point on the plane. Suppose it at B . Then a = R and bQ. Substituting these values in the equation, and it reduces to r a ZRrcos. v=Q. As there is no absolute term, r=0 will satisfy the equation and correspond to one point in the curve, and this is true, as P is supposed to be in the curve. Dividing by r, and r=2R cos. v. This value of r will be positive when cos. v. is positive, and neg ative when cos. v is negative ; but r being a radius vector can never be negative, and the figure shows this, as r never passes to the left of B\ but runs into zero at that point. When v=0, cos. v=I, then rBB . When v=9Q, cos. v=Q, and r becomes at B , and the variations of v from to 90, deter mine all the points in the semi-circumference BDB . 12 134 ANALYTICAL GEOMETRY. SCHOLIUM 2. If the pole be placed at B y then a=.-\-R and 6=0, which reduces the general equation to r= 2R cos. v. Here it is necessary that cos. v should be negative to make r pos itive, therefore v must commence at 90 and vary to 270 j that is, be on the left of the axis of Y drawn through B, and this corre sponds with the figure. APPLICATION. The polar equation of the circle in its most gen eral form is r -f 2(a cos. v+6 sin v)r+a?-\- b*=R*. (1) If we make 6=0, it puts the polar point somewhere on the axis of Xj and reduces the equation to r 2 -j-2a cos. ^.r-)-a 2 =^ 2 . (2) Now if we make v=0, then will cos. v=l, and the lines represented by r would refer to the points X } X, in the circle. This hypothesis reduces the last equa tion to r*+2ar=(R* a a ) (3) and this equation is the same in form as the common quadratic in algebra, or in the same form as x*px=q. Whence x=r, 2a=p, and R* a t =q These results show us that if we describe a circle with the radius Vq ~fip 2 > and place P on the axis of X at a distance from the cen ter equal to to p, then PX represents one value of x, and PX* the other. That is, Or x= - Jp-^ 2 +t= PX r , and this is the common solution. When p is negative, the polar point is laid off to the left from the center at P . The operation refers to the right angled triangle APM. THE CIRCLE. 135 =\p, PM= tfq, and AM |/$Hri/. Let the form of the quadratic be x*^ipx= q. Then comparing this with the polar equation of the circle, we have 2a=p. R^a > =q. Take AX=.R and describe a semi- circle. Take AP=$p and AP = %p. From P and 1 draw the lines PMj and P M to touch the circle; and draw AM, AM. Here AP is the hypotenuse of a right angled triangle. In the first case AP was a side. In this figure as in the other, PM= ^/q; but here it is inclined to the axis of X; in the first figure it was perpendicular to it. The figure thus drawn, we have PX for one value of x, and PX is the other, which may be determined geometrically. If c a -\-px= q or x= Observe that the first part of the value of x } is minus, correspond ing to a position from P to the left. If x* px= q, we take P for one extremity of the line x. q=, or x=p yp Here the first part of the value of x, (Jp), is plus, because it is laid off to the right of the point P f . Because R= |/lp a q R or AM becomes less and less as the numerical value of q approaches the value of ip 2 . When these two are equal, 7?=0, and the circle becomes a point. When q is greater than ip 2 , the circle has more than vanished, giving no real existence to any of these lines, and the values of x are said to be imaginary. We have found another method of geometrizing quad ratic equations, which we consider well worthy of notice, although it is of but little practical utility. 136 ANALYTICAL GEOMETRY. It will be remembered that the equation of a straight line passing through the origin of co-ordinates is y=ax, (!) and that the general equation of the circle is (x^c) 2 +(y^b} 2 =fi 2 . (2) If we make 6=0, the center of the circle must be some where on the axis of X. Let AM represent a line, the equation of which is y=ax, and if we take a=l, AM will in cline 45 from either axis, as rep- [E A[/^ c p| \s X resented in the figure. Hence ?/=x, and making 6=0, if these two values be substituted in eq. (2) and that equation re duced, we shall find (3) This equation has the common quadratic form. Equation (1) responds to any point in the straight line M M. Equation (2) responds to any point in the circum ference BMM f . Therefore equation (3) which results from the combina tion of eqs. (1) and 2) ? must respond to the points M and M f , the points in which the circle cuts the line. That is, PM and P M f are the two roots of equation (3), and when one is above the axis of X, as in this figure, it is the positive root, and P M f being below the axis of X, it is the negative root. When both roots of equation (3) are positive, the circle will cut the line in two points above the axis of X. When the two roots are minus, the circle will cut the line in two points below the axis of X. "When the two roots of any equation in the form of eq. (3) are equal and positive, the circle will touch the line above the axis of X. If the roots are equal and negative, THE CIRCLE. 137 the circle will touch the line below the axis of X. In case the roots of eq. (3) are imaginary, the circle will not meet the. line. We give the following examples for illustration : f %=5. To determine the values of y hy a geometrical construc tion of this kind, we must make c=_2, and ^Z^=5. a Whence .#=3.74, the radius of the circle. Take any distance on the axes for the unit of measure, and set off the distance c on the axis of X from the origin, for the center of the circle ; to the right, if c is negative, and to the left, if c is positive. Then from the center, with a radius equal to R= , describe a circumference cutting the line drawn midway between the two axes, as in the figure. In this example the center of the circle is at (7, the distance of two units from the origin A, to the right. Then, with the radius 3.74 we described the circumfer ence, cutting the line in M and M 1 , and we find by meas ure (when the construction is accurate) that JHP=4.44, the positive root, and M P r = 1.44, the negative root. For another example we require the roots of the following equation by. construction: IT. B. "When the numerals are too large in any equa tion for convenience, we can always reduce them in the following manner: Put y=nz, then the equation becomes Or *+ -*-. n w 12 138 ANALYTICAL GEOMETRY. Now let 7i= any number what ever. If 7i=3, then Here c=2. 2 "Whence At the distance of two units to the left of the origin, is the center of the circle. We see by the figure that 1 is the positive root, and 3 the neg ative root. But y=nz, n=3, 2=1, y=3 or 9. We give one more example. Construct the equation 7)2 _ fS. Here c=4, and - _ __= 6. Whence _K=2. 2 Using the same figure as before, the center of the cir cle to this example is at _D, and as the radius is only 2, the circumference does not cut the line M M, showing that the equation has no real roots. We have said that this method of finding the roots of a quadratic was of little practical value. The reason of this conclusion is based on the fact that it requires more labor to obtain the value of the radius of the circle than it does to find the roots themselves. Nevertheless this method is an interesting and instruct ive application of geometry in the solution of equations. When we find the polar equation of the parabola, we shall then have another method of constructing the roots of quad ratics which will not require the extraction of the square root. To facilitate the geometrical solution of quadratic equations which we have thus indicated, the operator should provide himself with an accurately constructed scale, which is represented in the following figure. It THE CIECLE. 139 23456 consists of two lines, or axes, at right angles to each other, and another line drawn through their intersection and making with them an angle of 45. On the axes, any con- ( venient unit, as the inch, the half, or the fourth of an inch, etc., is laid off a sufficient number of times, to the right and the left, above and below the origin, from which the divisions are numbered 1, 2, 3, etc., or 10, 20, 30, etc., or .1, .2, .3, etc. To use this scale, a piece of thin, transpa rent paper, through which the numbers may be distinctly seen, is fastened over it, and with the proper center and radius the circumference of a circle is described. The distances from the axis of JT of the intersections of this circumference, with the inclined line through the origin, will be the roots of the equation, and their numerical values may be determined by the scale. By removing one piece of paper from the scale and substituting another, we are prepared for the solution of another equation, and so on. EXAMPLES. 1. Given x 2 +llx=80, to find x. Ans. x=5, or 16. 2. Given z 2 3x=28, to find x. Ans. z=7, or 4. 3. Given x 2 x=2, to find x. Ans. x2, or 1. 4. Given x 2 12x= 32, to find x. Ans. z=4, or 8. 5. Given x 2 12x= 36, to find x. Ans. Each value is 6. 6. Given x 2 I2x= 38, to find x. Both values imag inary. 7. Given x 2 +6x= 10, to find x. Both values imag inary. 8. Given 2*= 81, to find x. Ans. x=9, or 9. 140 ANALYTICAL GEOMETEY. For example 8, c=o and __ZL_=81; "Whence, J2=9\/2. This method may therefore be used for extracting the square root of numbers. In such cases, the center of the circle is at the zero point. CHAPTER THE ELLIPSE. have already developed the properties of the El lipse, Parabola and Hyperbola by geometrical processes, and it is now proposed to re-examine these curves, and de velop their properties by analysis. As he proceeds, the student cannot fail to perceive the superior beauty and simplicity of the analytical methods of investigation; and, even if a knowledge of the conic sections were not, as it is, of the highest practical value, the mental discipline to be acquired by this study would, of itself, be a sufficient compensation for the time and labor given to it. As all needful definitions relating to these curves have been given in the CONIC SECTIONS, we shall not repeat them here, but will refer those to whom such reference may be necessary to the appropriate heads in that division of the work. PKOPOSITION I. To find the equation of the ellipse referred to its axes as the axes of co-ordinates, the major axis and the distance from the center to the focus being given. Let AA f be the major axis, F^F* the foci, and C the center of an ellipse. Make CJF=c CA=A. Take any THE ELLIPSE. 141 point on the curve, and from it let fall the perpendicular Pt on the major axis ; then, by our conventional notation, is Ctx, As F P+PF=2A, we may put F P= A+z, and PF= A z. Then the two right an- gled triangles F Pt, FPt, give us (1) (2) For the points in the curve which cause t to fall between C and F, we would have (cx) 2 +f=(Azy (3) But when expanded, there is no difference between eqs. (2) and (3), and by giving proper values and signs to x and y, eqs. (1) and (2) will respond to any point in the curve as well as to the point P. Subtracting eq. (2) from eq. (1), member from member, and dividing the resulting equation by 4, we find cx=Az, or z= c - (4) A This last equation shows that F P, the radius vector, varies as the abscissa x. Add eqs. (1) and (2), member to member, and divide the result by 2, and we have Substituting the value of z z from eq. (4), and clearing of fractions, we have Or, A 2 y 2 +(A 2 (?)x 2 =A 2 (A 2 c 2 ). (5) conceive the point P to move along describing the curve, and when it comes to the point Z), so that DC makes a right angle with the axis of JT, the two triangles DCF are right angled and equal. 142 ANALYTICAL GEOMETRY. DF f each is equal to A, and as C-F, CF , each is equal to c, we have It is customary to denote D C half the minor axis of the ellipse by B, as well as half the major axis by A, and ad hering to this notation jB 2 =JL 2 c 2 . (6) Substituting this in eq. (5), we have for the equation of the ellipse referred to its center for the origin of co-ordinates. If we wish to transfer the origin of co-ordinates from the center of the ellipse to the extremity A of its major axis, we must put x= A+x f , and y=y . Substituting these values of x and y in the last equa tion, and reducing, we have Or without the primes, we have for the equation of the ellipse when the origin is at the extremity of the major axis. Cor. 1. If it were possible for B to be equal to A, then c must be equal to 0, as shown by eq. (6). Or, while c has a value, it is impossible for B to equal A. If jB=J., then e=0, and the equation becomes A 2 y 2 +A 2 x 2 =A 2 A 2 . Or y*+3*=A*, the equation of the circle. Therefore the circle may be called an ellipse, whose eccentricity is zero, or whose eccen tricity is infinitely small. THE ELLIPSE. 143 Cor. 2. To find where the curve cuts the axis of JT, make y=0 in the equation, then showing that it extends to equal distances from the center. To find where the curve cuts the axis of F, make 2=0, and then Plus B refers to tha point D, B indicates the point directly opposite to ./), on the lower side of the axis of JT. Finally, let x have any value whatever, less than A, then an equation showing two values of ?/, numerically equal, indicating that the curve is symmetrical in respect to the axis of X. If we give to y any value less than jB, the general equa tion gives Showing that the curve is symmetrical in respect to the axis of Y. SCHOLIUM. The ordinate which passes through one of the foci, corresponds to x=c. But A 9 .Z? 8 =ic a . Hence A 3 c a or A y x t =B\ Or (J. 2 x^=B, and this value substituted in TP 272* the last equation, gives y== -- Whence _ is the measure of A A the parameter of any ellipse. PROPOSITION II. Every diameter of the ellipse is bisected in the center. Through the center draw the line DD f . Let x, and y, denote the co-ordinates of the point D, and x , # , the co-ordinates of the point D . 144 ANALYTICAL GEOMETRY. The equation of the curve is D The equation of a line passing through the center, must be of the form y=ax. This equation combined with the equation of the curve, gives AB aAB x= AB aAB These equations show that the co-ordinates of the point Z), are the same as those of the point D , except opposite in signs. Hence DD is bisected at the center. PROPOSITION III. The squares of the ordinates to either axis of an ellipse are to one another as the rectangles of their corresponding abscissas. Let y be any ordinate, and x its corresponding abscissa. Then, by the first proposition, we shall have Let y f be any other ordinate, and x its corresponding abscis sa, and by the same proposition we must have Dividing one of these equations by the other, omitting common factors in the numerator and denominator of the second member of the new equation, we shall have THE ELLIPSE. 145 f _ (2Ax)x y n (2Ax )x r Hence, y* : y l2 =(2Ax)x : (2Ax )x f . (1) By simply inspecting the figure, we cannot fail to per ceive that (2 A x), and x, are the abscissas corresponding to the ordinate y, and (2 A x ) and x are those corres ponding to y*. If we transfer the origin to the lower extremity of the conjugate axis, the equation of the ellipse may be put under the form and by a process in all respects similar to the above, we prove that ^ . ^ . ; (2 _ y)y ; (2jB _ yy . Therefore, the squares of the ordinates, etc. SCHOLIUM, Suppose one of these ordinates, as y to represent half the minor axis, that is, y B. Then the corresponding value of x will be A and (2 A z ,) will be A, also. Whence proportion (1) will become y 1 : B*=(2 A x)x : A*. In respect to the third term we perceive that if A His represented by x, AH will be (2A #), and if G is a point in the circle, whose diameter is A A } and GH the ordinate, then (2A x*)x= and the proportion becomes Or y : GH=B : A. Or A:B=GH:y=DH. If a circumference be described on the conjugate axis as a diam eter, and an ordinate of the circle to this diameter be denoted by X and the corresponding ordinate of the ellipse by x, it may be shown in like manner that 13 146 ANALYTICAL GEOMETRY. PROPOSITION IY. The area of an ellipse is a mean proportional between the areas of two circles, the diameter of the one being the major axis, and of the other the minor axis. On the major axis A A of the ellipse as a diameter describe a circle, and in the semicircle A D A inscribe a polygon of any num ber of sides. From the verti- ces of the angles of this polygon draw ordinates to the major axis, and join the points in which they intersect the ellipse by straight lines, thus constructing a polygon of the same number of sides in the semi-ellipse A D A. Take the origin of co-ordinates at A f , and de note the ordinates BE, CF, etc., of the circle by Y, F , etc., the ordinates B f E, C F, etc., of the ellipse by y, y r , etc., and the corresponding abscissas, which are common to ellipse and circle, by x, x f , etc. Then by the scholium to Prop. 3, we have TiynAiB and Y : y : : A : B, whence Y : Y f : : y : y , from which, by composition, we get Y+ Y : y+y :Y:y::A:B But the area of the trapezoid BEFC is measured by and that of the trapezoid B EFG by ( \(x Xj or \y~\~ therefore, trapez. BEFC Y+ Y A THE ELLIPSE. 147 That is, trapez. BEFC : trapez. B EFC :A:B , or, in words, any trapezoid of the semi-circle is to the corres ponding trapezoid of the semi-ellipse as A is to B. From this we conclude that the sum of the trapezoids in the semi-circle is to the sum of the trapezoids in the semi-ellipse as A is to B. But by making these trape zoids indefinitely small, and their number, therefore, in definitely great, the first sum will become the area of the semi-circle and the second, the area of the semi-ellipse. Hence, Area semi-circle : area semi-ellipse : : A : B or, area circle : area ellipse : : A : B That is, xA 2 : area ellipse : : A : B "Whence, area ellipse= "*!. _ ! __ __ TrA.B _/ But TrA.B is a mean proportional between nA z and Hence ; The area of an ellipse is a mean proportional, etc. SCHOLIUM. Hence the common rule in mensuration to find the area of an ellipse. RULE. Multiply the semi-major and semi-minor axes together, and multiply that product ty 3.1416. PROPOSITION Y. To find the product of the tangents of the angles that two supplementary chords through the vertices of the transverse axis of an ellipse make with that axis, on the same side. Let #, y, be the co-ordinates of any point, as P, and x , y r , the co ordinates of the point A . A I Then the equation of a line which passes through the two points A and P, (Prop. 3, Chap. 1,) will be 148 ANALYTICAL GEOMETRY. y y a(x x ). 00 The equation of the line which passes through the points A and P, will be of the form y y"=a (x x"}. (2) For the given point A , we have 2/ /= =0, and x f = A. "Whence eq. (1) becomes y=a{x+A). (3) For the given point A we have #"=0, and x"=A, which values substituted in eq. (2) give y=a (xA). (4) As y and x are the co-ordinates of the same point P in both lines, we may combine eqs. (3) and (4) in any man ner we please. Multiplying them member by member, we have y z =aa (x 2 A 2 ). (5) Because F is a point in the ellipse, the equation of the curve gives ^=J(^-**)=-JV-^). (6) Comparing eqs. (5) and (6), we find for the equation sought. SCHOLIUM 1. In case the ellipse becomes a circle, that is, in case A=JB, aa -f-l=:0, showing that the angle A PA would then be a right angle, as it ought to be, by (Prop. II, Chap. II.) 7?2 Because is less than unity, or aa f less than 1,* or radius ; A* the two angles PA A and PAA are together less than 90 ; there fore, the angle at P is obtuse, or greater than 90. SCHOLIUM 2. Since aa has a constant value, the sum of the two, , will be least when a=a f . * In trigonometry we learn that tan. x cot. =#2=1. That is, the pro duct of two tangents, the sum of whose arcs is 90, is equal to 1. When the sum is less than 90, the product will be a fraction. THE ELLIPSE. 149 Hence the angle at P will be greatest when P is at the vertex of the minor axis, and the supplementary chords equal ; and the angle at P will become nearer a right angle as P approaches A or A . PROPOSITION VI. To find the equation of a straight line which shall be tangent to an ellipse. Assume any two points, as P and , on the ellipse, and denote the co-ordinates of the first by x 1 , y 1 , and of the second by ", y". Through these points draw a line, the equation of which (Prop. 4, Chap. 1,) is yy =a(xx ) y (1) in which y x x" "We must now determine the value of a when this line becomes a tangent line to the ellipse. Because the points P and Q are in the curve, the co ordinates of those points must satisfy the following equa tions : By subtraction Or A*(y +y")(y -y")=B*(x f +x"}(x -x"}. (2) Whence n-jr ^f **(*+*) x x" A*(y +y") Now conceive the line to revolve on the point P until Q coincides with P, then PE will be tangent to the curve. But when Q coincides with P, we shall have y =y" and x =x". 13* 150 ANALYTICAL GEOMETKY. Under this supposition, we have A*y f The value of a put in eq. (1), gives ~y Reducing A*yy +B*xx =A*y *+B*tf** Or A*yy +B*xx =A*B*. This is the equation sought, x and y being the general co-ordinates of the line. SCHOLIUM 1. To find where the tangent meets the axis of X, we must make y=0. This gives x=^-= CT. In case the ellipse becomes a circle, J3=A, and then the equation will be- come yy -\-xx =A 2 , the equation for a tangent line to a cir cle; and to find where this tangent meets the axis of X> we make y= 0, and x^ CT, as before. In short, as these results are both independent of JB, the minor axis, it follows that the circle and all ellipses on the major axis AB have tangents terminating at the same point T on the axis of J5T, if drawn from the same ordinate, as shown in the figure. SCHOLIUM 2. To find the point in which the tangent to an ellipse meets the axis of T } we make #=0, then the equation for the tangent becomes y- y y As this equation is independent of A, it shows that all ellipses having the same minor axis, have tangents terminating in the same point on the axis of Y, if drawn from the same abscissa. SCHOLIUM 3. If from CTwe subtract CM, we shall have THE ELLIPSE. 151 a common subtangent to a circle, and all ellipses which have 2 A for a major diameter. That is x x We can also find RT by the triangle PRT, as we have the tan gent of the angle at T, / -) to the radius 1. \ A*y I Whence we have the following proportion : The minus sign indicates that the measure from T is towards the left. PROPOSITION VII. To find the equation of a normal line to the ellipse. Since the normal passes through the point of tangency, its equation will be in the form f ff f\ /^\ Because PN is at right angles to the tangent, oa +l=0. But by the last proposition a - Whence a = *JL, and this value of a put in eq. (1) gives JL> X for the equation sought. SCHOLIUM 1. To find where the normal cuts the axis of X, we must make ^=0, then we shall have 152 ANALYTICAL GEOMETRY. APPLICATION. Meridians on the earth are ellipses; the semi- major axis through the equator is A=39G3. miles, and the semi- minor axis from the center to the pole is .Z?r=3949.5. A plumb line is everywhere at right angles to the surface, and of course its prolongation would be a normal line like PJV. In latitude 42, what is the deviation of a plumb line from the center of the earth ? In other words, how far from the center of the earth would a plumb line meet the plane of the equator? Or, what would be the value of CNf As this ellipse differs but little from a circle, we may take CR for the cosine of 42, which must be represented by x . This being assumed, we have s =2945. ==C r .Ar Ans. A 2 J SCHOLIUM 2. To find JWR, the subnormal, we simply subtract (7.^ from CR, whence A* A* We can also find the subnormal from the similar triangles PR T, PNR, thus : TR:RP::RP:RN. A tf* : r. : ATR. Whence NR= PROPOSITION VIII. Lines drawn from the foci to any point in the ellipse make equal angles with the tangent line drawn through the same point. Let C be the center of the ellipse, PT the tangent line, and PF, PF , the two lines drawn to the foci. Denote the distance JB* by c, CF f THE ELLIPSE. 153 by c, the angle FPTby V, and the tangents of the angles P2!Z, PFT, by a and a . Now FPT=PTXPFT. By trigonometry, (Eq. 29, p. 253, Robinson s Geometry), we have Tan. ^PT=tan. (PTXPFT\ That is, tan. F= "". (!) 1-foa Prop. 6, gives us a=-^ . * ,?/ , being the co-ordi- 2p nates of the point P. Let x, y, be the co-ordinates of the point F, then from Prop. 4, Chap. 1, we have w y a!=- *L. x x But at the point _F, y=Q and xc. Whence a f =JL __ x f c These values of a and a f substituted in eq. (1) give B*x __ y f A*~y f x c - ~ A*y (x f c}B*x y r B*cx A* B* Tan. ^ =(AtB*)y?y A*cy f cy (cx A*) ctf observing that J.y 2 +^^ /2 =J. 2 ^ 2 ,and J. 2 JB*=c*. The equation of the line PF will become the equation of the line PF by simply changing +c to c, for then we shall have the co-ordinates of the other focus. We now have tan. cy But if c is made c, then tan. 2PPT= cy 154 ANALYTICAL GEOMETRY. As these two tangents are numerically the same, differ ing only in signs, the lines are equally inclined to the straight lines from which the angles are measured, or the angles are supplements of each other. Whence FPT+F PT=18Q. But F f PH+F PT=18Q. Therefore FPT=F PH. Cor. The normal being perpendicular to the tangent, it must bisect the angle made by the two lines drawn from the tangent point to the foci. SCHOLIUM. Any point in the curve may be considered as a point in a tangent to the curve at that point. It is found by experiment that light, heat and sound, after they approach to, are reflected off, from any reflecting surface at equal angles ; that is, for any ray, the angle of reflection is equal to the angle of incidence. . Therefore, if a light be placed at one focus of an ellipsoidal re flecting surface, such as we may conceive to be generated by revolv ing an ellipse about its major axis, the reflected rays will be con centrated at the other focus. If the sides of a room be ellipsoidal, and a stove is placed at one focus, the heat will be concentrated at the other. Whispering galleries are made on this principle, and all theaters and large assembly rooms should more or less approximate to this figure. The concentration of the rays of heat from one of these points to the other, is the reason why they are called the foci, or burning points. PROPOSITION IX. The product of the tangents of the angles that a tangent line to the ellipse and a diameter through the point of contact, make with the major axis on the same side, is equal to minus the square of the semi-minor divided by the square of the semi- major axis. THE ELLIPSE. 155 Let PT be the tangent line and PP f the diameter through the point of contact ///^ ^x^^^x^Xj X B$ and denote the co-ordi nates of P by x , y . The equation of the diameter is in which a f is the tangent of the angle PCT. Since this line passes through the point P, we must have y f a x f Whence a =^ (1) x f For the tangent of the angle P J!Xwe have Multiplying eqs. (1) and (2), member by member, we find , -- A* SCHOLIUM. The product of the tangents of the angles that a diameter and a tangent line through its vertex make with the major axis of an ellipse is the same (Prop. 5) as that of the tangents of the angles that supplementary chords drawn through the vertices of the major axis make with it. Hence, if a=a, then a =a . That is, if the diameter is paral lel to one of the chords, the tangent line will be parallel to the other chord, and conversely. This suggests an easy rule for drawing a tangent line to an ellipse at a given point, or parallel to a given line. OF THE ELLIPSE REFERRED TO CONJUGATE DIAMETERS. Two diameters of an ellipse are conjugate when either is parallel to the tangent lines drawn through the vertices of the other. 156 ANALYTICAL GEOMETEY. Since a diameter and the tangent line through its ver tex make, with the major axis, angles whose tangents satisfy the equation it follows that the tangents of the angles which any two conjugate diameters make with the major axis must also satisfy the same equation. Now let m he the angle whose tangent is a, and n be the angle whose tangent is a , then cos. m cos. n Substituting these values in the last equation, and re ducing, we obtain A 2 sin. m sin. n-\-J5 2 cos. m cos. 7i=0, which expresses the relation which must exist between A, B, m, and n, to fix the position of any two conjugate di ameters in respect to the major axis, and this equation is called the equation of condition for conjugate diameters. In this equation of condition, m and n are undeter mined, showing that an infinite number of conjugate di ameters might be drawn, but whenever any value is as signed to one of these angles, that value must be put in the equation, and then a deduction made for the value of the other angle. PBOPOSITION X. To find the equation of the ellipse referred to its center and conjugate diameters. The equation of the ellipse referred to its major and minor axes, is The formulas for changing rectangular co-ordinates THE ELLIPSE. 157 into oblique, the origin being the same, are (Prop. 9, Chap. 1,) x=x f cos. m+y f cos. n. yx 1 sin. m-\-y f sin. n. Squaring these, and substituting the values of x 2 and y 2 in the equation of the ellipse above, we have ( (A 2 siu 2 n+B 2 cos 2 n)y 2 +(A 2 sm 2 m+B 2 voa 2 m)x 2 } ^ijp \ +2(J. 2 siii.m sin.n+.B 2 cos.m coa.n)y f x f ) But if we now assume the condition that the new axes shall be conjugate diameters, then J. 2 sin. m sin. n+ IPcos. m cos. n=0, which reduces the preceding equation to (F) which is the equation required. But it can be simplified as follows : The equation refers to the two di ameters B"B and D"D as co-ordi nate axes. For the point B we must make ?/ ; =0, then X f2 = 2 . A * W _= -A 2 . (P) Designating CB f by A , and CD by B . For the point D f we must make x =Q. Then A1T& From (P) we have (J. 2 sin. 2 m+ J B 2 cos. 2 m)=i^-. xL A 2 * From (Q) (J. 2 sin. 2 n+^ 2 cos. 2 7i)=-g 7 2-- These values put in (F) give A 2 T>2 A2T)2 ^ 2 +^-x f2 =A 2 B 2 . B 2 y A 2 Whence A n y 2 +B 2 x 2 = A 2 B r2 . 14 158 ANALYTICAL GEOMETRY. We may omit the accents to x f and ?/ , as they are gen eral variables, and then we have for the equation of the ellipse referred to its center and conjugate diameters. SCHOLIUM. In this equation, if we assign any value to x less than A , there will result two values of y, numerically equal, and to every assumed value of y less than B 1 , there will result two corresponding values of x, numerically equal, differing only in signs, showing that the curve is symmetrical in respect to its conjugate diameters, and that each diameter bisects all chords which are paral lel to the other. OBSERVATION. As this equation is of the same form as that of the general equation referred to rectangular co-ordinates on the major and minor axis, we may infer at once that we can find equa tions for ordinates, tangent lines, etc., referred to conjugate diame ters, which will be in the same form as those already found, which refer to the axes. But as a general thing, it will not do to draw summary conclusions. PROPOSITION XI. As the square of any diameter of the ellipse is to the square of its conjugate, so is the rectangle of any two segments of the diameter to the square of the corresponding ordinate. Let CD be represented by A , and c 1^ CE by B , CH by x, and GH by y, then by the last proposition we have Which may be put under the form jy B f2 (A *x z \. & "Whence A 2 : B f * : : (A 2 x 2 ) : y\ Or (2A Y : (ZBJ : : (A +x)(A x) : y*. Now 2 A 1 and 2B f represent the conjugate diameters D D, E E, and since CH represents x, A +x=D H, and THE ELLIPSE. 159 A x=HD. Also y=GH. Hence the above propor tions correspond to (D D) 2 : (E E) 2 : : D HxHD : (GH}\ SCHOLIUM. As x is no particular distance from C, CF may represent x, then LF will represent y, and the proportion then be comes Comparing the two proportions, we perceive that D H-HD : D F-FD :: GH* : LF*. That is, The rectangle of the abscissas are to one another as the squares of the corresponding ordinates. The same" property as was demonstrated in respect to rectangular co-ordinates in Prop. 3. In the same manner we may prove that Eh-hE : Ef-fE :: (hg)* : PBOPOSITION XII. To find the equation of a tangent line to an ellipse referred to its conjugate diameters. Conceive a line to cut the curve in two points, whose co-ordinates are x* , ?/ , and x", y", x and y being the co ordinates of any point on the line. The equation of a line passing through two points is of the form yy f =a(xx r ), (i) an equation in which a is to be determined when the line touches the curve. From the equation of the ellipse referred to its conju gate axes we have A 2 y f2 + 2 x /2 =A 2 J3 f2 . Subtracting one of these equations from the other, and operating as in Prop. 6, we shall find B f2 x a= _ . A 2 y 160 ANALYTICAL GEOMETRY. This value of a put in eq. (1) will give T?/2/ y;u = X (XX \ 2^r Reducing, and A 2 y y+B /2 x x=A 2 B 2 , which is the equation sought, and it is in the same form as that in Prop. 6, agreeably to the observation made at the close of Prop. 10, PROPOSITION XIII. To transform the equation of the ellipse in reference to con jugate diameters to its equation in reference to the axes. The equation of the ellipse in reference to its conju gate diameter is A 2 y 2 +B 2 x 2 =A 2 B 2 . (l) And the formulas for passing from oblique to rectangu lar axes are (Prop. 10, Chap. 1,) ,_sin. n ycos.n ,_j/cos.ra xsm.m JU ~ -- . tJ ~~~ - ^ . sin. (ft m) sin. (n m) These values of x f and y f substituted in eq. (1) give } j 2( A 2 sin. m cos. m+B r2 sin. n cos. n)xy A f2 JS 2 sm. 2 (n m). This equation must be true for any point in the curve, x being measured on the major axis, and y the corres ponding ordinate at right angles to it. This being the case, such values of A f , B f , m, and n, must be taken as will reduce the preceding equation to the well known form Therefore we must assume A 2 cos. 2 m+B * cos. 2 n= A 2 . (1) A 2 sin. 2 m+B f * sin. 2 n=B*. (2) =0. (3) (4) THE ELLIPSE. 161 The values of m and n must "be taken so as to respond to the following equation, because the axes are in fact conjugate diameters. ^ 2 sin.msin.7i-f--5 2 cos.mcos.7i==0. (5) These equations unfold two very interesting properties. SCHOLIUM 1. By adding eqs. (1) and (2) we find Or 44 2 +4 / =< That is, the sum of the squares of any two conjugate diameters is equal to the sum of the squares of the axes. SCHOLIUM 2. Equation eq. (3) or (5) will give us m when n is given -, or give us n when m is given. SCHOLIUM 3. The square root of eq. (4) gives which shows the equality of two surfaces } one of which is obviously the rectangle of the two axes. Let us examine the other. Let n represent the angle NCB, >^M and m the angle PCB. Then the angle NCP will be represented by (n m). Since the angle MNK is the supplement of NCP, the two an gles have the same sine and In the right-angled triangle NKM, we have 1 : A : : sin.(n m) : MK. MK=A sm.(nm). But NC=B . Whence MK-NC=A B sm.(n m) the parallelogram NCPM. Four times this parallelogram is the parallelogram ML, and fonr times the parallelogram DOB II, which is measured by Ay^B, is equal to the parallelogram HF. Hence eq. (4) reveals this general truth : The rectangle which is formed by drawing tangent lines through 14* L 162 ANALYTICAL GEOMETRY. the vertices of the axes of an ellipse is equivalent to any parallelo gram which can be formed by drawing tangents through the vertices of conjugate diameters. NOTE. The student had better test his knowledge in respect to the truths embraced in scholiums 1 and 3, by an example : Suppose the semi-major axis of an ellipse is 10, and the semi-minor axis 6, and the inclination of one of the conjugate diameters to the axis of X is taken at 30 and designated ~by m. We are required to find A 2 and .Z? 2 , which together should equal AZ+BZ, or 136, and the area NCPM, which should equal AB, or 60, if the foregoing theory is true. Equation (5) will give us the value of n as follows : 100- Jtan./H-36-i-N/3=0. n 36v/3 Or tan.Tinr _ L 100 Log. 36 +i log. 3 log. 100 plus 10 added to the index to corres pond with the tables, gives 9.794863 for the log. tangent of the angle n, which gives 31 56 42", and the sign being negative, shows that 31 66 42" must be taken below the axis of JT, or we must take the sup plement of it, NCB, for n, whence 71=148 3 18", and (n m)=118 3 18". To find A * and .# , we take the formulas from Prop. 10. 100-36 = 3600 =69 2g 52 3600 99+25-92 66-77. And their sum=136. This agrees with scholium 1. As radius 10.000000 Is to 4 J(log.69.23) 0.920147 So is sine (n m) 61 56 42" 9.945713 log. MK= 0.865860 Log. B =.\ log. (66.77) 0.912290 60. log. 60= 1.778150 THE ELLIPSE. 163 PROPOSITION XIY. To find the general polar equation of an ellipse. If we designate the co-ordinates of the pole P, by a and 6, and es timate the angles v from the line PX parallel to the transverse axis, we shall have the following formu las : x=a+r cos.v. y=b+r sin v. These values of x and y substituted in the general equation A 2 y 2 +J3 2 x 2 =A 2 lF, will produce A 2 sin. 2 *; for the general polar equation of the ellipse. SCHOLIUM 1. When P is at the center, a 0, and b=Q, and then the general polar equation reduces to a result corresponding to equations (P) and ( Q) in Prop. 10. SCHOLIUM 2. When P is on the curve J. 2 5 2 - therefore "--"- sin.i> This equation will give two values of r, one of which is 0, as it should be. The other value will correspond to a chord, according to the values assigned to a, b, and v. Dividing the last equation by the equation r=0, and we have sin.v A The value of r in this equation is the value of a chord. When the chord becomes 0, the value of r in the last equation becomes also, and then 164 ANALYTICAL GEOMETRY. Or a result corresponding to Prop. 6, as it ought to do, because the radius vector then becomes tangent to the curve. SCHOLIUM 3. When P is placed at the extremity of the major axis on the right, and if vr^O, then sin. vmO, and cos. #=1 a=A f and 6 ; these values substituted in the general equation will re duce it to J2V-f2.Z? a ^4r=0, which gives r=Q, and r= 2A, obviously true results. When P is placed at either focus, then a=*/A* J3*=c, and 5=0. These values substituted, and we shall have It is difficult to deduce the values of r from this equation, therefore we adopt a more simple method. Let F be the focus, and FP any radi us, and put the angle PFD=v. By Prop. 1, of the ellipse, we learn that (!) an equation in which c -s/J. 8 *, and x any variably distance CD. Take the triangle PD F, and by trigonometry we have 1 : r :: cos.v : c-\-x. Whence x=:r cos.v c. This value of x placed in (1), will give cr. cos.v c a r=A+ J- Whence (A c cos.v}r=A* c a A 9 c 9 Or A c cos.v This equation will correspond to all points in the curve by giving to cos.v all possible values from 1 to 1. Hence, the greatest value of r is ( J.-j-c), and the least value (J. c), obvious results when the polar point is at F. THE ELLIPSE. 165 The above equation may be simplified a little by introducing the ecci iitririty. The eccentricity of an ellipse is the distance from the center to either focus, when the semi-major axis is taken as unity. Designate the eccentricity by e, then 1 : e= A : c. Whence c=eA. Substituting this value of c in the preceding equation, we have e a ) A eA cos. v 1 e cos. v This equation is much used in astronomy. PROPOSITION XV. PROBLEM. Given the relative values of three different radii, drawn from the focus of an ellipse, together with the angles between them, to find the relative major axis of the ellipse, the eccentricity, ami the position of the major axis, or its angle from one of the given radii. Let r, r , and r", represent the three given radii, m the angle be tween r and r , and n that between r and r". The angle between the radius r and the major axis is sup posed to be unknown, and we therefore, call it x. From the last proposition, we have 1 e cos. x r= 1 e cos. (x+m) A(le 2 ) / = _ ^ _ L _ (3\ 1 e cos. (x+rt) Equating the value of A(l e 2 ) obtained from eqs. (1) and (2), and we have r re cos. xr f r f e cos. (x+m) 166 ANALYTICAL GEOMETRY. r _ r t ~r cos. x r cos. (x+m). In like manner from eqs. (1) and (3), we have r re cos. x=r" r"e cos. (x+ri). _ rr" __ ~~ r cos. x r" cos. (x+ri) Equating the second members of eqs. (4) and (5), we have r r> _____ rr" _ r cos.x r 1 cos.(x+m)"~r cos.x r" cos.(x+ri) Whence, r ~ r = r coa. xr eoa. (x+m) r r " r CO s. x r" r cos. x r cos. x cos. m+r sin. x sin. m ~r cos. x r" cos. x cos. n+r" sin. x sin. n r r 1 cos. m+r sin. m tan. x ~~r r" cos. n+ r" sin. n tan. x For the sake of brevity, put r r f =d, r r"=d , the known quantity r r f cos. ma, and r r"cos.w= b. Then the preceding equation becomes d a+r sin.m tan.z d ~b+r"sm.n tan.z From which we get successively db+dr" sin. n tan. x=ad f +d f r l sin. m tan. # (dr" sin. TI e^r sin. m) tan. xad db, ad db tan. x=-j~r f -. - 17-7 : -- > dr sin. n d r sm.m The value of x from this equation determines the posi tion of the major axis with respect to that of r, which is supposed to be known, as it may be by observation. * Having x, eq. (4) or (5) will give e the eccentricity. If the values of e found from these equations do not agree, the discrepancy is due to errors of observation, and in such cases the mean result is taken for the eccentricity. THE ELLIPSE. 167 Equations (1), (2) jind (3) contain A, the semi-major axis, as a common factor in their second members. This factor, therefore, does not affect the relative values of r, r and r", and as it disappears in the subsequent part of the investigation, it shows that the angle x and the eccen tricity are entirely independent of the magnitude of the ellipse. To apply the preceding formulas, we propose the following EXAMPLE. On the first day of August, 1846, an astronomer observed the sun s longitude to be 128 47 31", and by comparing this observation with observations made on the previous and subse quent days, he found its motion in longitude was then at the rate of 57 24". 9 per day. By like observations made on the first of September, he determined the sun s longitude to be 158 37 46", and its mean daily motion for that time 58 6" 6 ; and at a third time, on the Wth of October, the observed longitude was 196 48 4", and mean daily motion 59 22". 9. From these data are required the longitude of the solar apogee, and the eccentricity of the apparent solar orbit. It is demonstrated in astronomy that the relative dis tances to the sun, when the earth is in different parts of its orbit, must be to each other inversely as the square root of the sun s apparent angular motion at the several points ; therefore, (r) 2 , (r ) 2 , and (r") 2 , must be in the proportion of J_ J_ ., and _ J_ 57 24" 9 58 6" 6 59 22" 9 Or as the numbers J_ J_ , and _!_ 3444.9 3486.6 3562.9 Multiply by 3562.9 and the proportion will not be changed, and we may put /3562.9U r , /3562.9U \ 3444.97 V 3486.6 / 168 ANALYTICAL GEOMETRY. By the aid of logarithms we soon find r=1.016982 r =1.010857 and / =!. Hence rr f =d= 0.006125, rr"=d = 0.016982. 158 37 46" 196 48 4" 128 47 31 128 47 31 m= 29 50 15 71= 68 33 To substitute in our formulas, we must have the natu ral sine and cosine of m and n. sin. m=sin. 29 50 15"= 0.497542, cos.= 0.867440. sin. n=sin. 68 33"=0.927238, cos. =0.374472. r r r cos.ra=a=0.140124. r r" cos. 71=6=0.642510. 0^=0.0023695, ^6=0.00393537. d r sin. m=0.008538616, dr" sin. 71=0.005679332. These values substituted in the formula x_ ad db db ad r - _^____^_ .7i ^V sin.m ^V sin.m dr" sin.w give tan y == - QQ156586 == 15.6586 .00285928 28.5928 Log. 15.6586 plus 10 to the index= 11.194746 Log. 28.5928 1.456224 Log. tan. 28 42 45" 9.738522 Long, of r 128 47 31" Long, apogee 100 4 46" According to observation, the longitude of the solar apogee on the 1st of January, 1800, was 99 30 r 8"39, and it increases at the rate of 61"9 per annum. This would give, for the longitude of the apogee on the 1st of January, 1861, 100 33 03"54. To find e, the eccentricity, we employ eq. (5), which is THE PARABOLA. 169 rr" g= __ . r cos.z r" cos.(x+ri) Whence, by substituting the values of r, r", cos. x, etc., we find 0.016982 .016982 = ~ r cos. 2842 / 45 / cos. 96 4318" .891891-f .11694 1.0088 CHAPTER IV. THE PARABOLA. To describe a parabola. Let CD be the directrix, and F the focus. Take a square, as D-B(7, and to one side of it, GB, attach a thread, and let the thread be of the same length as the side GB of the square. Fasten one end of the thread at the point 6r, the other end at F. Put the other side of the square against CD, and with a pencil, P, in the thread, bring the thread up to the side of the square. Slide one end of the square along the line CD, and at the same time keep the thread close against the other side, permitting the thread to slide round the pencil P. As the side of the square, _RD, is moved along the line CD, the pencil will describe the curve represented as passing through the points V and P. GP+PF= the thread. GP+P= the thread. By subtraction PFP=0, or PF=PB. This result is true at any and every position of the point P ; that is, it is true for every point on the curve. Hence, FV=VH. 15 170 ANALYTICAL GEOMETRY. If the square be turned over and moved in the oppo site direction, the other part of the parabola, on the other side of the line FH may be described. PROPOSITION I. To find the equation of the parabola. Take the axis of the parabola for the axis of abscissas and the line at right angles to it through the vertex for the axis of ordinates. The perpendicular distance from the "H V . F D focus F to the directrix BH, is called \ p, a constant quantity, and when this constant is large, we have a parabola on a large scale, and when small, we have a parabola on a small scale. By the definition of the curve, V is midway between F and the line BH, and PF=PB. Put VD=x and PD=y, and operate on the right an gled triangle PDF. (FD) 2 +(PD) 2 =(PF) 2 . That is, (XT- %p)*+f=(x+p) 2 . Whence y 2 =2px, the equation sought. Cor. 1. If we make cc=0, we have y=Q at the same time, showing that the curve passes through the point "F, cor responding to the definition of the curve. As ?/= v/ ^j9x, it follows that for every value of x there are two values of y, numerically equal, one -f , the other , which shows that the curve is symmetrical in respect to the axis of X. Cor. 2. If we convert the equation y 2 =2px into a pro portion, we shall have x : y : : y : 2p, THE PARABOLA. 171 a proportion showing that the parameter of the axis is a third proportional to any abscissa and its corresponding ordi- nate. Cor. 3. If we substitute \p for x in the equation y 2 2px we get y=p or 2y=2p. That is the parameter of the axis of the parabola is equal to the double ordinate through the focus, or, it is equal to four times the distance from the vertex to the directrix. PROPOSITION II. The squares of ordinates to the axis of the parabola are to one another as their corresponding abscissas. Let x, y, be the co-ordinates of any point P, and the co-ordinates of any other point in the curve. Then by the equation of the curve we must have y*=2px. (1) y 2 =2px , (2) By division ^7i M T Whence y 2 : y 2 : : x : d . PROPOSITION III. To find the equation of a tangent line to the parabola. Draw the line SPQ intersecting the parabola in the two points P and Q. Denote the co-ordinates of the first point by a/, y , and of the sec ond, by x", y". The equation of the straight line T~"~ passing through these points is (1) 172 ANALYTICAL GEOMETEY. y -y" in which a is equal to x , x n It is now required to find the value of a when the point Q unites with P, or, when the secant line "becomes a tangent line at the point P. Since P and Q are on the parabola we must have y 2 =2px f And y" 2 =2px" Whence y *y"*= 2p(x x") or (y-yw +/)=%>(* -* ) y u" 2pX Therefore a= V-^/= -TIT/ x x y +y Substituting this value of a in eq. (1) we have for the equation of the secant line. Now if this line he turned about P until Q coincides with P we shall have y"=y f and the line becomes tangent to the curve at the point P. 73 Under this supposition the value of a becomes j and equation (2) reduces to Or y y y * ^ But y 2 2px , substituting this value y 12 in the last equation, transposing and reducing, we have finally yy =p(x+x ) (3) for the equation of the tangent line. Cor. To find the point in which the tangent meets the axis of JT, we must make y=0, this makes Or x = x. THE PARABOLA. 173 That is, VD= VT, or the sub-tangent is bisected by the vertex. Hence, to draw a tangent line from any given point, as P, we draw the ordinate PD, then make TV= VD, and from the point T draw the line 2P, and it will be tan gent at P, as required. PROPOSITION IV. To find the equation of a normal line in the parabola. The equation of a straight line passing through the point P is y-y =a(x-x }. (1) . Let #!, 3/j, be the general co-ordinates of another line passing through the same point, and a the tangent of the angle it makes with the axis of the parabola, its equation will then be y l y =a f (x l x r }. (2) But if these two lines are perpendicular to each other, we must have aa f = 1. (3) But since the first line is a tangent, This value substituted in eq. (3) gives il a =^-. P And this value put in eq, (2) will give for the equation required. 15* 174 ANALYTICAL GEOMETRY. Cor. 1. To find the point in which the normal meets the axis of X, we must make y , = 0. Then by a little reduction we shall have p=x l x f . But VC=x l9 and VD=x f . Therefore DC=p, that is, The sub-normal is a constant quantity, double the distance between the vertex and focus. Cor. 2. Since TV= VD, and VF=$DC, TF=FC. Therefore, if the point F be the center of a circle of which the radius is FC, the circumference of that circle will pass through the point P, because TPC is a right angle. Hence the triangle PFTis isosceles. Therefore, If from the point of contact of a tangent line to the parabola a line be drawn to the focus it will make an angle with the tan gent equal to that made by the tangent with the axis. Cor. 3. Now as V bisects TD and VB is, parallel to PZ>, the point B bisects TP. Draw FB, and that line bisects the base of an isosceles triangle, it is therefore perpendicular to the base. Hence, we have this general truth : If from the focus of a parabola a perpendicular be drawn to any tangent to the curve, it will meet the tangent on. the axis of Y. Also, from the two similar right-angled triangles, FB V and FB T, we have TF-.FB:: FB : FV. Whence ~BF*= TF - FV. But FV is constant, therefore (ft}?} 2 varies as TF, or as its equal PF. SCHOLIUM. Conceive a line drawn par allel to the axis of the parabola to meet the curve at P; that line will make an angle with the tangent equal to the angle FTP. But the angle FTP is equal to the angle FPT; hence the L LPA=ih& THE PARABOLA. 175 [_ FPT. Now, since light is incident upon and reflected from sur faces under equal angles, if we suppose LP to be a ray of light in cident at P, the reflected ray will pass through the focus F, and this will be true for rays incident on every point in the curve; hence, if a reflecting mirror have a parabolic surface, all the rays of light that meet it parallel with the axis will be reflected to the focus ; and for this reason many attempts have been made to form perfect parabolic mirrors for reflecting telescopes. If a light be placed at the focus of such a mirror, it will reflect all its rays in one direction j hence, in certain situations, parabolic mirrors have been made for lighthouses for the purpose of throwing all the light seaward. PROPOSITION Y. If two tangents be drawn to a parabola at the extremities of any chord passing through the focus, these tangents will be perpendicidar to each other, and their point of intersection will be on the directrix. Let PP f be any chord through the focus of the parabola, and P T, P r T the tangents drawn through its extremities. Through T, their intersection, draw BB f perpendic ular to the axis HF, and from the focus let fall the perpendiculars Ft, Ft 1 upon the tangents producing them to intersect BB at B and B . Draw, also, the lines PB, P B , and it . First. The equation of the chord is (1) and of the parabola f=2px (2) Combining eqs. (1) and (2) and eliminating x 9 we find that the ordinates of the extremities of the chord are the roots of the equation 176 ANALYTICAL GEOMETBY. "Whence _ y ,_P+pS*+J and , f= P-P^+i a a Therefore the tangents of the angles that the tangent lines at the extremities of the chord make with the axis are P a The product of these tangents is a a _ 1 tf+~ Whence we conclude that the tangent lines are perpen dicular to each other. Second. Because the AtFt f is right-angled and FV is a perpendicular let fall from the vertex of the right angle upon the hypothenuse, we have (Th. 25, B. II, Geom.) Ff : Ft * : : Vt : Vt and because W and BB f are parallel, (Cor. 3, Prop. 4), we also have Ff .Fi *i:FB i : FB>* nHB: HB But (Cor. 3, Prop. 4,) J? : Ft * : : FP : FP Therefore FP : FP f ::HB: HB Hence the lines PB, P B f are parallel to the axis of the parabola, and (Cor. 2, Prop. 4,) the angles BPt and tPF are pqual. Therefore the right-angled triangles BPt and tPF are equal, and PB=PF. In the same way we prove that P B =P f F. The line BB f is therefore the directrix of the parabola. Cor. Conversely: If two tangents to the parabola are per pendicular to each other, the chord joining the points of contact passes through the focus. THE PARABOLA. 177 For, if not, draw a chord from one of the points of contact through the focus, and at the extremity of this chord draw a third tangent. Then the second and third tangents being both perpendicular to the first, must be parallel. But a tangent line to a parabola, at a point whose or- dinate is ?/ , makes with the axis an angle having ^ for & its tangent ; and as no two ordinates of the parabola are algebraically equal, it is impossible that the curve should have parallel tangent lines. PROPOSITION VI. To find the equation of the parabola referred to a tangent line and the diameter passing through the point of contact as the co-ordinate axes. Let Vbe the vertex and "PLI the axis of the parabola. Through Y any point of the curve, as P, draw the tangent PFand the diameter PR, and take these lines for a sys tem of oblique co-ordinate axes. From a point M, assumed at plea sure, on the parabola, draw MR parallel to P Y and MS perpendicular to VX ; also, draw PQ perpendicular to VX. Let our notation be VQ=c, PQ=b, VS =x, MS =y, PR=x , MR=y f and [__MRS=[_MIl S =m; then the formulas for changing the reference of points from a sys tem of rectangular to a system of oblique co-ordinate axes having a different origin, give, by making [_n=0, x= c-\- x + ?/ r cos.w M 178 ANALYTICAL GEOMETRY. These values of x and y substituted in the equation of the parabola referred to V as the origin which is y*=2px (1) will give b 2 +2by 8iu.m+y 2 sm. 2 m < 2 i pc+2px f +2py f cos.m (2) Because P is on the curve, 6 5 =2pc, and because JRM is parallel to the tangent P Y, we also have (Prop. 3,) cos.m b Whence 26?/ sin.m=2p/ cos.ra By means of these relations we can reduce eq. (2) to Or If we denote -- by 2p the equation of the curve sm. 2 m J referred to the origin Pand the oblique axes PJT, PY, becomes y"=2pV an equation of the same form as that before found when the vertex V was the origin and the axes rectangular. Cor. 1. Since the equation gives y =^2p x f , that is for every value of x two values of y 1 , numerically equal, it follows that every diameter of the parabola bisects all chords of the curve drawn parallel to a tangent through the vertex of the diameter. Cor. 2. The squares of the ordinates to any diameter of the parabola are to each other as their corresponding abscissas. Let x, y and x , y be the co-ordinates of any two points in the curve, then Whence IL=5_ yfZ X f THE PARABOLA. 179 Or y* : y f2 : : x : x f Cor. 3. By a process in no respect differing from that followed in proposition 3 we shall find for the equation of a tangent line to the parabola when referred to any diameter and the tangent drawn through its vertex as the co-ordinates axes. If, in this equation, we make y=0 we get x+x =Q or x= x 1 . T]jat is, the subtangent on any diameter of the parabola is bisected at the vertex of that diameter. SCHOLIUM. Projectiles, if not disturbed by the resistance of the atmosphere, would describe parabolas. Let Pbe the point from which a projec tile is thrown in any direction PH. Undis turbed by the atmosphere and by gravity, it would continue to move in that line, describ ing equal spaces in equal times. But grav ity causes bodies to fall through spaces pro portional to the squares of the times. From P draw PL in the direction of a plumb line, the direction in which bodies fall when acted upon by gravity alone, and draw from A, T, H, etc., points taken at pleasure on PH, lines parallel to PL. Make AB equal to the distance through which a body starting from rest, would fall while the undisturbed projectile would move through the space PA, and lay off TV to correspond to the proportion PA* : PT*::AB: TV (1) Also lay off HK to correspond to the proportion PA* iPffr.AB: EK (2) In the same way we may construct other distances on lines drawn from points of PH parallel to PL. Now through the points B } V, K, etc., draw parallels to PH, intersecting PL in (7, D, L, etc., and through the points B, V, 180 ANALYTICAL GEOMETRY. Kj etc., trace a curve. This curve will represent the path de scribed by a projectile in vacuo, and will be a parabola. Because AB is parallel to PC, and PA parallel to JBC, the figure PAJSOis a parallelogram, and so are each of the other figures, PTVD, PHKL, etc. Let PA=y, PT=y r , PH=y" etc. and PC=x, PD=x f , PL=x" etc. Then proportions (1) and (2) become respectively y* :y *::x : x f y z :y"*\:x : x" But by corollary 2 of this proposition, the curve that possesses the property expressed by these proportions is the parabola, and we therefore conclude that the path described by a projectile in vacuo is that curve. PROPOSITION VII. The parameter of any diameter of the parabola is four times the distance from the vertex of that diameter to the focus. We are to prove that 2p =4PF. Let the angle YPR=m as before. Then by (Prop. 3,) sin. m=r p m a) cos. w b The co-ordinates of the point P being c, by as in the last proposition, we have b 2 =2pc. (2) From eq. (1) 6 2 sin. 2 ra=> 2 cos. 2 m. =p 2 (l sin . 2 m) = Or sin. 2 m=_^ = ^ - b z +p 2 2pc+p 2 But in the last proposition _ _ =2 . Whence sin. 2 m sin. *m=-?~. p> THE PARABOLA. 181 Therefore ^/ 2c+p. Or 2p =4(c+) \ 2i I But ( c+^) =PF. (Prop. 1.) Hence 2p , the param- \ ^/ eter of the diameter PJR, is four times the distance of the vertex of the diameter from the focus. SCHOLIUM. Through the focus F draw a line parallel to the tangent PY. Designate PR by x, and RQ by y. Then, by (Prop. 6), But PF=FT, (Prop. 4, Cor. 2.) And PR=TF, because is a parallelogram. Whence PR=PF; and, since PR=x, and P.#=c+, Therefore 4;r=4f c-f-- j =2z/. or <* P V 2/ 2 This value of a; put in the equation of the curve gives That is, the quantity 2p , which has been called the parameter of the diameter PR, is equal to the double ordinate passing through the focus. PROPOSITION VIII. If an ordinate be drawn to any diameter of the parabola, the area included between the curve, the ordinate and the cor responding abscissa, is two-thirds of the parallelogram con structed upon these co-ordinates. Let V P PQ be a portion of a parabola included between the arc PP P, and the co-ordinates WQ, PQ of the extreme point P, re ferred to the diameter V Q and the ^ tangent through its vertex. 16 182 ANALYTICAL GEOMETRY Take a point, P , on the curve between P and V ; draw the chord PP f and the ordinates PQ, P f Q . Through N, the middle point of PP , draw the diameter NS, and at P and P draw tangents to the parabola intersecting each other at M and the diameter V 7 Q produced at T and T f . The tangents at the points P and P f have a common sub- tangent on the diameter VS, because these points, when referred to this diameter and the tangent at its vertex, have the same abscissa, VJN, (Cor. 3, Prop. 6). The point M is therefore common to the two tangents and the di ameter VS produced. By this construction we have formed the trapezoid PQQ P within, and the triangle TMT without, the par abola, and we will now compare the areas of these figures. From ^draw NL parallel to PQ, and from Q draw QO perpendicular to P , and let us denote the angle YV Q that the tangent at V makes with the diameter V Q by m. By the corollary just referred to we have V T= V Q and V T = V Q . "Whence T T= Q r Q ; and because N is the middle point of PP we also have Therefore (Th. 34, B. I, Geom.,) the area of the trap ezoid PQQ P is measured by NLx QO=NLx Q Qsin.m=Q QxNL$m.m. But NL sin.ra is equal to the perpendicular let fall from j^Vupon Q f Q which is equal to that from M upon the same line. Hence the area of the triangle TMT is measured" The area of the trapezoid is, therefore, twice that of the triangle. If another point be taken between P and V 7 , and we make with reference to it and P the construction that THE PARABOLA. 183 has just been made with reference to P f and P, we shall have another trapezoid within, and triangle without, the parabola, and the area of the trapezoid will be twice that of the triangle. Let us suppose this process continued until we have in scribed a polygon in the parabola between the limits P and V ; then, if the distance of the consecutive points P, P , etc., be supposed indefinitely small, it is evident that the sum of the trapezoids will become the interior curvilinear area PP V Q, and the sum of the triangles the exterior curvilinear area TPV V. Since any one of these trapezoids is to the correspond ing triangle as two is to one, the sum of the trapezoids will be to the sum of the triangles in the same propor tion. But the interior and exterior area together make up the triangle PQT. Therefore interior area=f &PQT, and APT=j:rxPsin.m=F xPsin.w. But VQxPQaiTi. m measures the area of the parallel ogram constructed upon the abscissa V r Q and the ordi- nate PQ. We will denote VQ\>jx and PQ by y. Then the expression for the area in question becomes frj/.sin.m Cor. "When the diameter is the axis of the Q, parabola, then m=90, and sin. m=l, and the expression for the area becomes fry. That is, every segment of a parabola at right angles with the axis is two-thirds of its circumscribing rec tangle. PROPOSITION IX. To find the general polar equation of the parabola. Let Pbe the polar point whose co-ordinates referred to the principal vertex, T 7 ", are c and b. Put VD=x, and D M 184 ANALYTICAL GEOMETRY. =y ; then by the equation of the curve we have y 2 =2px. (1) Put PM=R, the angle MPJ=m, then y / we shall have \ F VD=x=c+ R cos. m. DM=y=b+R sin. m. These values of x and y substituted in eq. (1) will give (b+R sin. m) 2 =2p(c+R cos. m). (2) Expanding and reducing this equation, (R being the variable quantity), we find R 2 sm. 2 m+2R(b sin. m p cos. m)=2pc b* for the general polar equation of the parabola required. Cor. 1. When P is on the curve, b 2 =2pc, and the gen eral equation becomes R 2 ain.*m+2R(b sin.m p cos.m)=-=0. Here one value of R is 0, as it should be, and the other value is D_2(p cos. m b sin. m) jfl i . : - When m = 270, cos. m = and sin. m= 1. Then this last equation becomes a result obviously true. Cor. 2. When the pole is at the focus F, then 6=0, and c=P, and these values reduce the general equation to But sin. 2 m=l cos. 2 m. Whence R 2 R 2 cos. 2 m 2Rpcos.mp 2 . Or R 2 =p 2 -}-2Rpeos.m+R 2 cos. 2 m. Or R=p+R cos.m. Whence R= ?. 1 cos. m and this is the polar equation when the focus is the pole. THE PAEABOLA. 185 When m=0, cos.m= 1Z= P 1, and then the equation becomes or ?.= infinite, 11 showing that there is no termination of the curve at the right of the focus on the axis. When m=90, cos.ra=0, then R=p, as it ought to be, because p is the ordinate passing through the focus. When m=180, cos.m= 1, then R=p; that is, the distance from the focus to the vertex is \p. As in can be taken both above and below the axis and the cos. m is the same to the same arc above and below, it follows that the curve must be symmetrical in respect to the axis. SCHOLIUM 1. If we takep for the unit of measure, that is, as sume p=]-, then the general polar equation will become J? 2 sin. 2 m-|-2 J R(6sin.m cos.m)=2c 6 2 . Now if we suppose m=90, then sin.m=l, cos.m=0, and R would be represented by the line PM , and the equation would be come and this equation is in the common form of a quadratic. Hence, a parabola in which jp=l will solve any quadratic equa tion by making c= VB, JBP=b, then PM will give one value of the unknown quantity. To apply this to the solution of equations, we construct a parabo la as here represented. Now, suppose we require the value of *S y, by construction, in the following equa- +2 tion, . Here 25=2, and 2c Z> 2 =8. Whence 6=1, and c=4.5. Lay off c on the axis, and from the ex tremity lay off b at right angles, above the axis if b is plus, and below if minus. This being done, we find P is the polar point corresponding to 16* -1 -2 -3 1234 M 186 ANALYTICAL GEOMETRY. this example, and PM T =2 is the plus value of y, and PM 4 is the minus value. Had the equation been j,*-2y=8,_ then P would have been the polar point, and P M =4 the plus value, and P M= 2 the minus value. For another example let us construct the roots of the following equation : y* $y= 1. Here b= 3, and 2c 6 2 = 7. Whence c=l. From 1 on the axis take 3 downward, to find the polar point P". Now the roots are P"m and P"m , both plus.- P"m=1.58, and P"m =4.4l4. Equations having two minus roots will have their polar points above the curve. When c comes out negative, the ordinates caiinot meet the curve, showing that the roots would not be real but imaginary. The roots of equations having large numerals cannot be con structed unless the numerals are first reduced. To reduce the numerals in any equation, as we proceed as follows : Puty nz, then = 146 n n 2 Now we can assign any value to n that we please. Suppose 71=10, then the equation becomes The roots of this equation can be constructed, and the values of y are ten times those of z. SCHOLIUM 2. The method of solving quadratic equations em ployed in Scholium 1 may be easily applied to the construction of the square roots of numbers. Thus, if the square root of 20 were required, and we represent it by y, we shall have THE PARABOLA. 187 an incomplete quadratic equation; but it may be put under the form of a complete quadratic by introducing in the first number the term xy, and we shall then have ^0x^=20. Here 25=0, and2c 2 =20; whence c=10; which shows that the ordinate corresponding to the abscissa 10 on the axis of the pa rabola will represent the square root of 20. In the same way the square roots of other numbers may be determined EXAMPLES. 1. What is the square root of 50 ? Let each unit of the scale represent 10, then 50 will be repre sented by 5. The half of 5 is 2. An ordinate drawn from 2 on the axis of X will be about 2.24, and the square root of 10 will be represented by an ordinate drawn from 5, which will be about 3, 16. Hence, the square root of 50 cannot differ much from (2.24) (3.16) = 7,0786. ANOTHER SOLUTION. 50=25x2, v/50=W2^ From 1 on the axis of X draw an ordinate ; it will measure 1.4-)-. Hence, ^50=5(1.4+)=7,+. "What is the square root of 175? 175=25 x 7, x/175 = 5v/ 7. An ordinate drawn from 3.5 the half of 7 will measure 2.65. Therefore ^175=5(2.65)=13.25 nearly. 3. Given x 2 T \z=8 to find x. Ans. sc=2.9.-j- 4. Given %x 2 -\-$x= T \to&nd x. * Ans. #=0.60 -f. 5. Given \y 2 y=2 to find y. Ans. y=3.17, or 2.5-f. 188 ANALYTICAL GEOMETRY. CHAPTER V. THE HYPERBOLA. To describe an hyperbola. The definition of this curve suggests the following method of describing it mechanically : Take a ruler F H, and fasten one end at the point F 9 on which the ru ler may turn as a hinge. At the other end of the ruler attach a thread, and let its length be less than that of the ruler by the given line A A. Fasten the other end of the thread at F. With a pencil, P, press the thread against the ruler and keep it at equal tension between the points H and F. Let the ruler turn on the point F 1 , keeping the pencil close to the ruler and letting the thread slide round the pencil; the pencil will thus describe a curve on the paper. If the ruler be changed and made to revolve about the other focus as a fixed point, the opposite branch of the curve can be described. In all positions of P, except when at A or A , PF f and PF will be two sides of a triangle, and the difference of these two sides is constantly equal to the difference be tween the ruler and the thread ; but that difference was made equal to the given line A 1 A ; hence, by definition, the curve thus described must be an hyperbola. PROPOSITION I. To find the equation of the hyperbola referred to its center and axes. THE HTPEEBOLA. 189 Let C be the center, F and F r the foci, and AA r the transverse axis of an hyperbola. Draw CO at right angles to AA f , and take these lines _<s for the co-ordinate axes. From P, any point of the curve, draw PF, PF f to the foci, and PH perpendicular to AA f . Make CF=c, CA=A, CH=x, and PH=y; then the equation which expresses the relation between the vari ables x and y, and the Constances c and A, will be the equation of a hyperbola. By the definition of the curve we have r f r=2A. (1) The right-angled APffiPgives r*=(xcy+y*. (2) The right-angled &PHF gives r /a=( a; + c )a + y3. (8) Subtracting eq. (2) from eq. (3) we get Dividing eq. (4) by eq. (1) we have . J. Combining eqs. (1) and (5) we find r =J.+-, and r= ^L+^. .A -a. This value of r substituted in eq. (2) gives A 2 2cx+=x* A 2 Reducing, we find for the equation sought. SCHOLIUM. As c is greater than A, it follows that (A 2 c 2 ) must be negative ; therefore we may assume this value equal to J5 2 . Then the equation becomes 190 ANALYTICAL GEOMETRY. This fdrm is preferred to the former one on account of its simi larity to the equation of the ellipse, the difference being only in the negative value of 2 . Because A 2 c*=B* Now to show the geometrical mag nitude of Bj take C as a center, and CF as a radius, and describe the circle FI1F . From A draw AH at right angles to CF. Now CH=c, OA=A, and if we put AH=B, we shall have A 2 +B 2 =c 2 , as above. Whence AH must equal B. PROPOSITION II. To determine the figure of the hyperbola from its equation. Kesuming the equation and solving it in respect to y, we find If we make =0, or assign to it any value less than A, the corresponding value of y will be imaginary, showing that the curve does not exist above or below the line A A. If we make x=A, then y showing two points in the curve, both at A. If we give to x any value greater -Pt than Ay we shall have two values of y, numerically equal, showing that the curve is symmetrically divided by the axis A r A produced. If we now assign the same value to x taken negatively, that is, make x ( x\ we shall have two other values of y, the same as before, corresponding to the left branch of the curve. Therefore, the two branches of the curve are THE HYPEBBOLA. 191 equal in magnitude, and are in all respects symmetrical but op posite in position. Hence every diameter, as DD , is bisected in the center, for any other hypothesis would be absurd. SCHOLIUM 1. If through the center, C, we draw CD, CD , at right angles to A A, and each equal to J3, we can have two opposite branches of an hyperbola passing through D and D above and below C. as the two others which pass through the points A and A, at the right and left of C. The hyperbola which passes through D and D is said to be con jugate to that which passes through A and A , or the two are con jugate to each other. DD is the conjugate diameter to A A, and DD may be less than, equal to, or greater than A A, according to the relative values of c and A in Prop. 1. When B is numerically equal to A, the equation of the curve becomes y*-x*=A*, and DD = A A . In this case the hyperbola is said to be equilateral. SCHOLIUM 2. To find the value of the double ordinate which passes through the focus, we must take the equation of the curve and make x=c, then But we have shown that A*+B*c*, or .B 2 =c 2 A*. Whence A*y*=E 4 . Or Ay=B\ or 2y=2^!. A That is, 2 A : 2B : : 2B : 2y, showing that the parameter of the hyperbola is equal to the double ordinate t to the major axis, that passes through the focus. SCHOLIUM 3. To find the equation for the conjugate hyper bola which passes through the points D, D f , we take the general equation 192 ANALYTICAL GEOMETRY. and change A into E and x into y, the equation then becomes JB*x 2 A*y*= A**, which is the equation for conjugate hyperbola. PROPOSITION III. To find the equation of the hyperbola when the origin is at the vertex of the transverse axis. When the origin is at the center, the equation is And now, if we move the origin to the vertex at the right, we must put Substituting this value of x in the equation of the hy perbola referred to its center and axes, we have AyB*x 2 2JB 2 Ax = 0. We may now omit the accents, and put the equation under the following form : which is the equation of the hyperbola when the origin is the vertex and the co-ordinates rectangular. PROPOSITION IY. To find the equation of a tangent line to the hyperbola, the origin being the center. In the first place, conceive a line cutting the curve in two points, P and Q. Let x and y be co-ordinates of any point on the line, as $, x r and y co-ordinates of the point P on the curve, and x" and y" the co- s / ordinates of the point Q on the . curve. THE HYPERBOLA. 193 The student can now work through the proposition in precisely the same manner as Prop. 6, of the ellipse was worked, using the equation for the hyperbola in place of that of the ellipse, and in conclusion he will find for the equation sought. Cor. To find the point in which a tangent line cuts the axis of JL, we must make y=0, in the equation for the tangent ; then x If we subtract this from CD (x f ) we shall have the sub tangent = A * x 2 A* ~ PRO POSITION Y. To find the equation of a normal to the hyperbola. Let a be the tangent of the angle that the line TP makes with the transverse axis, (see last figure), and a the same with reference to the line PN. Then if PN is a normal, it must be at right angles to PT, and hence we must have oa +l=0. (1) Let x and/ be the cor-ordinates of the point P on the curve, and x, ?/, the co-ordinates of any point on the line PN, then we must have y -. y r=a f (x x }. (2) In working the last proposition, for the tangent line PTwe should have found This value of a put in eq. (1) will show us that a!~ 17 194 ANALYTICAL GEOMETRY. And this value of a put in eq. (2) will give ns for the equation of the normal required. Cor. To find the point in which the normal cuts the axis of X, we must make y=0. This reduces the equation to Whence = CN. If we subtract CD, (x ) 7 from CN 9 we shall have DN, the sub-normal. That is, , the sub-normal. PROPOSITION VI. A tangent to the hyperbola bisects the angle contained by lines drawn from the point of contact to the foci. If we can prove that F P: PF: :F T:TF, (1) it will then follow (Th. 24, B. IE, Geom.,) that the angle F PT=t~hQ angle TPF. In Prop. 1, of the hyperbola, we find that =r =^L+.-, and A and by corollary to Prop. 4 F>T=F f C+CT=c+ We will now assume the proportion , and TF= x x THE HYPERBOLA. 195 Multiply the terms of the first couplet by A, and those of the last couplet by x, then we shall have (A* + cx) : ( A*+cx) : : (cx+A 2 ) : xz. Observing that the first and third terms of this propor tion are equal, therefore xzcx A 2 . Or z=c ^-=TF.- x Now the first three terms of proportion (2) were taken equal to the first three terms of proportion (1), and we have proved that the fourth term of proportion (2) must be equal to the fourth term of proportion (1), therefore proportion (1) is true, and consequently F PT=TPF. Cor. 1. As TT is a tangent, and PN its normal, it follows that the angle TPN= the angle FPN, for each is a right angle. From these equals take away the equals TPF, T PQ, and the remainder FPNwuti equal the re mainder QPN. That is, the normal line at any point of the hyperbola bisects the exterior angle formed by two lines drawn from the foci to that point. A 2 Cor. 2. The value of CT we have found to be , and x the value of CD is x, and it is obvious that A * A A x . ~cL . . yi .X, X is a true proportion. Therefore (A) is a mean proportional between CT and CD. A tangent line can never meet the axis in the center, because the above proportion must always exist, and to make the first term zero in value, we must suppose x to be infinite. Therefore a tangent line passing through the center cannot meet the hyperbola short of an infinite distance there from. Such a line is called an asymptote. 196 ANALYTICAL GEOMETRY. OF THE CONJUGATE DIAMETERS OF THE HYPERBOLA. DEFINITION. Two diameters of an hyperbola are said to be conjugate when each is parallel to a tangent line drawn through the vertex of the other. According to this definition, GGr and jET.fi" in the ad joining figure are conjugate diameters. EXPLANATION. 1. The tangent line which passes through the point H is par allel to CG. Hence CG makes the same angle with the axis as that tangent line does. If we designate the co-ordinates of the point If, in reference to the center and axes by x and y , and by a the tangent of the angle made by the inclination of CG with the axis, then in the in vestigation (Prop. 6,) we find Now if we designate the tangent of the angle which CH makes with the axis by a , the equation of OH must be of the form because the line passes through the center. Whence a =. __ . (2) Multiplying eqs. (1) and (2) together member by member, and we find to which equation all conjugate diameters must correspond. EXPLANATION 2. If we designate the angle GOB by n, and HOB by m, we shall have And sin. m , sin. n =a , =a. cos. m cos. n tan. m tan. n= , THE HYPERBOLA. 197 PROPOSITION VII. To find the equation of the hyperbola referred to its center and conjugate diameters. The equation of the curve referred to the center and axes is , to change rectangular co-ordinates into oblique, the origin being the same, we must put x~*f cpB.m+y cp B .n } And y=x f sin. m+y sin. n J These values of x and y, substituted in the above gen eral equation, will produce -f 2(sin. m sin. nA* cos. m cos. n = A**. (1) Because the diameters are conjugate, we must have sin. m sin. n B 2 cos. m cos. n A 2 Whence (sin. m sin. n A 2 ccs. m cos. nB 2 )=Q (k) This last equation reduces eq. (1) to which is the equation of the hyperbola referred to the center and conjugate diameters. If we make 2/ r =0, we shall have (3) If we make x =0, we shall have J2J?2 -,=CG (4) If we put A 2 to represent CH*, and regard it as posi tive, the denominator in eq. (3) must be negative, the nu- 198 ANALYTICAL GEOMETRY. merator being negative. That is, ^ 2 sin. 2 m must be less than J9 2 cos. 2 w. That is, .4 2 sin. 2 ra< 2 cos. 2 w. B tan. ra<-7. 7>a B ut tan. m tan. n -p. T) Whence tan. n>-f, r ? ^ 2 sin. 2 ft>.5 2 cos. 2 %. Therefore the denominator in eq. (4) is positive, but the numerator being negative, therefore CG* must be negative. Put it equal to B". Now the equations (3) and (4) become A * = j-^-^T ? f , B* - a . a ^ 2 - a Or (^ 2 sin. 2 m Comparing these equations with eq. (2) we perceive that eq. (2) may be written thus : "Whence A *y n Omitting the accents of x f and y* , since they are gene ral variables, we have A yB f2 x 2 =A 2 2 , for the equation of the hyperbola referred to its center and conjugate diameters. SCHOLIUM 1. As this equation is precisely similar to that re ferred to the center and axes, it follows that all results hitherto de termined in respect to the latter will apply to conjugate diameters by changing A to A and B to B , For instance, the equation for a tangent line in respect to the center and axes has been found to be THE HYPERBOLA. Therefore, in respect to conjugate diameters it must be 199 and so on for normals, sub-normals, tangents and sub-tangents. SCHOLIUM 2, If we take the equation and resolve it in relation to y, we shall find that for every value of x greater than A r we shall find two values of y numerically equal, which shows that ON bisects MM and every line drawn parallel to MM, or parallel to a tangent drawn through L, the vertex of the diameter LL . Let the student observe that these several geometrical truths were discovered by changing rectangular to oblique co-ordinates. We will now take the reverse operation, in the hope of discovering other geometrical truths. Hence the following : PROPOSITION VIII. To change the equation of the hyperbola in reference to any system of conjugate diameters, to its equation in reference to the axes. The equation of the hyperbola referred to conjugate diameters is A 2 y 2 _B 2 x f2 =A IS 2 . To change oblique to rectangular co-ordinates, the for mulas are (Chap. 1, Prop. 10,) ,_xain.n ?/cos.n ,_?/cos.w xsin. m sin. (n m) sin. (n m) Substituting these values of x and y f in the equation, we shall have A \y cos. m x sin. m) 2 B \x sin. n y cos. rif_ j, 2 M sin. F (w m) sin. 2 (n m) By expanding and reducing, we shall have 200 ANALYTICAL GEOMETRY. (J. /2 cos. 2 w 2( = A 2 B f2 sin. 2 (ft m). which, to be the equation of the hyperbola when referred to the center and axes, must take the well known form, A 2 fB 2 x 2 =A 2 B 2 . Or *in other words, these two equations must be, in fact, identical, and we shall therefore have A 2 cos. 2 mB 2 co8 2 n=A 2 . (1) A 2 8in. 2 mB 2 sin. 2 /*= B 2 . (2) A n sin. ra cos. m-\-B f2 sin , n cos. n= 0. (3) A 2 B fz $in.\nm)=A 2 B 2 . (4) By adding eqs. (1) and (2), observing that (eos. 2 w-J- 8in. 2 m)=l, we shall have Or 4J/ 2 4B f2 =4A y which equation shows this general geometrical truth : That the difference of the squares of any two conjugate di ameters is equal to the difference of the squares of the axes. Hence, there can be no equal .conjugate diameters un less AB y and then every diameter will be equal to its con jugate : that is, A B . T>2 Equation (3) corresponds to tan.mtan.n= _ ? the -equa- _^L tion of condition for conjugate di ameters. Equation (4) reduces to A B &m.(nm)=AB. The first member is the measure of the parallelogram GCHT, and it being equal to A X B, shows this ge ometrical truth : That the parallelogram formed by drawing tangent lines through the vertices of any system of conjugate diameters of THE HYPERBOLA. 201 the hyperbola, is equivalent to the rectangle formed by drawing tangent lines through the vertices of the axes. REMAKK. The reader should observe that this propo sition is similar to (Prop. 13,) of the ellipse, and the gen eral equation here found, and the incidental equations (1), (2), (3), and (4), might have been directly deduced from the ellipse by changing B into B */ i ? and B r into OF THE ASYMPTOTES OF THE HYPERBOLA. DEFINITION. If tangent lines be drawn through the vertices of the axes of a system of conjugate hyperbolas, the diagonals of the rectangle so formed, produced inde finitely, are called asymptotes of the hyperbolas. Let AA , BB , be the axes of conjugate hyperbolas, and through the vertices A , A , B, B , let tan gents to the curves be drawn form ing the rectangle, as seen in the figure. The diagonals of this rect angle produced, that is, DD f and EE , are the asymptotes to the curves corresponding to the definition. If we represent the angle -DOJTby m, E CX will be m also, for these two angles are equal because CB= CB . It is obvious that T> tan. m A sin. 2 m B 2 whence 2 = -^ cos 2 , m A 2 But cos. 2 m=l sin.^ra. Therefore sin. 2 m _-B 2 1 sin. 2 m A 2 202 ANALYTICAL GEOMETRY. Consequently sin. 2 m= 2 P2 > and cos. 2 w=-_ __, .4. ~T -D -A ~T-O which equations furnish the value of the angle which the asymptotes form with the transverse axis. PROPOSITION IX. To find the equation of the hyperbola, referred to its center and asymptotes. Let CM=x, and PMy. Then the equation of the curve referred to its center and axes is A*&. (1) From P draw PH parallel to CE, and PQ parallel to CM. Let CH=x , and ~Now the object of this proposition is to find the values of x and y in terms of x and?/ , to substitute them in eq. (1). The resulting equation reduced to its most simple form will be the equation sought. The angle HCMis designated by m, and because IIP is parallel to CE, and PQ parallel to CM, the angle HPQ is also equal to m. Now in the right angled triangle CHh we have Hh =x sin. m, and Chx cos. m. In the right angled triangle PQH we have HQ =y r sin. m, and PQ=y cos. m. Whence Hh HQ=Qh=PM=y=x sin. m y r sin. m. Or y=(% j/0 sin. m. ( 2 ) Ch+ QP= CM=x=x f cos. m-f y cos. m. Or z=(a: -HO cos. m. (3) These values of y and x found in eqs. (2) and (3) sub stituted in eq. (1) will give THE HYPERBOLA. 203 A\x yJ sin. 2 ra W (x +y } 2 cos. 2 w= J. 2 JB 2 . Placing in this equation the values of sin. 2 mandcos. 2 w, previously determined, we have Dividing through by A Z B 2 , and at the same time mul tiplying by (J. 2 -f -B 2 ), we get Or or . which is the equation of the hyperbola referred to its center and asymptotes. Cor. As x and y f are general variables, we may omit the accents, and as the second member is a constant quantity, we may represent it by M 2 . Then xy= M 2 , or x=^l y This last equation shows that x increases as y decreases ; that is, the curve approaches nearer and nearer the asymptote as the distance from the center becomes greater and greater. But x can never become infinite until y becomes 0; that is, the asymptote meets the curve at an infinite distance, corres ponding to Cor. 2, Prop. 6. PROPOSITION X. All parallelograms constructed upon the abscissas, andordi- nates of the hyperbola referred to its asymptotes are equivalent, each to each, and each equivalent to JAB. Let x and y be the co-ordinates corresponding to any point in the curve, as P. Then by the equation of the curve in relation to the center and asymptotes, we have x=M z . (1) 204 ANALYTICAL GEOMETRY. Also let x , y 9 represent the co-ordinates of the point Q. Then x y =M\ (2) The angle p CD between the asymptotes we will represent by 2m. Now multiply both members of equations (1) and (2) by sin. 2m. Then we shall have xy sin. 2m= M 2 sin. 2m. (3) x y 1 sin. 2m= M 2 sin. 2m. (4) The first member of eq. (3) represents the parallelo gram (7P, and the first member of eq. (4) represents the parallelogram CQ ; and as each of these parallelograms is equivalent to the same constant quantity, they are equiv alent to each other. Now A is another point in the curve, and therefore the parallelogram AH CD is equal to (M 2 sin. 2m), and there fore equal to CQ, or CP. Hence all parallelograms bounded by the asymptotes and terminating in a point in the curve, are equivalent to one another, and each equiv alent to the parallelogram AHCD, which has for one of its diagonals half of the transverse axis of A. We have now to find the analytical expression for this parallelogram. The angle HCA=m, ACD=m, and because AH is pa rallel to <7D, CAH=m. Hence, the triangle CAH is isosceles, and CH=HA. The angle AHq=2m. Now by trigonometry sin. 2m : A : : sin. m : CH. But sin. 2m=2 sin. m cos. m. Whence 2 sin. m cos. m : A : : sin. m : CH. 2cos.m Multiply each member of this equation by CA A 9 and sin.m, then THE HYPERBOLA. 205 . m. 2 cos.m 2 The first member of this equation represents the area 75 of the parallelogram CHAD, and the tan. w=_. Hence, A. A2 T> the parallelogram is equal _-_=J^-B, which is the value 2t A. also of all the other parallelograms, as CQ, CP, etc. PROPOSITION XI. To find the equation of a tangent line to the hyperbola re ferred to its center and asymptotes. Let P and Q be any two points on the curve, and denote the co-ordinates of the first by x , # , and of the second by #", y". The equation of a straight line pass ing through these points will be of the form y y =a(x x ) (1) in which a=^ ^ . x x" We are now to find the value of a when the line be comes a tangent at the point P. Because P and Q are points in the curve, we have x f y =x"y". From each member of this last equation subtract then Or Whence x (y -y")=-y(x -x"). =^ * ** x x\ x This value of a put in eq. (1) gives y /= ^(x-x f ). (2) ANALYTICAL GEOMETRY. Now if we suppose the line to revolve on the point P as a center until Q coincides with P, then the line will be a tangent, and x f =x", and y =y", and eq. (2) will be come ?/ y y = ?-(x * )> which is the equation sought. Cor. To find the point in which the tangent line meets the axis of JT, we must make y=Q ; then x=2x . That is, Ct is twice CR, and as RP and CT are parallel, tP=PT. A tangent line included between the asymp totes is bisected by the point of tangency. SCHOLIUM. From any point on the asymptote, as Z>, draw D G parallel to Tt, and from C draw CP, and produce it to S. By scholium 2 to Prop. 7 we learn that CP produced will bisect all lines parallel to tT and within the curve; hence gd is bisected in S. But as CP bisects tT, it bisects all lines parallel to tT within the asymptotes, and DG is also bisected in S ; hence dD= Gg. In the same manner we might prove dh=kv, because hk is par allel to some tangent which might be drawn to the curve, the same as D G is parallel to the particular tangent t T. Hence, If any line be drawn cutting the hyperbola, the parts be tween the asymptotes and the curve are equal. This property enables us to describe the hyperbola by points,, when the asymptotes and one point in the curve are given. Through the given point d, draw any line, as D G, and from G set off Gg=dD, and then g will be a point in the curve. Draw any other line, as hk, and set off kv=dh ; then v is another point in the curve. And thus we might find other points between v and g, or on either side of v and g. THE HYPERBOLA. 207 PROPOSITION XII. To find the polar equation of the hyperbola, the pole being at either focus. Take any point P in the hyperbola, and let its distance from the nearest focus be represented by r, and its dis tance from the other focus be repre- p> A: C A! F H sented by r . Put CH=x, OF=c, and CA=A. Then, by Prop. 1, we have , (1) A. (2) A * "Now the problem requires us to replace the symbol x, in these formulas, by its value, expressed in terms of r and r , and some function of the angle that these lines make with the transverse axis. First. In the right-angled triangle PFH, if we desig nate the angle PFH by v, we shall have 1 : r : : cos. v : FH=r cos. v. CH= CF+FH. That is, x=c+r cos. v. The value of x put in eq. (1) gives A .(?+cr cos. v T Whence r= ^^ _. (3) A c cos. v Second. In the right-angled triangle F PH, if we des ignate the angle PF H^j v f , we shall have 1 : r : : cos. v f : F f H=r cos. v . But F H=F 0+ CH. That is, r cos. v =c+x. Or x=r cos. v r c* 208 ANALYTICAL GEOMETKY. and this value of x put in eq. (2) gives cr r cos. v f (? Whence r = . A c cos. v 1 Equations (3) and (4) are the polar equations required. Let us examine eq. (3). Suppose fl=0, then cos.v=l, and Ac But a radius vector can never be a minus quantity, therefore there is no portion of the curve on the axis to the right of F. To find the length of r when it first strikes the curve, we find the value of the denominator when its value first becomes positive, which must be when A becomes equal to c cos. v ; that is, when the denominator is 0. the value of r will be real and infinite. If A ccos.v=0, A then cos. v _ . c This equation shows that when r first meets the curve it is parallel to the asymptote, and infinite. When v=90, cos.v=0, and then r is perpendicular at the point F, and equal to c _ or __ half the paranie- A A ter of the curve, as it ought to be. "When #=180, then cos.v= 1, and ccos.v=c; then c+A a result obviously true. As v increases, the value of r will remain positive, and, consequently, give points of the hyperbola until cos.-y again becomes equal to _ , which will be when the radius c THE HYPERBOLA. 209 vector makes with the transverse axis an angle equal to 360 minus that whose cosine is __. Equation (3) will c therefore determine all points in the right hand branch of the hyperbola. Now let us examine equation (4). If we make i/=0, then A c as it ought to be. To find when r f will have the greatest possible value, we must put A ccos.?/=0. Whence cos.t/= __ . c This shows that v is then of such a value as to make r 1 parallel to the asymptote and infinite in length. If we increase the value of v from this point, the denominator will become positive, while the numerator is negative, which shows that then / will become negative, indicating that it will not meet the curve. The value of r will continue negative until the radius vector falls below the transverse axis, and makes with it an angle having -f _ for its cosine. Values of v between c . this and 360 will render r positive and give points of the hyperbola. Equation (4) will, therefore, also determine all the points in the right hand branch of the hyperbola-. By changing the sign of c, we change the pole to the focus F , and eqs. (3) and (4), which then determine the left hand branch of the hyperbola, become (3 ) . v and r^ A* c* (4/) A+c cos. v 10* o 210 ANALYTICAL GEOMETRY. GENERAL REMARKS. When the origin of co-ordinates is at the circumference of a circle, its equation is When the origin of a parabola is at its vertex, its equation is ya=2pa;. When the origin of co-ordinates of the ellipse is at the vertex of the major axis, the equation of the curve is When the origin of co-ordinates is at the vertex of the hyper bola, the equation for that curve is But all of these are comprised in the general equation In the circle and the ellipse, q is negative ; in the hyperbola it is positive, and in the parabola it is 0. CHAPTER VI. ON THE GEOMETRICAL REPRESENTATION OF EQUATIONS OF THE SECOND DEGREE BETWEEN TWO YARIABLES. 1. It has been shown in Chap. 1, that every equa tion of the first degree between two variables may be represented by a straight line. It has also been shown that the equations of the circle, the ellipse, the parabola and the hyperbola were all some of the different forms of an equation of the second de gree between two variables. It is now proposed to prove that, when an equation of the second degree between two variables represents any geometrical magnitude, it is some. one of these curves. The limits assigned to this work compel us to be as brief in this* investigation as is consistent with clearness. "We shall, therefore, restrict ourselves to a demonstration INTERPRETATION OF EQUATIONS. 211 of this proposition ; the determination of the criteria by which it may be decided in every case presented, to which of the conic sections the curve represented by the equa tion belongs, and the indication of the processes by which the curve may be constructed. 2. The equation of the second degree between two variables, in its most general form, is Ay*+Bxy+ Cx*+Dy+Ex+F=Q, for, by giving suitable values to the arbitrary constants, A 9 B, Cj etc., every particular case of such equation may be deduced from it. The formulas for the transformation of co-ordinates being of the first degree in respect to the variables, the degree of an equation will not be changed by changing the reference of the equation from one system of co-or dinate axes to another. We may therefore assume that our co-ordinate axes are rectangular without impairing the generality of our investigation. The resolution, in respect to ?/, of the general equation gives ~2A X 2^L 2^\|. 4AC "Now let A X, A Y be the co-ordinate axes, and draw the straight line M Q, whose equation is _B_ X __I>_ 2 A 2 A* For any value, AD, of x, the or- dinate, DC, of this line, is ex pressed by B D E X 2T 2T AA ^J^l. x and this ordinate, increased and diminished successively by what the radical part, when real, of the general value of y becomes for the same substitution for x, will give 212 ANALYTICAL GEOMETRY. two ordinates; DP, DP , corresponding to the abscissa AD. Since P and P are two points whose co-ordinates, when substituted for x and ?/, will satisfy the equation, Ay*+Bxy+Gx?+, etc., =0, they are points in the line that this equation represents. By thus constructing the values of y answering to assumed values of x, we may determine any number of points in the curve. In getting the points P and P , we laid off, on a par allel to the axis of y, equal distances above and below the point C; PP is, therefore, a chord of the curve par allel to that axis, and is bisected at the point C. The solution of the general equation in respect to x, gives - 2(7" 2(7 4GP 4 OF The equation _ 20 2C" is that of a straight line, making, with the axis of #, an T) angle whose tangent is _ , and intersecting the axis 2t G -rji of JT at a distance from the origin equal to - . A G As above, it may be shown that any value of y that makes the radical part of the general value of x real, re sponds to two points of the curve, and that the straight line whose equation is x^-Zy-Z IC 2 20 bisects the chord, parallel to the axis of Jf, that joins these points. By placing the quantity under the radical sign in the value of y equal to 0, we have an equation of the second degree in respect to #, which will give two values for x. INTEKPRETATION OF EQUATIONS. 213 If these values are real the corresponding points of the curve are on the line M Q ; that is, they are the intersec tions of this line with the curve, since, for each of these values, there will be but one value of y, which, in con nection with that of x, will satisfy the general equation, and these values also satisfy the equation, 2A 2JL In like manner, placing the quantity under the radical sign in the value of x equal to 0, we shall find two values of y, to each of which there will respond a single value of x, and the points of the curve answering to these val ues of y will be the intersections of the curve with the line whose equation is *=-_V-_^ 2(7 2(7 A diameter of a curve is defined to be any straight line that bisects a system of parallel chords of the curve. From the preceding discussion we therefore conclude, 1. That if an equation of the second degree between two variables be resolved in respect to either variable, the equation that results from placing this variable equal to that part of its value which is independent of the radical sign will be the equa tion of that diameter of the curve which bisects the system of chords parallel to the axis of the variable. 2. The values of the other variable found from the equation which results from placing the guantity under the radical sign equal to zero, in connection with the corresponding values of the first variable, will be the co-ordinates of the vertices of the diameter. 3. The formulas for changing the reference of points from a system of rectangular co-ordinate axes to any other system having a different origin are x=a-\-x coQ. m-f^/ cos. n. y=b+ z sin. m-fy sin. n. 214 ANALYTICAL GEOMETKY. Substituting these values of x and y in the equation developing, and arranging the terms of the resulting equation with reference to the powers of y f and x f and their product, we find (A sin. 2 n+B sin. n cos. n+ C cos. 2 ft) y n +(A sin. 2 m+ B sin. m cos. m-\- C cos 2 m) x +\%A sin. m sin. n+B (sin. m cos. n -fsin ft cos. m) +2(7 cos. m cos. ft] +l(2Ab+Ba+D) =0 (1) COS. ft]?/ +[2^16+^a+D) sin. cos. m]x +Ab z +Bab+ Ca?+Db+Ea+F. Since we have four arbitrary quantities, a, 6, m, and ft entering this equation we may cause them to satisfy any four reasonable conditions. Let us see if, by means of them, it be possible to reduce the coefficient of the first powers, and of the product of the variables, separately to zero. "We should then have f 2A sin. m sin. n+B (sin. m cos. ft-fsin. ftl __Q j cos. m) +2 C cos. m cos. ft. j sin. n+(2Ca+Bb+E) cos. ft=0 (3) sin. m+(2Ca+Bb+E) cos. m=0 (4) These equations may be put under the form 2J. tan. m tan. n+B (tan. m+tan. ft)+2<7=0 (20 (2^6+^a+D) tan. n+SGH-^+JE^O ( 3/ ) (2Ab+Ba+D) tan. m+2(7a+J56+^=0 W Now, since it is necessary that m and ft should differ in value, it is evident that, in order to satisfy eqs. (3 ) and (4 ), we must have 2Ab+Ba+D=0 (5) And 2Ca+Bb+E=Q (6) INTEKFKETATION OF EQUATIONS. 215 Whence a J^AK-BD W-A.AG And b These values of a and b are infinite when B 2 41(7=0, and it will then be impossible to satisfy both eqs. (3 ) and (4 ), and consequently to destroy the co-efficients of the first powers of the two variables in eq. (1) ; we shall, for the present, assume that B 2 41 (7 is either greater or less than zero. By transposition and division eqs. (5) and (6) become , B D o = a 21 21 f) IT 1 J3 i Mi ~~~2C 2C the first of which, if a and b be regarded as variables, is the equation of the diameter that bisects the chords of the curve which are parallel to the axis of ?/, and the sec ond, that of the diameter which bisects the chords which are parallel to the axis of X. The values of a and b, given above, are, therefore, the co-ordinates of the inter section of these diameters. Since eq. (2 ) contains both of the undetermined quan tities, m and ft, we are at liberty to assume the value of either, and the equation will then give the value of the other. Let us take for the new "axis of X the diameter whose equation is 2A 2A T> then tan. ra= This value of tan. m substituted in 2A eq. (2 ) gives 2A(B B) tan. n=5 2 41(7, Or l tan. 216 ANALYTICAL GEOMETRY. That is, the new axis of y is at right angles to the primitive axis of X. The values of a, 6, and tan. n which we have thus found, in connection with the assumed value of tan. m, will reduce the co-efficients of the first powers and of the product of the variables in -eq. (1) to zero. To find what the co-efficients of y 2 and x f2 become, we must first get the values of the sines and cosines of the angles m and n from the values of tan. m and tan. n. T> Since tan. m= , and 71=90 we have 2JL sin. m=db cos. m=qc sin. n=~L cos. n=Q. The sign is written before the value of sin. w, and the sign rp before that of cos. m, because if the essential sign of tan. m is minus, which will be the case when A and B have the same sign, sin. m and cos. m must have opposite signs ; but if the essential sign of tan. m is plus, then A and B have opposite signs, and sin. m and cos. m must have like signs. Making these substitutions in eq. (1) it will become, whether the signs of A and B are like or unlike, Ay 2 A \ x 2 = (Ab 2 +Bab+ Ca 2 +Db+Ea > 4^1 ~"J-JL> / +F. (! ) E"ow, since the first term of the general equation may always be supposed positive, the two terms in the first member of equation (I/) will have like signs when B 2 4JLC<0, and unlike signs when B 2 4J.(7>0. In the first case the form of the equation is that of the equation of the ellipse, and in the second, the form is that of the equation of the hyperbola, referred in either case, to the center and conjugate diameters. INTERPRETATI N OF EQUATIONS. 217 Hence, when the transformation by which eq. (I/) was derived from the general equation is possible, we conclude that the latter equation will represent either the ellipse, or hyperbola, according as 4. Let us now examine the case in which B 2 4-40=0. Since, under this hypothesis, the co-efficients of the first powers of both variables in eq. (1) cannot be de stroyed, we will see if it be possible to destroy the abso lute term of the equation, and the co-efficients of the product of the variables, the second power of one varia ble and the first power of the other variable. Then the equations to be satisfied are f 2J.sin.wsin.7i-f.B(sin.mcos.tt-fsin.ncos.w) \ -f-2(7cos.mcos.n A&\n. 2 m-\-B sin.m cos.m-f <7cos. 2 m=0. (8) (2Ab+Ba+D)aiu.n+(2Ga+Bb+I!)coB.n==Q. (3) when it is required that the co-efficients of x * and y f should reduce to zero in connection with the absolute term and the co-officient of x y , in eq. (1). To reduce the co-efficients of y fz and x to zero, instead of those of x 2 and y , it would be necessary to replace eqs. (8) and (3) by A sin. 2 ft-f B sin.n cos.n+ (7cos. 2 n=0. (9) Equations (2) and (8) may be written Atan. 2 m+B tan.m-f (7=0. (8 ) From eq. (8 ) we find B 1 |~~ B 18 218 ANALYTICAL GEOMETKY. and this value of tan. m substituted in eq. (2 ) gives or tan. n=. 73 That is, when tan. m is equal to , eq. (2 ) and, therefore, eq. (2), will be satisfied independently of the angle n. Equation (7), being what the general equation becomes when a and b take the place of x and y respectively, shows that the new origin of co-ordinates must be on the curve. Solving this equation with reference to 6, and introducing the condition B 2 4J.(7=0, we find Now, because the imposed conditions require that the transformed equation shall be of the form it follows that every value of x* must give two numeri cally equal values of y ; hence, the new axis of Y must be parallel to the system of chords bisected by the new axis of X. That is, n must be equal to 90, and, conse quently, sin.ft=l, cos. 7i=0. Equation (3) will therefore become Whence b= _ a _ , and the radical part of the 2A 2A value of b will disappear, or we shall have 2(_RD From which we get = These values of a and b place the new origin at the vertex of the diameter whose equation is y -- ^-^, 2A 2A INTERPRETATION OF EQUATIONS. 219 and make the new axis of Y a tangent line to the curve at the vertex of this diameter. The values of a, 6, m and n which we have now found, substituted in eq. (1), will reduce it to Or JO. Denoting the co-efficient of x f by 2p , this last equa tion becomes y 3 =2p z , (10) which is of the form of the equation of the parabola re ferred to a tangent line and the diameter passing through the point of contact. The transformation by which eq. (10) was derived from the general equation is always possible when JS 2 4AC =0, unless we also have BD %AE=Q. If we suppose that both of these conditions are satisfied, the general value of y, which is reduces to whence and which are the equations of two parallel straight lines. Under the suppositions just made, the general equa tion may be written under the form which may be satisfied by making, first one, then the other factor of the first member, equal to zero. Each of 220 ANALYTICAL GEOMETRY. the equations thus obtained, being of the first degree in respect to x and y, will represent a right line. If the further condition, D 2 4J.J7 r <0, be imposed, the right lines will have no existence, and the general equa tion can be satisfied by no real values of x and y. The value of 2p , the parameter of the diameter which becomes the new axis of JT, will be found by substituting in the expression the values of a, b and cos. m. These values ajre cos. m= To reduce eq. (1) to the form z 2 =2/y (11) we must satisfy equations (7), (2), (9) and (4). T> From eq. (9) we find tan. n -, and this value of 2iA tan. n substituted in eq. (2 r ) gives tan. w=-^> resu ^s which might have been anticipated, since eqs. (3) and (4) are the same, except that m in the former takes the place of n in the latter. Because eq. (11) will give two numerically equal val ues of x for every value of y f , the new axis of JTmust be parallel to the system of chords bisected by the new axis of Y; hence m=0, sin. w=0, cos. w=l, and equa tion (4) therefore reduces to -r> ~ffl Whence a= __ b _ - 2(7 2(7 Solving eq. (7) with reference to a we have INTERPRETATION OF EQUATIONS. 221 T) ~fjl -I a= By comparing this value of a with that which precedes we find Whence b= _E*-4CF These values of a and 6 place the new origin at the vertex of the diameter whose equation is **-. 20 2(7 ,-?!-! The transformation by which eq. (4) is derived from eq. (1) will be impossible when b is infinite ; that is when It may be easily proved that when W 4 A (7=0, the condition BD 2AE=Q necessarily includes the condi tion BE 2(7.D=0 ; that is, when we cannot transform eq. (1) into eq. (10), it will also be impossible to trans form it into eq. (11). For BD 2AJE=Q And ^4^(7=0 gives j 2(7 E Whence ""= > or 5. "We have now established the following criteria for the interpretation of any equation of the second degree between two variables, viz : For the ellipse, B 2 4AC<0. For the hyperbola, B 2 4AC>0. For the parabola, B 2 4J.G=0. It remains for us to indicate the construction of any of these curves from its equation, and in doing this, we 19* 222 ANALYTICAL GEOMETRY. shall follow the order in which the conditions are given above. First, B* 4AC<0, the ellipse. 6. Let us resume the" formulas. a JlAEBD B*-4AC 2 A A ( B2 ^ AC \r 2=(A b *+Bab+ Ca and suppose, for a particular case, J3=0, and A=C. E i D "We shall then have a= ;r-p b HI And y f2 +x f2 That is, the general equation, under the suppositions made, represents a circle having a= r-j> ^ = ir-4^ or ^ e &-A- Z-A co-ordinates of its center, and I ^AU f OT ^ s ra , \ 4A 2 dius. Draw AX^ A Y for the primitive co-ordinate axes, lay off AB 77T T\ , AD= - , and through the points B and D draw the parallels jB(7and DC to the axes. Their intersection, (7, is the center of the circle, and the circumference de scribed with CE= l-Lr+H A F ag a ra ^i ug w in "be N 4A 2 that represented by the given equation. The general equation gives INTERPRETATION OF EQUATIONS. 223 B D 1 Placing the quantity under the radical sign, in this value of y, equal to zero, we have and denoting the roots of this equation by x f and z", the value of y may be written Now x and x" are the abscissas of the vertices of the diameter whose equation is 2A 2A The corresponding values of y are f== _Bx +D Bx"+D 2A Substituting these values of # , x" and # , y" in the for mula we have x x \B 2 -\-4A 2 ^ or ^ ne l en gth of the diameter. The diameter which is conjugate to this is that which is parallel to the axis of y. We find the ordinates of its x f +x" vertices by substituting a= - for x in eq. (q), which 2 then becomes B(x +x"} Z> ,x y= ~^A~ "lA-lA-^ -^ Tb-oL &-c\- rt-n. ^^ Denoting these two values of y by y ^ y^ their differ ence, which is the length of the conjugate diameter, is x x" 224 ANALYTICAL GEOMETRY. To find the angle that the con jugate diameters make with each other, let VV r be the first diameter and QQ the second. The angle that W makes with the axis of X is equal to V VR, and its cosine B D E VE vv> x s and the [_QCV f =t^Q t_BVV f =$Q+the \_V VE. "When the roots of eq. (p) are equal, the vertices of the first diameter, and also those of its conjugate, coincide, and the ellipse reduces to a point. Equation (q) may then be put under the form Because JB 2 &AC is negative, this value of y will be imaginary for every value of x except the particular one, x=x r , which causes the radical to disappear. When the roots of eq. (p) are real and unequal, that one of the factors (x x \ (x x") under the radical in eq. (q), which corresponds to the root which is algebraically the greater, will be negative, while the other will be pos itive, for all values of x included between the limits of the smaller and greater roots. The quantity under the radical, being then composed of the product of three factors, two of which are negative and one positive, will itself be positive and the corresponding values of y will therefore be real. All values of x which exceed the greater, and, also, all values of x which are less than the smaller, of these roots, will render the quantity under the radical negative and the corresponding values of ?/ imaginary. The roots x and x" are therefore the limits within which we would INTERPRETATION OF EQUATIONS. 225 select values of x to substitute in the equation to get the co-ordinates of points of the curve. When the roots of eq. (p) are imaginary, the product of the factors (x x \ (x x"] under the radical in eq. (q) will remain positive for all real values of x; and because the other factor is IP 4:A (7<0, the radical will always be imaginary : that is, no real value of x which will give a real value for y. There is, then, in this case, no point in the plane of the co-ordinate axes whose co-ordinates will satisfy eq. (q), and, consequently, the equation from which it was derived, and the curve, has no existence, or it is imaginary. By the solution of eq. (p) it will be found that when the expression is positive, the roots of the equation are real and unequal ; when the expression is zero the roots are real and equal, and when negative the roots are imaginary. If we solve the general equation with reference to x instead of y, and place the quantity under the radical sign equal to zero, we shall find that when the expression (BEZ CD) 2 ( 2 4AC) (E 2 CF) is positive, the roots of the resulting equation are real and unequal ; when zero, these roots are real and equal, and when negative they are imaginary. It might be inferred that if these roots are real and unequal, equal, or imaginary when the general equation is resolved with reference to one variable, they would be like characterized when it is resolved with reference to the other. To prove this, we develope the first of the above expressions and find that it becomes 4J. (A(fi)*+ C(D} 2 +F(B) 2 BDE-4A OF.) The development of the second is 226 ANALYTICAL GEOMETRY. + C(D} z +F(BfBDE-4ACF.\ The only difference in these developments is that the coefficient of the parenthesis in the first is 4A, and in the second it is 4(7; but when jB 2 4J.(7<0, A and C must have the same sign, hence these expressions must be posi tive, negative, or zero at the same time. Second, B 2 4JL(7>0, the hyperbola. 7. "We will begin by supposing B=Q, and A C. The formulas for a, b and tan. m will then give Tjl J~\ a , b= _ . tan.ra=0, 2A* 2A and eq, (! ) will become 4J. This is the equation of an equilateral hyperbola whose semi-axis is the square root of the numerical value of the expression . Since tan. m=0, m=0, and one of the axes of the hyperbola is parallel and the other perpendicular to the primitive axis of X. If the sign of iF - is negative, the transverse is the parallel axis ; if negative, it is the perpendicular axis. To construct the curve, let AX and A Fbe the primitive co-ordinate axes. Lay off the positive abscissa jji = = n. ) and the negative ordinate ZuoL = ; the parallels to the axes 2A drawn through D and E will be the axes of the hyper bola, and C will be its center. On these axes, lay off from the center, the distances CV, CV> CR, CR , each INTERPRETATION OF EQUATIONS. 227 equal to \ PIP 4 AF and we have ^ e ^ eQ of con _ ~ ~~ jugate equilateral hyperbolas. The foci may be found by describing a circumference with C as a center and CH, the hypothenuse of the isosceles right-angled triangle CVH, as a radius ; the circumference will intersect the axes at the foci. For another case, let us suppose -4=0 and C=0 ; then T) the value -which was assumed for tan. m becomes 2J. infinite, or the new axis of X is perpendicular to the primitive axis of X, and since tan. n is also infinite, the new co-ordinates axes would coincide ; in other words, with this value of tan. m, it would be impossible, under the hypothesis, to transform the original equation into eq. (V}. But if J.=0, and (7=0, the co-efficient of x y in eq. (1) becomes -B(sin. m cos. 71+ sin. n cos. m). Placing this equal to zero, and dividing through by B cos. m cos. 7i, we have tan. m-f tan. 7i=0, Or tan. m= tan. n. Since we are at liberty to select a value for either m or 7i, let us make 71=45 ; then m= 45. The values of a and 6, which will destroy the co-efficients of x r and y f D , E a=-_,6=-_. Substituting these values in eq. (1), reducing and trans posing, we have which is also the equation of the equilateral hyperbola, D E the co-ordinates of whose center are a= ^, 6 -- -> JD JL> 228 ANALYTICAL GEOMETRY. and whose semi-axis is the square root of the numerical value of -i - -- L The asymptotes of this hyperbola O/ 7~) TT 7-? 77^ are parallel to the primitive axes, and if 3 _ --- is negative, the transverse axis makes a negative angle with the primitive axis of JT, if positive, it makes a positive angle with that axis. There is another case in which the transformation by which eq. (V) was obtained, cannot be made with the 73 value for tan m. It is that in which A becomes zero, 2-<4_ and C does not. We then assume for tan. m the tangent of the angle that the diameter whose equation is B _E ~~W y 20 makes with the axis of X. That is, we make 2 G tan. m= 75 Proceeding with this as with the value , we shall 2A find for the transformed equation By making A=Q, this equation becomes which is that of an hyperbola referred to a system of conjugate diameters, one of which bisects the chords which are parallel to the primitive axis of X. In the general case the course to be pursued for the hyperbola differs so little from that already indicated for the ellipse, that it is unnecessary to dwell upon it at length. INTERPRETATION OF EQUATIONS. 229 The quantity under the radical in the general value of y placed equal to zero gives the equation > The roots of this equation are the abscissas of the ver tices of the diameter, whose equation is y=-^x-. 2A 2A When these roots are real and unequal, the diameter terminates in the hyperbola; when imaginary, it termi nates in the conjugate hyperbola. Denoting these abscissas, when real, by x f and x", and the corresponding ordinates by y and y, we have Bx +D y 1 - f- 2A Bx"+D 2A By placing these values of a/, x" and y , y" in the for mula we shall have the length of the diameter, and the angle included between it and its conjugate will be found pre cisely as in the ellipse. If x f be the smaller and x" the greater abscissa, then all values of x between x and x" will give imaginary values for ?/, and will answer to no points of the curve ; but all values of x less than x 1 , and also all values of x greater than x" will give real values for # , and such values of x with the corresponding values of y will be the co-ordi nates of points of the hyperbola. When the roots x , x" are imaginary, the diameter whose equation is 2A 2A 20 230 ANALYTICAL GEOMETRY. terminates in the hyperbola which is conjugated to that represented by the given equation, and the diameter which is conjugate to this diameter will terminate in the given hyperbola. The conjugate diameter may be found in the case of both the ellipse and hyperbola by making first ?/ =0 in eq. (I ), and taking the square root of the corresponding numerical value of x 2 , and then =(), and taking the square root of the corresponding numerical value of y n . 8. In the transformation of co-ordinates by which the original equation was changed into eq. (1) had the condi tion, that the new co-ordinate axes should be rectangular, been imposed, as it might, we would have had n m=90, n= 90 +m. Sin. 71= cos. m, cos. n sin. m. These values being substituted in eq. (2) will give 2A sin. m cos. m B sin. 2 m-fJ3cos. 2 m 2(7sin.mcos.m=0, which, by dividing through by cos. 2 m, and denoting sin m by *, becomes COS. ill Whence ^ Since the product of these two values of t is equal to 1, they are the tangents of the angles that two straight lines at right angles to each other make with the axis of X. Now, if eqs. (5) and (6) are satisfied at the same time ; that is, if the new origin be placed at the point of which the co-ordinates are the values of t just found will be the tangents of the angles that the axes of the ellipse, or hyperbola, as the case may be, make with the primitive axis of X. De noting these tangents by t r and t", we shall have INTERPRETATION OF EQUATIONS. 231 yb=t (xa), for the equations of the axes, and by combining the equations of the axes with the original equation, we may find the co-ordinates of their vertices, and, consequently, their length. 9. When the roots x f and x" become equal, the value of y may be written JBx+D, x x y= _ __ __ Mo2 4 A n 2A 2A >p For the hyperbola, HP 4JL(7>0, and these values of y are real. We therefore have These equations represent two right lines, and, since the co-efficients of x 9 when the second members are ar ranged with reference to it, are different, these lines will intersect. We see that by making x=x , the two equa tions will give the same value for y. Hence, x=x , and y= are the co-ordinates of the intersection of 2A the lines. The line BE, whose equation is y=- *^, 2A 2A still has the property of bisecting all lines drawn parallel to the axis of Y", which are limited by the lines BC and BD, whose equations are eqs. (r) and (s). Third, B 2 4L4C==0, the parabola. 10. The equation of the diameter that bisects the chords of the curve which are parallel to the axis of Y is 232 ANALYTICAL GEOMETRY. and that of the diameter which bisects the chords paral lel to the axis of JTis 2C E y= --^- ff Since a tangent line drawn through the vertex of a di ameter is parallel to the chords that the diameter bisects, it follows that the diameters represented by the above equations are perpendicular to each other, and, therefore, (Prop. 5, Chap. 4), their intersection, in the case of the parabola, is on ftie directrix. The abscissa of the vertex of the first diameter is the value of x given by the equation , 2(J3D ZAE}x+D* 4^=0, the first member of which is the quantity under the radical in the general value of ?/, after we have made HP 4./i(7=0. Denoting this abscissa by x we have If we denote the co-ordinates of the vertex of the sec ond diameter by x" and y, we have ,, = ^ 2 4 CF 2C Let Pand P f be the two vertices thus found. Through the first draw PT parallel to the axis of F, and through the second, P f T parallel to the axis of X. These lines will be tangent to the parabola at P and P r respectively, INTERPRETATION OF EQUATIONS. 233 and their intersection, T, will be a point of the directrix. The lines CM, BN, drawn through P and P ; , making, with the axis of Jfj angles having for their common tangent _B_ JLC_ 2A B* \ are diameters of the curve, and BC drawn through T perpendicular to these diameters, is the directrix. With P as a center and PC as a radius, or with P as a center and P B as a radius, describe an arc of a circle. This arc will cut the chord PP f at the focus F. The perpendicular FD, drawn through F to the directrix, is the axis, and the middle point, V, of FD, is the vertex of the parabola. EXAMPLES. It will aid in the construction of the curve represented by any equation to find the points in which it is inter sected by the co-ordinate axes. If we make either vari able equal to zero in the equation, the values of the other variable given by the resulting equation will be the dis tances from the origin to the intersections of the curve, with axis of the latter variable. When the roots of the equation which we solve are real and unequal, there will be two intersections, where real and equal, the axis will be tangent to the curve at the point thus determined, and when imaginary, the curve and the axis will have no common points. 1. Construct the curve represented by the equation Whence Here J 20* _ 2x(x 2). , JB=2, (7=3; therefore B 2 4AC<0, and 234 ANALYTICAL GEOMETRY. the curve is an ellipse which passes through the origin of co-ordinates, since the equation has no absolute term. y=x is the equation of a diameter of the curve and the co-ordinates of its ver tices are x =0, y f = and x"= 2, /= 2. By making x=~L in the original equa tion, we find y=+. 41-f, or 2.41 for the ordinates of the vertices of the diameter conjugate to the first. The length of the first diameter is equal to ^8=2.82+, and the length of the second is +.41+2.41=2.82. 2. Determine the curve that corresponds to the equation y*+2xy+x 2 6y-f9=0. Here^=l, .=2, (7=1, hence B 2 4^(7=0, and the curve is a parabola. We find And z= yy 9. The diameter whose equation is y= +3 has x =0, and ?/ =3 for the co-ordinates of its vertex. The axis of y is therefore tangent to the curve. The co-ordinates of the vertex of the diameter whose equation is x= y are, x"= 1J, and y=l|, and a line drawn through this point parallel to the axis of X will be tangent to the curve. Let P be the vertex of the first diameter and P that of the second. The chord PP passes through the focus. P S , PS making with the axis of X, on the negative side? angles of 45 are diameters of the curve, and B T a perpendicular to P$is the directrix. X INTERPRETATION OF EQUATIONS. 235 3. Determine the curve of which the equation is In this case A=l, 5=2, C= 2 ; hence W 4AC>0, and the curve is an hyperbola. The equation gives The abscissas of the vertices of the diameter whose equation is 2/= x+2 are the roots of the equation Whence x f = 1, and x"=2, and the corresponding val ues of y are y =S and ?/"=0. The diameter which is parallel to the axis of y is conjugate to PP\ and terminates in the conjugate hy perbola. The co-ordinates of its vertices are imaginary and may be found by making x=-| in the original equation. We would thus find 2 The conjugate diameter will therefore be about 5.2. The point E in which the curve intersects the axis of X is on the left of the origin and at a distance from it equal to 2J units. 4. Determine the curve represented by the equation In this, the condition B 2 4AC=Q is satisfied, and the curve is the parabola ; but it answers to the case in which the parabola reduces to two parallel lines. In fact the equation may be put under the form Whence Or 2/4-3z=5 or 3. 236 ANALYTICAL GEOMETRY. The first member of the equation may therefore be re solved into the factors y+Sx 5, and y+3x+&, which, placed separately equal to zero, give for the parallel lines the equations And y= 3x 3. 5. Determine the curve of which the equation is In this we have -B 2 4 A (7<0, and the curve is an ellipse, but it answers to the case in which the curve becomes imaginary. For, resolving the equation in relation to y, we find (x 2) 2 . The quantity under the radical in this value of y will be negative for every real value of cc, hence, al] values of y are imaginary ; that is, there is no point whose co-ordi nates will satisfy the given equation. By inspection we may also discover that the first mem ber of the equation can be placed under the form ( 2/ _2z-l) 2 +(r-2) 2 , which is the sum of two squares, and must therefore re main positive for all real values of x and y. 6. What kind of a curve corresponds to the equation Ans. It is an hyperbola. The axis of Y is midway be tween the two branches. One branch of the curve cuts the axis of X at the point 1 ; the other branch cuts the same axis at the point -f 3. 7. Determine the curve represented by the equation Eesolving, we find (yx) 2 +(x 1) 2 +3=0. INTERSECTION OF LINES. 237 The condition for the ellipse is satisfied, but the curve is imaginary. 8. What kind of a curve corresponds to the equation Ans. It is a parabola passing through the origin and ex tending without limit, in the direction of x and y negative. 9. What kind of a curve corresponds to the equation Ans. It is a parabola, cutting the axis of X at the dis tance of 1 and -f 1 from the origin, and extending in definitely in the direction of plus x and plus y. 10. What kind of a curve corresponds to the equation Ans. It is a straight line passing through the origin, making an angle of 26 34 with the axis of Y. 11. What kind of a curve corresponds to the equation Ans. It is an ellipse limited by parallels to the axis of Y drawn through the points 1, and +1, on the axis of X. CHAPTER VII. ON THE INTERSECTIONS OF LINES AND THE GEOME TRICAL SOLUTION OF EQUATIONS. "We have seen that the equation of a straight line is ytx-\-c^ And that the general equation of a circle is (xa)*+(yb)*=E 2 . The first is a simple, the second a quadratic equation, 238 ANALYTICAL GEOMETRY. and if the value of x derived from the first be substituted in the second, we shall have a resulting equation of the second degree, in which y cannot correspond to every point in the straight line, nor to every point in the cir cumference of the circle, but it will correspond to the two points in which the straight line cuts the circumference, and to those points only. And if the straight line should not cut the circumfer ence, the values of y in the resulting equation must neces sarily become imaginary. All this has been shown in the application of the polar equation of the circle, in Chap. 2. Let us now extend this principle still further. The equation of the parabola is an equation of the second degree, and the equation of a circle is also an equation of the second degree. But when two equations of the second degree are combined, they will produce an equation of the fourth degree. But this resulting equation of the fourth degree can not correspond to all points in the parabola, nor to all points in the circumference of the circle, but it must cor respond equally to both ; hence, it will correspond to the points of intersection, and if the two curves do not in tersect, the combination of their equations will produce an equation whose roots are imaginary. Let us take the equation y 2 =2px, and take p for the unit of measure, (that is, the distance from the directrix to the focus is unity,) then z=^_, and this value of x 2t substituted in the equation of the circle, will give INTERSECTION OF LINES. 239 Let the vertex of the parabola Y be the origin of rectangular co ordinates. Take AP=x, and let it refer to either the parabola or the circle, and let PM=y, AF=^ AH=a, JHC=b, and CM=E. ISTow in the right angle triangle */ CMD, we have and corresponding to this particular figure, we shall have in lieu of the preceding equation "Whence ^/ 4 +(4 4% 2 8%=4(^ 2 a 2 b\) (F) This equation is of the fourth degree, hence it must have four roots, and this corresponds with the figure, for the circle cuts the parabola in four points, M, M f , M", and M" r . The second term of the equation is wanting, that is, the co-efficient to y* is 0, and hence it follows from the theory of equations, that the sum of the four roots must be zero. The sum of two of them, which are above the axis of A X, (the two plus roots,) must be equal to the sum of the two minus roots corresponding to the points M" and M ". The values of a and b and R may be such as to place the center C in such a position that the circumference can cut the parabola in only two points, and then the result ing equation will be such as to give two real and two imaginary roots. Indeed, a circumference referred to the same unit of measure and to the same co-ordinates, might not cut the 240 ANALYTICAL GEOMETRY. parabola at all, and in that case the resulting equation would have only imaginary roots. In case the circle touches the parabola, the equation will have two equal roots. Now it is plain that if we can construct a figure that will truly represent any equation in this form, that figure will be a solution to the equation. For instance, a figure correctly drawn will show the magnitude of PM, one of the roots of the equation. We will illustrate by the following EXAMPLES. 1. Find the roots of the equation y_-11.14?/ 2 6.74^+9.9225=0. This equation is the same in form as our theoretical equation (F), and therefore we can solve it geometrically as follows : Draw rectangular co-ordinates, as in the figure, and take AF=^ and construct the parabola. To find the center of the circle and the radius, we put 4 4a= 11.14, (1) 86= 6.74, (2-) and 4(^ 2 a 2 6 2 )= 9.9225. (3) From eq. (1), a=3.78. From eq. (2), 6=0.84. And these values of a and 6, substituted in eq. (3), give .=3.34, nearly. Take from the scale which cor- y responds to AF=, ^LZT=a=3.78, HC= 0. 84, and from C as a center, with a radius equal to 3.34, des cribe the circumference cutting the parabola in the four points, M, M f , C M", and M ". The distance of M ^ from the axis of X is +3.5, of M it is +0.7, of M" it is 1.5, and of M " it is 2.7, and these are the four roots of the equation. H ! INTERSECTION OF LINES. 241 Their sum is 0, as it ought to be, because the equation contains no third power of y. 2. Find the roots of the equation y*+y*+fy 2 +l2y 72=0. This equation contains the third power of y ; therefore this geometrical solution will not apply until that term is removed. But we can remove that term by putting (See theory of transforming equations in algebra). This value of y substituted in the equation, it becomes and this equation is in the proper form. Nowput 4 4a=5f, 86=9 J, and 4(jft 2 a 2 6 2 )=74jf |. Whence a= Jf, 6= |f, and -R=4.485. These values of a and b designate the point C f for the center of the circle. From this center, with a radius =4.485, we strike the circumference, cutting the parabola in the two points m and m . The point m is 2J units above the axis A X, and the point m f is 2f units from the same line, and these are the two roots of the equation. The other two roots are imaginary, shown by the fact that this circumference can cut the parabola in two points only. If we conceive the circumference of a circle to pass through the vertex of the parabola A, then will a?+b*=It 2 , and this supposition reduces the general equation (F) to Here y=0 will satisfy the equation, and this is as it should be, for the circumference actually touches the par abola on the axis of X. Now divide this last equation by this value of y, and we have 21 242 ANALYTICAL GEOMETKY. Here is an equation of the third degree, referring to a parabola and a circle ; the circumference cutting the par abola at its vertex for one point, and if it cuts the par abola in any other point, that other point will designate another root in equation (G). It is possible for a circle to touch one side of the par abola within, and cut at the vertex A and at some other point. Therefore it is possible for an equation in the form of eq. (G) to have three real roots, and two of them equal. The circumferences of most circles, however, can cut the parabola in A and in one other point, showing one real root and two imaginary roots. Equation (G) can be used to effect a mechanical solu tion of all numerical equations of the third degree, in that form.* "We will illustrate this by one or two EXAMPLES. 1. Given y 3 -f 4y=39, to find the value of j by construc tion. (See fig. following page) Put 4 4a=4, and 86=39. Whence a=0, and &=4f. These values of a and b designate the point C on the axis of Y for the center of the circle, CA=4-J, the radius. The circle again cuts the parabola in P, and PQ mea sures three units, the only real root of the equation. 2. Given y 3 75y=250, to find the values of y by con struction. When the co-efficients are large, a large figure is re quired ; but to avoid this inconvenience, we reduce the co-efficients, as shown in Chap. 2. * Observe that the second term, or 7/ 2 , in a regular cubic is wanting. Hence, if any example contains that term, it must be removed before a geometrical solution can be given. INTERSECTION OF LINES. 243 Thus put y=nz. Then the equation becomes n B z 3 75ftz= Now take n=5, then we have z*3z=2. In this last equation the co-effi cients are sufficiently small to apply to a construction. Put 4 4a= 3, and 86=2. Whence a== ^%, and 6=J. These values of a and b designate the point D for the center of the circle. D A is the radius. The circle cuts the parabola in t, and touches it in T 7 , showing that one root of the equation is +2, and two others each equal to 1. But y=nz. That is, #=5x2, or 5, 5. Or the roots of the original equation are +10, 5, 5. When an equation contains the second power of the unknown quantity, it must be removed by transforma tion before this method of solution can be applied. 3. Given y 3 48y=128 to find the values of y by con struction. Ans. +8, 4, 4. 4. Given y 3 13y= 12, to find the values of y by con struction. Ans. +1, +3, and 4. Conversely we can describe a parobola, and take any point, as H, at pleasure, and with HA as a radius, de scribe a circle and find the equation to which it belongs. This circle cuts the parabola in the points m, n and o, indicating an equation whose roots are +1, -f 2.4, and 3.4. We may also find the particular equation from the general equation 244 ANALYTICAL GEOMETEY. observing the locality of H, which corresponds to a=3-3 and 6= 1, and taking these values of a and b, we have f 9.2?/= 8, for the equation sought. EEMAEKS ON THE INTEEPEETATION OF EQUATIONS. In every science it is important to take an occasional retrospective view of first principles, and the conviction that none demand this more imperatively than geometry will excuse us for reconsidering the following truths so often in substance, if not in words, called to mind before. An equation, geometrically considered, whatever may be its degree, is but the equation of a point, and can only designate a point. Thus, the equation y=ax-\-b designates a point, which point is found by measuring any assumed value which may be given to x from the origin of co-ordinates on the axis of X, and from that extremity measuring a distance represented by (ax-\-b) on a line parallel to the axis of Y. The extremity of the last measure is the point designated by the equation. If we assume another value for x, and measure again in the same way, we shall find the point which now corresponds to the value of x. Again, as sume another value for x, and find the designated point. Lastly, if we connect these several points, we shall find them all in the same right line, and in this sense the equa tion of the first degree, y=ax+b, is the general equation of a right line, but the right line is found by finding points in the line and connecting them. In like manner the equation of the second degree only designates a point when we assume any value for x, (not inconsistent with the existence of the equation), and take the plus sign. It will also designate another point INTERPRETATION OF EQUATIONS. when we take the minus sign. Taking another value of x, and thus finding two other points, we shall have four points, still another value of x and we can find two other points, and so on, we might find any number of points. Lastly, on comparing these points we shall find that they are all in the circumference of the same circle, and hence we say that the preceding equation is the equation of a circle. Yet it can designate only one, or at most, two points at a time. If we assume different values for y, and find the cor responding values of x, the result will be the same circle, because the x and y mutually depend upon each other. Now let us take the last practical example y z 13y= 12, and, for the sake of perspicuity, change y into x, then we shall have x 3 13x4-12=0. Now we can suppose ?/=0 to be another equation ; then will y=tf 13x+12 (A) be an independent equation between two variables, and of the third degree. The particular hypothesis that y=0, gives three values to x, (+1, +3, and 4), that is, three points are designated: the first at the distance of one unit to the right of the axis of Y; the second at the distance of three units on the same side of the axis of Y; and the third point four units on the opposite side of the same axis, and this is all the equation can show until we make another hypothesis. Again, let us assume ?/=5, then equation (A) becomes 5=x 3 13x4-12, or x 3 13x4-7=0, and this is, in effect, changing the origin five units on the axis of Y. A solution of this last equation fixes three other points on a line parallel to the axis of X. Again, let us assume J/=10, then equation (A) becomes x 3 13x4-2=0, 21* 246 ANALYTICAL GEOMETRY. 7/=25. =1.1 y=20. =0.40 y=I5. =0.20 </=10. =+0.14 and a solution of this equation gives three other points. And thus we may proceed, assigning different values to y, and deducing the corresponding values of , as ap pears in the following table, commencing at the origin of the co-ordinates, where y=0, and varying each way. ?/=30.0388 =2.2814 +4.1628 2.0814 +4.03 + 3.80 + 3.70 +3.52 I r\ fr er __[__ o o "When ?/=0. then will =+1. #=5 =+1.66 y= 6.0388 =+2.0814 Taking ?/=0, a solution of the equation ?/= 3 13x+12, gives the three points a, a, a, on the axis of X. Then taking y=5, and a solution gives three points 6, b, b, on a line X parallel to the axis of JT, and at the distance of 5 units above said axis. Again, taking ?/=10, and another solution gives the three points c, c, c. Now joining the three points (a, b, c,) (a, b, c), and (a, b, c), we shall have apparently three curves corresponding to the equation of the third degree, and thus, we might hastily conclude that every equation of the third degree would give three curves, and every equa tion of the fourth degree four curves, etc., etc., but this is not true. If we continue finding points as before, we shall find that the three curves (a, b, c,) (a, b, c,) and (a, b, c,) are but different portions of the same curve, and w r e can now venture to draw this general conclusion : That in an equation involving y, the ordinate, to the first power, INTERPRETATION OF EQUATIONS. 247 and the abscissa, x, to the third power, the axis of X, or lines parallel to that axis, may cut the curve in three points. From analogy, we also infer that if we have an equa tion involving x to the fourth power, the axis of X, or its parallels, Will cut the curve in four points ; and if we have an equation involving x to the fifth power, that axis or its parallels will cut the curve in five points, and so on. In the equation under consideration, (y=x* 13x+12), if we assume y greater than 30.0388, or less than 6.0388, we shall find that two values of x in each case will be come imaginary, and on each side of these limits the parallels to JTwill cut the curve only in one point. Two points vanish at a time, and this corresponds with the truth demonstrated in algebra, " that imaginary roots enter equations in pairs." The points m, m, the turning points in the curve, are called maximum points, and can be found only by approx imation, using the ordinary processes of computation, but the peculiar operation of the calculus gives these points at once. To find the points in the curve we might have assumed different values of x in succession, and deduced the cor responding values of y, but this would have given but one point for each assumption ; and to define the curve with sufficient accuracy, many assumptions must be made with very small variations to x. , We solved the equa tions approximately and with great rapidity by means of the circle and parabola as previously shown. We conclude this subject by the following example : Let the equation of a curve be (a 2 x 2 )(x 6) 2 =ry, from which we are required to give a geometrical deline ation of the curve. From the equation we have 248 ANALYTICAL GEOMETRY. The following figure represents the curve which will be recognized as corresponding to the equation, after a little explanation. If =0, then y becomes infinite, and therefore the ordinate at A is an asymptote to the curve. If AB=b, and P be taken between A and _B, then FM and Pm will be equal, and lie on different sides of the abscissa AP. If x=b, then the two values of y vanish, because x 6=0; and consequently, the curve passes through B, and has there a duplex point. If AP be taken greater than AB, then there will be two values of y, as before, having contrary signs, that value which was positive before, now becomes negative, and the nega tive value becomes positive. But if AD be taken =a, and P come to D, then the two values of y vanish, because +/ a z X vQ f And if A P is taken greater than AD, then a 2 x 2 becomes negative, and the value of y impossible ; and therefore, the curve does not extend beyond D. If x now be supposed negative, we shall find y= zb^a 2 "-^? x (b +x)~-x? If x vanish, both these values of y become infinite, and consequently, the curve has two infinite arcs on each side of the asymptote AK. If x increase, it is plain y dimin ishes, and if x becomes = a,y vanishes, and consequently the curve passes through E, if AE be taken = AD, on the opposite side. If x be supposed, numerically, greater than a, then y becomes impossible ; and no part of the curve can be found beyond E. This curve is the conchoid of the ancients. STRAIGHT LINES IN SPACE. 249 CHAPTER VUL STRAIGHT LIKES DT SPACE. Straight lines in one and the same plane are referred to two co-ordinate axes in that plane, but straight lines in space require three co-ordinate axes, made by the inter section of three planes. To take the most simple view of the subject, conceive a horizontal plane cut by a meridian plane, and by a per pendicular east and west plane. The common point of intersection we shall call the origin or zero point, and we might conceive this point to be the center of a sphere, and about it will be eight quad rangular spaces corresponding to the eight quadrants of a sphere, which extended, would comprise all space. The horizontal east and west line of intersection of these planes, we shall call the axis of X. The horizon tal intersection in the direction of the . meridian, the axis of T; and that perpendicular to it in the plane of the meridian, the axis of Z. Distances estimated from the zero point horizontally to the right, as we look towards the north, we shall designate as plus, to the left minus. Distances measured on the axis of Y and parallel thereto, towards us from the zero point, we shall call plus; those in the opposite direction will therefore be minus. Perpendicular distances from the horizontal plane up wards are taken as plus, downward minus. The horizontal plane is called the plane of xy, the me ridian plane is designated as the plane of yz, and the per pendicular east and west plane the plane of xz. Now let it be observed that x will be plus or minus, ac cording to its direction from the plane of yz, y will be plus or minus, according to its direction from the plane 250 ANALYTICAL GEOMETRY. xZy and z will be plus or minus, according as it is above or below the horizontal place xy. PROPOSITION I. To jmd the equation of a straight line in space. Conceive a straight line passing in any direction through space, and conceive a plane coinciding with it, and per pendicular to the plane xz. The intersection of this plane with the plane xz, will form a line on the plane xz, and this is said to be the projection of the line on the plane xz, and the equation of this projected line will be in the form x=az-\-7r. (Chap. 1, Prop. 1.) Conceive another plane coinciding with the proposed line, and perpendicular to the plane yz, its intersection with the plane yz is said to be the projection of the line on the plane yx, and the equation of this projected line is in the form These two equations taken together are said to be equations of the line, because the first equation is a gen eral equation for all lines that can be drawn in the first projecting plane, and the second equation is a general equation for all lines that can be drawn in the second projecting plane ; therefore taken together, they ex press the intersection of the two planes, which is the line itself. For illustration, we give the following example : Construct the line whose equations are =32 2 STRAIGHT LINES IN SPACE. 251 Make 2=0, then rc=l, and ?/= 2. Now take J.P=1, and draw Pm parallel to the axis of Y, making Pm= 2 ; then m is the point in the plane xy, through which the / T7~ X line must pass. /- IL. Now take z equal to any num ber at pleasure, say 1, then we shall have z=3 and y=l. Take J.P^=3, P m =-f 1, and from the point m r in the plane xy erect m r n perpendicular to the plane xy> and make it equal to 1, because .we took 2=1, then n is an other point in the line. Draw n m and produce it, and it will be the line designated by the equations. PROPOSITION II. To find the equation of a straight line lohich shall pass through a given point. Let the co-ordinates of the given point be represented by # , j/ , z f . The equations sought must satisfy the general equa tions x=az+x. The equations corresponding to the given point are x =az +K. y =bz f +fi. Subtracting eq. (1) from these, respectively, we have x r x=a(z f 2), and y f y=b(z f z\ the equations required. PROPOSITION III. To find the equations of a straight line which shall pass through two given points. 252 ANALYTICAL GEOMETRY. Let the co-ordinates of the second point be x", y", z", Now by the second proposition, the equations which ex press the condition that the line passes through the two points, will be x X =a(z" z ) 9 Alid yy> = b(z"Z f }. "Whence a-^JZ^, b= y "~~ y -. 2"_2/ Z Z > Substituting the values of a and 6 in the equations of a line passing through a single point (Prop. 2,) we have for the equations required. PROPOSITION IV. To find the condition under ivhich two straight lines intersect in space, and the co-ordinates of the point of intersection. Let the equation of the lines be x=a z+n . y=b f z+p f . If the two lines intersect, the co-ordinates of the com- mojo. point, which may be denoted by x 9 y, z, will satisfy all of these four equations, therefore by subtraction, we have Whence, by eliminating 2, we find 7T-- 7r = /9 p which is the condition under which two lines intersect. Now 2= 7r ^ and this value of z being substituted a a in the first equations, we obtain an +~ctn , b8 b> and y ^ a a STRAIGHT LINES IN SPACE. 253 for the value of the co-ordinates of the point of inter section. Cor. If a~a , the denominators in the second mem ber will become 0, making x and y infinite ; that is, the point of intersection is at an infinite distance from the origin, and the lines are therefore parallel. PROPOSITION V. PROBLEM. To express analytically the distance of a given point from the origin. Let P be the given point in space ; it is in the perpendicular at the point N 9 which is in the plane xy. The angle AMN=9Q. Also, the angle JJVP=90. Let AMXj MN=y, NP=z. Then JJ? 2 =, But T~P 2 "Now if we designate AP by r, we shall have r 2 =x 2 -f;?/ 2 +2 2 for the expression required. PROPOSITION YI. PROBLEM To express analytically the length of a line in space. Let PP =D be the line in question. z Let the co-ordinates of the point P be x, ?/, z 9 and of the point P f be x 9 y , z . Now MM =x x=NQ. QN =y y. 22 254 ANALYTICAL GEOMETRY. In the triangle PEP f we have Or &=(x f x)*+(y f y^+y zf, (1) which is the expression required. SCHOLIUM. If through one extremity of the line, as P, we draw PA to the origin, and from the other extremity P , we draw P $ parallel and equal to PA, and draw AS, it will be parallel to PP f , and equal to it, and this virtually reduces this proposition to the previous one. This also may be drawn from the equation, for if A is one extremity of the line, its co-ordinates x } y t and z are each equal to zero, and PROPOSITION VII. PROBLEM. To find the inclination of any line in space to the three axes. From the origin draw a line z ** parallel to the given line ; then the inclination of this line to the axes will be the same as that of the given line. The equations for the line pass- ing from the origin are x=az, and y=bz. (1) Let X represent the inclination of this line with the axis of x, Y its inclination with the axis of y, and Z its inclination with the axis of z. The three points P, N 7 M, are in a plane which is par allel to the plane zy, and A M is a perpendicular between" the two planes. AMP is a right-angled triangle, the right angle being at M. Let APr and A M=x. Then, by trigonometry, we have As r : sin. 90 : : x : cos. X. Whence x=r cos. X. Also, as r : sin. 90 : : y : cos. Y. Whence y=r cos. F. STRAIGHT LINES IN SPACE. 255 Also, as r : sin. 90 : : z : cos. Z. Whence zr cos. Z. From Prop. 5 we have r * = x 2+y2+ Z *. (2) Substituting the values of x, y, and z, as above, we have r 2=r 2 cos. 2 JT"-f r 2 cos. 2 Y+r 2 coa. 2 Z. Dividing by r 2 will give cos.sJT-fcos. 2 r+cos. 2 =l, (3) an equation which is easily called to mind, and one that is useful in the higher mathematics. If in eq. (2) we substitute the values of x 2 and y 2 taken from eq. (1), we shall have But we have three other values of r 2 as follows : r 2 = x * , r 2 = ^ 2 _ and r 2 = _ ?__. cos. 2 X cos. 2 Y cos. 2 Z "Whence ~^^ cos. A And _ = v 1+a8+ 5 2 . (7) COS. ^ In eq. (5) put the value of x drawn from eq. (1), and in eq. (6) the value of y from eq. (1), and reduce, and we shall obtain The analytical expressions for the inclination of a line in space to the three co-or dinates. COS. ^= The double sign shows two angles supplemental to each other, the plus sign corresponds to the acute angle, and the minus sign to the obtuse angle. 256 ANALYTICAL GEOMETRY. PKOPOSITION VIII. To find the inclination of two lines in terms of their sepa rate inclinations to the axes. Through the origin draw two lines respectively paral lel to the given lines. An expression for the cosine of the angle between these two lines is the quantity sought. Let AP be parallel to one of the given lines, and AQ parallel to the other. The angle PA Q is the angle sought. Let the equations of one of these lines be x=az, y=bz, and of the other x*=a z 9 y =b z . Let AP=r, AQ=r f , PQ=D, and the angle PAQ= V. Now in plane trigonometry (Prop. 8, p. 260, Geom.,) we have 2rr From Prop. 6 we have Expanding this, it becomes 2 x x2y y2z z. But by Prop. 5 we have Q " P and x f *+y f2 +z 2 =r /2 . Whence 2x x+2y y+2z z=r*+r s D*. This equation applied to eq. (I) reduces it to cos. V rr But r and r r may have any values taken at pleasure; their lengths will have no effect on the angle V. There fore, for convenience, we take each of them equal to unity. Whence cos. V=*x r x+y y+z f z. (2) STKAIGHT LINES IN SPACE. 257 But in Prop. 7 we found that x=rGos.^ y=r cos. Y", etc., and that x r r cos.^T , y f =r j cos. Y , etc. ; and since we have taken r=l and r =l, x=cos. X, etc., and x = cos.JT , etc. Hence cos. 7=cos.Xcos. J^ +cos. Fcos. Y +cos.^cos.Z . (3) But by Prop. 7 we have CL cos. JT= . r=. and cos. X= Substituting these values in eq. (3) we have l+ao +W cos. y=- for the expression required. The cos. V will be plus or minus, according as we take the signs of the radicals in the denominator alike or un like. The plus sign corresponds to an acute angle, the minus sign to its supplement. Cor. 1. If we make 7=90, then cos. 7=0, and the equation becomes which is the equation of condition to make two lines at right angles in space. Cor. 2. -If we make 7=0, the two straight lines will become parallel, and the equation will become 1= Squaring, clearing of fractions, and reducing, we shall find (a! _a)+ (fi/_ &)2+ (aV a b?= 0. Each term being a square, will be positive, and there fore the equation can only be satisfied by making each term separately equal to 0. "Whence a =a, b =6, and ab f =a f b. The third condition is in consequence of the first two. 22* E 258 ANALYTICAL GEOMETRY. CHAPTER IX. ON THE EQUATION OF A PLANE. An equation which can represent any point in a line is said to be the equation of the line. Similarly, an equation which can represent or indicate any point in a plane, is, in the language of analytical ge ometry, the equation of the plane. PROPOSITION I. To find the equation of a plane. Let us suppose that we have a plane which cuts the axes of JT, Y and Z at the points JB, C and D, respec tively ; then, if these points be connected by the straight lines BCj CD and DJ3, it is evi dent that these lines are the inter sections of the plane with the planes of the co-ordinate axes. Now a plane may be conceived as a surface generated by moving a straight line in such a manner that in all its positions it shall be parallel to its first position and intersect another fixed straight line. Thus the line DC, so moving that in the several positions, D C", D"C", etc., it remains parallel to DO and constantly intersects DjB, will generate the plane determined by the points D, C and B. The line DB being in the plane xy, its equations are 2/=0, z=mx+b, (1) and for the line DC we have z=0, z=ny+b. (2) The plane passed through the line U C parallel to the EQUATION OF A PLANE. 259 plane zy, cuts the axis of JTat the point p. Denoting Ap by Cj the equations of the line D f C f become x=c, z=ny+b . (3) It is obvious that eqs. (3) can be made to represent the moving line in all its positions by giving suitable values to c and b f , and that, for any one of its positions, the co ordinates of its intersection with the line DB must satisfy both eqs. (1) and (3). That is, c and b , in the first and second of eqs. (3), must be the same as x and , respec tively, in the second of eqs. (1). Hence b =z ny, and b =mx+b. Equating these two values of b , we have z ny=mx+b, or z=mx+ny+b. (4) This equation expresses the relation between the co-or dinates x, y and z for any point whatever in the plane generated by the motion of the line DC, and is, there fore the equation of this plane. Cor. 1. Every equation of the first degree between three variables, by transposition and division, may be re duced to the form of eq. (4), and will, therefore, be the equation of a plane. Cor. 2. In eq. (4), m is the tangent of the angle which the intersection of the plane w T ith the plane xz makes with the axis of X, n the tangent of the angle that the inter section with the plane yz makes with the axis of Y, and b the distance from the origin to the point in which the plane cuts the axis of Z. Hence, if any equation of the first degree between three vari ables be solved with respect to one of the variables, the co-effi cient of either of the other variables denotes the tangent of the angle that the intersection of the plane represented by the equa- tionjjoith the plane of the axes of the first and second variables, makes with the axis of the second variable. 260 ANALYTICAL GEOMETKY. SCHOLIUM. If we assume ro= =-:. =- C C and substitute these values in eq. (4), it will become, by reduction and transposition, which is the form under which the equation of the plane is very often presented. From this equation we deduce the following general truths : First. If we suppose a plane to pass through the origin of the co-ordinates for this point, #=0, y=0, and 2= 0, and these values substituted in the equation of the piano will give D=Q also. There fore, when a plane passes through the origin of co-ordinates, the general equation for the plane reduces to Ax+y+Cz=Q. /Second. To find the points in which the plane cuts the axes, we reason thus : The equation of the plane must respond to each and every point in the plane ; the point P, therefore, in which the plane cuts the axis of X, must correspond to y=0 and 2=0, and these values, substituted in the equation, reduces it to Or *=!: A For the point Q we must take x=Q and =0. And y = ~L.~OQf. For the point R, z= _=OJ2. Third. If we suppose the plane to be perpendicular to the plane XY, PR , its intersection with, or trace on, the plane XZ, must be drawn parallel to OZ, and the plane will meet the axis of Z at the distance infinity. That is, OR, or its equal, ( - ) , must be infi nite, which requires that (7=0, which reduces the general equation of the plane to EQUATION OF A PLANE. 261 Ax-\- By -\-D-Qj which is the equation of the trace or line PQ on the plane XY. If the plane were perpendicular to the plane ZX, the line Q, or its equal, ( -- j, must be infinite, which requires that .#=0, and this reduces the general equation to which is the equation for the trace PR, and hence we may conclude in general terms, That when a plane is perpendicular to any one of the co-ordinate planes, its equation is that of its trace on the same plane. PKOPOSTION II. PEOBLEM. To find the length of a perpendicular drawn from the origin to a plane, and to find its inclination with the three co-ordinate axes. Let JRPQ be the plane, and from the origin, 0, draw Op perpendicular to the plane ; this line will be at right-angles to every line drawn in the* plane from the pointy. Whence Op =90, Op^=90, and QpP=90. Let Op p. Designate the angle pOP by X, pOQ by Y 9 and by^. By the preceding scholium we learn that =-, 06=- and OR=- A, 13, C and D being the constants in the equation of a plane. Now, in the right-angled triangle OpP 9 we have OP : 1 : : Op : cos. X. That is, -. :l::p: cos. Z. (1) A 262 ANALYTICAL GEOMETRY. The right-angled triangle OpQ gives :?:!::: cos. Y. B The right-angled triangle OpR gives _2>. C" Proportion (1) gives us P. :l::pi cos. r. (2) _B : 1: :2>: coa.Z. 0) 5 s (6) (2) gives cos. 2 Y=^B and (3) gives cos. 2 ^==^ (7 : Adding these three equations, and observing that the sum of the first members is unity, (Prop. 7, Chap. 8), and we have (5) Whence y?= P CO This value of p placed in eqs. (4), (5) and (6), by re duction, will give cos. JT= =b A __ (8) COS. F=dr __ . (9) cos. Z= . (10) Expressions (7), (8), (9) and (10) are those sought. PROPOSITION III. PROBLEM. To find the analytical expressions for the inclination of a plane to the three co-ordinate planes respectively. EQUATION OF A PLANE. 263 Let Ax+Ey+ Cz+D=0 be the equa tion of the plane, and let PQ represent its line of intersection with the co-ordi nate plane (xy}. From the origin, 0, draw OS per pendicular to the trace PQ. Draw pS. OpS is a right-angled triangle, right- angled at p, and the angle OSp measures the inclination of the plane with the horizontal plane (xy]. Our object is to find the angle OSp. In the right-angled triangle POQ we have found D *** JD * OP=- D Whence Now PS, a segment of the hypothenuse made by the perpendicular OS, is a third proportional to PQ and PO. Therefore : B : : -r : PS= A Or The other segment, QS, is a third proportional to PQ and OQ. Therefore D . D D . AB Or AD But the perpendicular, OS, is a mean proportional be tween these two segments. Therefore we have OS= !N"ow, by simple permutation, we may conclude that the perpendicular from the origin to the trace PR is 264 ANALYTICAL GEOMETRY. D and that to the trace QR is D "We shall designate the angle which the plane makes with the plane of (xy] by (xy)< and the angle it makes with (xz) by (xz\ and that with (yz} by (yz). !N"ow the triangle OpS gives OS : sin. 90 : : Op : sin. OSp. D D That is, , : 1 : : Whence But by trigonometry we know that cos. 2 =l sin. 2 . A 2 4- 7? 2 r7 2 mence 003.^)=!---=, eta Whence c C 2 _u A cos.(yz) C 2 > Expressions sought. C 2 ^ Squaring, and adding the last three equations, we find cos. 2 (r?/) + cos. 2 (#2) -f cos. 2 ( yz)=l. That is, the sum of the squares of the cosines of the three angles which a plane forms with the three co-ordinate planes, is equal to radius square, or unity. EQUATION OF A PLANE. 265 PKOPOSITION IV. PROBLEM. To find the equation of the intersection of two planes. Let Ax+By+Cz+D=Q, (1) A x+B y+ C 2+jD =0, (2) be the equations of the two planes. If the two planes intersect, the values of , y and z will be the same for any point in the line of intersection. Hence, we may combine the equations for that line. Multiply eq. (1) by C f and eq. (2) by (7, and subtract the products, and we shall have (A C A 1 C}x+ (B O B f C)y+ (D C D <?)= 0, for the equation of the line of intersection on the plane (xy). If we eliminate y in a similar manner, we shall have the equation of the line of intersection on the plane (xz) ; and eliminating x will give us the equation of the line of intersection on the plane (yz). PKOPOSITION Y. PROBLEM. To find the equation to a perpendicular let fall from a given point (x , y , z ,) upon a given plane. As the perpendicular is to pass through a given point, its equations must be of the form xx =a(zz ) 9 (1) yy f =b(zz \ (2> in which a and b are to be determined. The equation of the plane is The line and the plane being perpendicular to each other, by hypothesis, the projection of the line and the trace of the plane on any one of the co-ordinate planes will be perpendicular to each other. For the traces of the given plane on the planes (xz) and (yz\ we have Ax+ Cz+D=Q and By+ Cz+D=0. 23 266 ANALYTICAL GEOMETKY. From the former x z _. ( 3 ) A A From the latter y= ~z :?. () JD JD ISTow eqs. (1) and (3) represent lines which are at right angles with each other. Also, eqs. (2) and (4) represent lines at right angles with each other. But when two lines are at right angles, (Prop. 5, Chap. 1), and a and a 1 are their trigonometrical tangents, we must have (aa -f-l=0). That is, <+l=0, or a=^. ^JL O T> Like reasoning gives us b= , and these values put in C eqs. (1) and (2) give xx =~(zz ) | for ,; y-y>=* (z - z .) PROPOSITION VI. PROBLEM. To find the angle included by two planes given by their equations. Let Ax+By+Cz+D=0 9 (1) And A x+3y +C z+iy=0 9 (2) be the equations of the planes. Conceive lines drawn from the origin perpendicular to each of the planes. Then it is obvious that the angle contained between these two lines is the supplement of the inclination of the planes. But an angle and its supple ment have numerically the same trigonometrical ex pression. EQUATION OF A PLANE. 267 Designate the angle between the two planes by V, then Proposition 8, in the last chapter gives Jr I+aa +bb COS. V= - - . (3) The equations of the two perpendicular lines from the origin must be in the form x=az, y=bz, xa z y=b z. But because the first line is perpendicular to the first plane, we must have a=4, and 6=:?, (Prop. 5.) o u And to make the second line perpendicular to the sec ond plane requires that - and V -. ^These values of a, 6, and a , V, substituted in eq. (3) will give, by reduction, cos. T^+ for the equation required. Cor. When two planes are at right angles, cos. F=0, which will make AA +BB + PROPOSITION VII. PROBLEM. To find the inclination of a line to a plane. Let MN be the plane given by its equation and let PQ be the line given by its equations 268 ANALYTICAL GEOMETRY. x=az+a. 1VP Take any point P in the given line, and let fall PR, the perpendicular, up on the plane ; RQ is its projection on the plane, and PQR, which we will denote by F, is obviously the least an- gle included between the line and the plane, and it is the angle sought. Let xa f z+7r y and y=b f z+f?, be the equation of the perpendicular PR, and because it is perpendicular to the plane, we must have (by the last proposition) a =^, and & =:* Because PQ and PR are two lines in space, if we des ignate the angle included by F, we shall have cos. F= . ( p rop< 8? But the cos. F is the same as the sin. PQR, or sin. v, as the two angles are complements of each other. Making this change, and substituting the values of a and & , we have Bn., for the required result. Cor. When #=0, sin. #=0, and this hypothesis gives for the equation expressing the condition that the given line is parallel to the given plane. "We now conclude this branch of our subject with a few practical examples, by which a student can test his knowledge of the two preceding chapters. EQUATION OF A PLANE. 269 EXAMPLES. 1. What is the distance between two points in space of which the co-ordinates are 3=3, y=5, *= 2, = 2, /= 1, * =6. .An*. 11.180+. 2. (y w/McA the co-ordinates are x=I, y= 5, z=3, 3 =4, y =-4, * =!. Ans. 5^3 nearly. 3. TAe equations of the projections of a straight line on the co-ordinate planes (xz), (yz), are 3=2*+l, 2/=Jz 2, required the equation of projection on the plane (xy). Ans. y\x 2J. 4. 7%e equations of the projections of a line on the co-ordi nate planes (xy) and (yz) are 2yx 5 and 2y=z 4, required the equation of the projection on the plane (xz). Ans. xz-\-1. 5. Required the equations of the three projections of a straight line which passes through two points whose co-ordinates are z/=2, # =1, 2 =0, and z"= 3, /=0, z"= 1. What are the projections on the planes (xz) and (yz) ? Ans. x=5z+2, y=z-\-\. And from these equations we Und the projection on the plane (xy\ that is, 5?/=-f 3. (See Prop. 3, Chap. 8.) 6. Required the angle included between two lines whose equations are I of the 1st, and x = z + 2 I of the 2d. j y=z+I j Ans. F= (See Prop. 8, Chap. 8.) 23* 270 ANALYTICAL GEOMETRY. 7. Find the angles made by the lines designated in the pre ceding example, with the co-ordinate axes (See Prop. 7, Chap. 8.) ( 36 42 with JT, f 5444 with -J, Ans. The 1st line 1 57 41 20" F,2d ? line^ 125 16 F, (74 29 54" ^, (54 44 Z. 8. Having given the equation of two straight lines in space, as f the j and ;?/= to find the value of /3 , so that the lines shall actually intersect, and to find the co-ordinates of the point of intersection. j x== < (See Prop. 4, Chap. 8.) 9. Given the equation of a plane 8x Zy+z 4=0, to find the points in which it cuts the three axes, and the perpen dicular distance from the origin to the plane. (Prop. 2.) Ans. It cuts the axis of X at the distance of J from the origin ; the axis of Y at 1 J ; and the axis of Z at +4. The origin is .4649+ of unity below the plane. 10. Find the equations for the intersections of the two planes (Prop. 4.) r-1-0, (On the plane (xy) 17z 10y+9=0. ( On the plane (xz) 13x 102+23=0. 11. Find the inclination of these two planes. (Prop. 6.) Ans. 41 27 41". EQUATION OF A PLANE. 271 12. The equations of a line in space are x=2z+l, and y=32+2. find the inclination of this Line to the plane represented by the equation (Prop. 7.) Sx 3y+z 4=0. Ans. 48 13 13" 13. Find the angles made by the plane whose equation is Sx Zy+z 4=0, with the co-ordinate planes. (Prop. 3.) f 83 19 27" with (xy\ Ans.{ 110 24 38" with (xz). ( 21 34 5" with (yz). 14. The equation of a plane being Required the equation of a parallel plane whose perpendicu lar distance is (a) from the given plane. Ans. Because the planes are to be parallel, their equa tions must have the same co-efficients, J., B, and C. In Prop. 2, we learn that the perpendicular distance of the origin from the given plane may be represented by Now, as the planes are to be a distance a asunder, the distance of the origin from the required plane must be or "Whence the equation required is Ax+By+ 15. Find the equation of the plane which will cut the axis of Z at 3, the axis of 5 at 4, and the axis of Y at 5. Ans. 272 ANALYTICAL GEOMETRY. 16. Find the equation of the plane which will cut the axis of X at 3, the axis of Z at 5, and which will pass at the perpendicular distance 2 from the origin. At what distance from the origin will this plane cut the axis of Y ? Ans. The equation of the plane is 10z+^89?/+62: 30=0. 30 The plane cuts the axis of Y at . 17. Find the equations of the intersection of the two planes whose equations are 3x 2j/ z 4= 0, The equation of the projection of the inter section on the plane (xy) is 10z+?/ 6=0. A ry\ Q 23# z 16=0, and that on the plane (yz) is 23^+10^+22=0. 18. Find the inclination of the planes whose equations are expressed in example 17. Ans. 7=60 50 55" or 119 9 5". 19. A plane intersects the co-ordinate plane (xz) at an in clination of 50, and the co-ordinate plane (yz) at an inclina tion of 84. At what angle will this plane intersect the plane Ans. F=4038 6". MISCELLANEOUS PROBLEMS. 273 MISCELLANEOUS PROBLEMS. 1. The greatest diameter or major axis of an ellipse is 40 feet, and a line drawn from the center making an an gle of 36 with the major axis and terminating in the el lipse is 18 feet long ; required the minor axis of this el lipse, its area and excentricity. NOTE. The excentricity of an ellipse is the distance of either focus from the center, when the semi major axis is taken as unity. ( The minor axis is 30.8752. AnsA Area of the ellipse, 969.972 sq. feet. ^ Excentricity .63575. 2. If equilateral triangles be described as the three sides of any plane triangle and the centers of these equilateral triangles be joined, the triangle so formed will be equilat eral ; required the proof. Let ABC represent any plane triangle, A, B and C denoting the angles, and a, b and c the respect ive sides, the side a being opposite the angle A, and so on. On A C, or 6, suppose an equilat eral triangle to be drawn, and let P be its center. Make the same suppositions in regard to the sides c and a, finding P l and P 2 . Draw PP l , P, P 2 and PP 2 ; then is PP l P 2 an equilateral triangle, as is to be proved. We shall assume the principle, which may be easily demonstrated, that a line drawn from the center of any equi lateral triangle to the vertex of either of the angles, is equal to times the side of the triangle. Hence we have - \/3 \/3 V3 x/3 Also, the angles PJ.<7=30, P lV l_B=30 , ANALYTICAL GEOMETRY. and so on. Now it is obvious that the angle PAP i is expressed by (J.-f 60), the angle P^P 2 by (5+60), and PCP 2 by ( (7+60). We must now show that the analyt ical expressions for PP l and P 1 P 2 are the same. In an alytical trigonometry it was found that the cosine of an an gle, A, of a plane triangle would be given by the equation cos. A= Whence, a 2 =6 2 +c 2 2bc cos. A. That is, The square of one side is equal to the sum of the squares of the other two sides, minus twice the rectangle of the other two sides into the cosine of the opposite angle. Applying this to the triangle PAP l we have Z +3 3" cos _ , c 2 , a 2 2ac Also, PjP,, =2+3 -- 3~ cos - (-#+60) (2) _ 2 a b 2 2ab And PP/=3+g -- g- cos. ((7+60) (3) By trigonometry, cos. (J.+60)=cos. A cos. 60 sin. A sin. 60. But cos. 60=}, and sin. 60=^3 -v/3 Whence, cos. (J.+60)=J cos. A -- - sin. A 2 This value substituted in eq. (1) that equation becomes _ 2 b 2 c 2 be be PPi K= 3~+3 3 cos - b 2 +c 2 a 2 be But cos. A= QA~ . Whence -o-cos. A= This value of ~ cos. A placed in eq. (4), gives _ 2 26 2 ,2c 2 b 2 c 2 ,a 2 ,bc . pp -TT-+ - - - + - + sm. A r * 6 6 6 6 6 v Or, MISCELLANEOUS PROBLEMS. 275 By a like operation equation (2) becomes But by the original triangle A B C we have sin. J._sin. B a ^- g > or sin - A== b sin. B Placing this value of sin. A in equation (5) that equa tion becomes ._aH-y+? + *c_ Bin> B (7) 1 6 v/8 We now observe that the second members of (6) and (7) are equal ; therefore, PP l =P 1 P 2 And in like manner we can prove PPj=PP 2 . There fore the triangle PP t P 2 has been shown to be equilateral. PKOBLEM. G-iven, the excentricity of an Ellipse, to find the difference between the mean and true place of the planet, corres ponding to each degree of the mean angle, reckoned from the major axis; the planet describing equal sectors or areas in equal times, about one of the foci, the center of the attractive force. Let AB be the major axis of an ellipse, of which CB CA=A\ is the semi-transverse axis, and also let C be the common center of the ellipse and of the circle of which CB is the radius. Then FC=e, and F is the focus of the ellipse. Suppose the planet to be at _B, the apogee point of the orbit, (so called in Astronomy). Also, conceive another planet, or material point, to be at _B, at the same time. Now, the planet revolves along the ellipse, describing equal areas in equal times, and the hy pothetical planet revolves along the circle BPQ, describ- 276 ANALYTICAL GEOMETRY. ing, in equal times, equal areas and equal angles about the center C. It is obvious that the two bodies will arrive at A in the same time. The other halves of the orbits will also be described in the same time, and the two bodies will be to gether again at the point B. But at no other points save at A and at B (the apogee and perigee points) will these two bodies be in the same line as seen from F, and the difference of the directions of the two bodies as seen from the focus F is the equation of the center. For instance, suppose the planet to start from B and describe the ellipse as far as p. It has then described the area BFp of the ellipse, about the focus F. In the same time the fictious planet in the circle has mov ed along the circumference BPto Q, describing the sector BCQ about the center C. Now the areas of these two sectors must be to each other as the area of the ellipse is to the area of the circle. That is, sector BFp : sector BCQ : : area Ell. : area Cir. Through p draw PD at right angles to J.J5, and repre sent the arc of the circle BP by x. Then (7-D=cos. x, and PD=sin. x. Draw Op and CP. But, denoting the semi-conjugate axis by B, we have area DpB : area DPB : : area Ell. : area Cir. : : B :A :: pD : PD Also we have ACpD : ACPD : : pD : PD Hence, area DpB : ACpD : : area DPB : ACPD Therefore, area DpB+A.CpD : area DPB+ACPD : : B : A or, sector CpB : sector CPB : : B: A : : area Ell. : area Cir. Hence it follows that sector FpB : sector CpB : : sector CQB : sector CPB "Whence sector FpB sect. CpB : sect. CQB sect. CPB : :B:A MISCELLANEOUS PROBLEMS. 277 or, &FpC: sector QCP : : B : A : : area Ell. : area. Cir. But the area of tlie ellipse is xAB and the area of the circle is A 2 x. But A=l and B=^\ c 2 . The area of the triangle FCp is \e (pD\ and the area of the sector is Jy, representing the arc QP by y. Whence JE(pD) iyii ^l^e^: 1. (1) But we have PD : pD : : A : B : : 1 : 1 e 2 , and PD=sin. x. Hence, sin. x : pD : : 1 : ^1 e 2 ; pD=smx^l e 2 This value of pD placed in (1) that proportion becomes e sin. x^l e 2 : y : : ^1 e 2 : 1 Or, e sin. x : y : : 1 : 1. y=e sin. x. ( 2 ) DEFINITIONS. 1st. The angle x, in astronomy, is called the excentric anomaly. 2d. The angle QCB, or (x+y) is called the mean anomaly. 3d. The angle p FB is called the true anomaly. 4th. The difference between QCB or nCB (of the tri angle FnC) and nFC (which is the angle n of the trian gle CFn) is the equation of the center. The angle QCB, the mean anomaly, is an angle at the center of the ellipse, which is equal to the sum of the an gles at n and F; that is, n taken from the angle at the center will give the true angle at the focus, F. "We will designate the angle pFB by t. Now, by the polar equation of an ellipse, we have Again, by the triangle FDp, we find, But _ ~FD 2 =(e+coa. x) 2 =e 2 +2e cos. z-f cos. 2 z And pJD =sin. 2 x (1 e 2 )=sin. 2 x e 2 sin. 2 x Therefore, FD 2 +pD 2 =e 2 +2e cos. x+le 2 sm. 2 x But e 2 sin. 2 xe, 2 2.4 278 ANALYTICAL GEOMETBY. Substituting this value of & sin. 2 - in the preceding expression we have cos. "Whence Fp=</ FD 2 +pD z =l+e cos. x. Equaling these two values of Fp and we obtain 1 e 2 =(l-j-c cos. x) (1 e cos. t) e-f-cos. x Whence cos. ^- Here we have a value of t in terms of x and e, but the equation is not adapted to the use of logarithms. By equation (27) Plane Trigonometry, we have 1 cos. t If the value of cos. t from equation (3) be placed in this we shall have e+cos. x .. _ 1+e cos. x i_|_g C os. x e cos. x 1-fecos.a; Or e+coa.x l+ecos.z+e-t os.z (l+e)+(l+6) cos. a; (l+c) (1+cos.z) That is, tan. J^= ( l ~ e ) * tan. Jx. (4) From eq. (2) we obtain .Mean Anomaly =-{- sin. x. (5) By assuming a:, equation (5) gives the Mean Anomaly. Then equation (4) gives the corresponding True Anomaly. To apply these equations to the apparent solar orbit, the value of e is .0167751 the radius of the circle being unity. But y=e sin. x, and as y is a portion of the circumfer ence to the radius unity, we must express e in some known part of the circumference, one degree, for exam ple, as the unit. Because 180 is equal to 3.14159265, therefore the value of e, in degrees, is found by the following proportion. MISCELLANEOUS PROBLEMS. 79 3.14159265 : 180 : : .0167751 : d degrees. By log., log. 0167751 2.2246652 log. 180 2.2552725 0.4799377 log. n 0.4971499 Log. e, in degrees, of arc, 1,9827878 Add log. 60 1.7781513 Log. e, in min. of arc, 1.7609391 constant log. lZ7 / 0.9832249 -1-992714 cons. log. "We are now prepared to make an application of equa tions (4) and (5) For example, we require the equation of the center for the solar orbit, corresponding to 28 of mean anom aly, reckoning from the apogee. The excentric anomaly is less than the mean by about half of the value of the equation of the center at any point; and x must be as sumed. Thus, suppose z=27 32 ; then Jz=13 46 sin. z=sin. 27 32 9.664891 Constant, 1.760939 e sin. x= 26 6518 1.425830 Add x 27 32 Mean Anomaly=27 58 39"! Tan. Jx 13 46 9.389178 Const. 1.992714 tan. J* 13 32 59" 9.381892 2 True anomaly 27 5 58" Mean Anomaly 27 58 39"! Equation of center 52 r 41"! corresponding to the mean anomaly of 27 58 39"1, not to 28 as was required. 280 ANALYTICAL GEOMETRY. Now let us take z=27 40 ; then ^=13 50 sin. x 27 40 9.666824 Con. 1.760939 e sin. x 26 777 1.427763 Add x 27 40 Mean Anomaly, 28 6 46"6 tan. }z=13 50 9.391360 Con. 1.992714 tan. \t 13 36 43" 9.384074 2 *=27 13 26" Mean anomaly 28 6 46" 6 Eq. center, 53 20 "6 corresponding to 28 6 46"6. Now, we can find the equation corresponding to 28 by the following obvious proportion : 28 6 46"6 53 20"6 28 00 00" 27 58 39 1 52 41 1 27 5 39 1 8 7"5 : 39"5 : : V 20"9 : 4"7 Add 52 41"! Equation or value sought, 52 45"! In like manner we can find the value of the equation of the center of any and every other degree of the mean anomaly in the orbit of the sun, or any other orbit, when the excentricity is known. LOGARITHMIC TABLES ALSO A TABLE OF NATURAL AND LOGARITHMIC SINES, COSINES, AND TANGENTS, TO EVERY MINUTE OF THE QUADRANT. LOGARITHMS OF NUMBERS FROM 1 TO 10000, N. Log. N. Log. N. Log. N. Log. 1 000000 26 414973 51 1 707570 76 1 880814 301030 27 431364 52 1 716003 77 1 886491 3 477121 28 447158 53 1 724276 78 1 892095 4 6020GO 29 462398 54 1 732394 79 1 897627 5 698970 30 477121 55 1 740363 80 1 903090 6 778151 31 491362 56 1 748188 81 1 908485 7 845098 32 505150 57 1 755875 82 1 913814 8 903090 33 6J8514 58 1 763428 83 1 919078 9 954243 34 531479 59 1 770852 84 1 924279 10 1 000000 35 544068 60 1 778151 85 1 929419 11 041393 36 556303 61 1 785330 86 1 934498 12 079181 37 568202 62 1 792392 87 1 939519 13 113943 38 679784 63 1 799341 88 1 944483 14 146128 39 1 591065 64 1 808180 89 1 949390 15 176091 40 1 602060 65 1 812913 90 1 954243 16 204120 41 1 612784 66 1 819544 91 1 959041 17 230449 42 1 623249 67 1 826075 92 1 963788 18 255273 43 1 633468 68 832509 93 1 968483 19 278754 44 1 643453 69 838849 94 1 973128 20 301030 45 1 653213 70 845098 95 1 977724 21 1 322219 46 662578 71 851258 96 1 982271 22 1 342423 47 672098 72 857333 97 1 986772 23 1 361728 48 681241 73 863323 98 1 991226 24 1 380211 49 690196 74 869232 99 1 995635 25 1 397940 50 698970 75 8750ol 100 2 000000 NOTE. In the following table, in the last nine columns of each page, where the first or leading figures change from 9 s to O s, points or dots are now introduced instead of the O s through the rest of the line, to catch the eye, and to indicate that from thence the corresponding natural number in the first column stands in the next lower line, and its annexed first two figures of the Logarithms in the second column. LOGARITHMS OF NUMBERS. 3 N. 1 2 3 4 5 6 7 8 9 100 000000 0434 0868 1301 1734 2166 2598 3029 3461 3891 101 4321 4750 5181 5609 6038 6466 6894 7321 7748 8174 102 8 JOO 9026 9451 9876 .300 .724 1147 1570 1993 2416 103 01-2.S37 3259 3680 4100 4521 4940 5360 5779 6197 6616 104 7033 7451 7868 8284 8700 9116 9532 9947 .361 .775 105 021 1S9 1603 2016 2428 2841 3252 3664 4075 4486 4896 103 530 J 5715 6125 6533 6942 7350 7757 8164 8571 8978 107 9384 9789 .195 .600 1004 1408 1812 2216 2619 3021 108 033424 3826 4227 4628 5029 5430 5830 6230 6629 7028 109 7426 7825 8223 8620 9017 9414 9811 .207 .602 .998 110 041393 1787 2182 2676 2969 3362 3755 4148 4540 4932 111 5323 5714 6105 6495 6885 7275 7664 8053 8442 8830 112 9218 9606 9993 .380 .766 1153 1538 1924 2309 2694 113 053078 3463 3846 4230 4613 4996 5378 5760 6142 6524 114 6905 7286 7666 8046 8426 8805 9185 9563 9942 .320 115 050898 1075 1452 1829 2206 2582 2958 3333 3709 4083 116 4458 4832 5206 5580 5953 6326 6699 7071 7443 7815 117 8186 8557 8928 9298 9668 ..38 .407 .776 1145 1514 118 071882 2250 2617 2985 3352 3718 4085 4451 4816 6182 119 5547 6912 6276 6640 7004 7368 7731 8094 8457 8819 120 9181 9543 9904 .266 .626 .987 1347 1707 2067 2426 121 082785 3144 3503 3861 4219 4576 4934 5291 5647 6004 122 6360 6716 7071 7426 7781 8136 8490 8845 9198 9552 123 9905 .258 .611 .963 1315 1667 2018 2370 2721 3071 124 093422 3772 4122 4471 4820 5169 5518 5866 6215 6562 125 6910 7257 7604 7951 8298 8644 8990 9335 9681 1026 136 100371 0715 1059 1403 1747 2091 2434 2777 3119 3462 127 3804 4146 4487 4828 5169 5510 5851 6191 6531 6871 128 7210 7549 7888 8227 8565 8903 9241 9579 9916 .253 129 110590 0926 1263 1599 1934 2270 2605 2940 3275 3609 130 3S43 4277 4611 4944 5278 6611 5943 6276 6608 6940 131 7271 7603 7934 82G5 8595 8926 9256 9586 9915 0245 132 120574 0903 1231 1560 1888 2216 2544 2871 3198 3525 133 3852 4178 4504 4830 5156 5481 5806 6131 6456 6781 134 7105 7429 7753 8076 8399 8722 9045 9368 9690 ..12 135 130334 0655 0977 1298 1619 1939 2260 2580 2900 3219 136 3539 3858 4177 4496 4814 5133 5451 5769 6086 6403 137 6721 7037 7354 7671 7987 8303 8618 8934 9249 9564 138 9879 .194 .508 .822 1136 1450 1763 2076 2389 2702 139 143015 3327 3630 3951 4263 4574 4885 6196 5507 6818 140 6128 6438 6748 7058 7367 7676 7985 8294 8603 8911 141 9219 9527 9835 .142 .449 .756 1063 1370 1676 1982 142 152288 2594 2900 S205 S510 3815 4120 4424 4728 5032 143 5336 5640 6943 G246 6549 6852 7154 7457 7759 8061 144 8362 8G64 8965 9266 9567 9868 .168 .469 .769 10G8 145 161308 1667 1967 2266 2564 2863 3161 3460 3758 4055 146 4353 4650 4947 5244 5541 5838 6134 6430 6726 7022 147 7317 7613 7908 8203 8497 8792 9086 9380 9674 9968 148 170262 0555 0848 1141 1434 1726 2019 2311 2603 2895 149 3186 3478 37G9 4060 4351 4641 4932 5222 5512 5802 18 4 LOGARITHMS N. 1 2 3 4 5 6 7 8 9 150 176091 6381 6670 6959 7248 7536 7825 8113 8401 8689 151 8977 9264 9552 9839 .126 .413 .699 .985 1272 1558 152 181844 2129 2415 2700 2985 3270 3555 3839 4123 4407 153 4691 4975 5259 5542 5825 6108 6391 6674 6956 7239 154 7521 7803 8084 8366 8647 8928 9209 9490 9771 ..51 281 155 190332 0612 0892 1171 1451 1730 2010 2289 2567 2846 156 3125 3403 3681 3959 4237 4514 4792 5069 5346 5623 157 5899 6176 6453 6729 7005 7281 7556 7832 8107 8382 158 8657 8932 9206 9481 9755 ..29 .303 .577 .850 1124 159 201397 1670 1943 2216 2488 2761 3033 3305 3577 3848 273 160 4120 4391 4663 4934 5204 5475 5746 6016 6286 6556 161 6826 7096 7365 7634 7904 8173 8441 8710 8979 9247 162 9515 9783 ..51 .319 .586 .853 1121 1388 1654 1921 163 212188 2454 2720 2986 3252 3518 3783 4049 4314 4579 164 4844 5109 5373 5638 5902 6166 6430 6694 6957 7221 264 165 7484 7747 8010 8273 8536 8798 9060 9323 9585 9846 166 220108 0370 0631 0892 1153 1414 1675 1936 2196 2456 167 2716 2976 3236 3496 3755 4015 4274 4533 4792 5051 168 6309 5568 5S26 6084 6342 6600 6858 7115 7372 7630 169 7887 8144 8400 8657 8913 9170 9426 9682 9938 .193 257 170 230449 0704 0960 1215 1470 1724 1979 2234 2488 2742 171 2996 3250 3504 3757 4011 4264 4517 4770 5023 5276 172 5528 5781 6033 6285 6537 6789 7041 7292 7544 7795 173 8046 8297 8548 8799 9049 9299 9550 9800 ..50 .300 174 240549 0799 1048 1297 1546 1795 2044 2293 2541 2790 249 175 3038 328G 3534 3782 4030 4277 4525 4772 5019 5266 176 5513 5759 6006 6252 6499 6745 6991 7237 7482 7728 177 7973 8219 8464 8709 8954 9198 9443 9687 9932 .176 178 250420 0664 0908 1151 1395 1638 1881 2125 2368 2610 179 2853 3096 3338 3580 3822 4064 4306 4548 4790 5031 242 180 5273 5514 5755 5996 6237 6477 6718 6958 7198 7439 181 7679 7918 8158 8398 8637 8877 9116 9355 9594 9833 182 260071 0310 0548 0787 1025 1263 1501 1739 1976 2214 183 2451 2688 2925 3162 3399 3636 3873 4109 4346 4582 184 4818 5054 5290 5525 5761 5996 6232 6467 6702 6937 235 185 7172 7406 7641 7875 8110 8344 8578 8812 9046 9279 186 9513 9746 9980 .213 .446 .679 .912 1144 1377 1609 187 271842 2074 2306 2538 2770 3001 3233 3464 3696 3927 188 4158 4389 4G20 4850 5081 5311 5542 5772 6002 6232 189 6462 6692 6921 7151 7380 7609 7838 8087 8296 8525 229 190 8754 8982 9211 9439 9667 9895 .123 .351 .578 .806 191 281033 1261 1488 1715 1942 2169 2396 2622 2849 3075 192 3301 3527 3753 3979 4205 4431 4656 4882 5107 5332 193 5557 5782 6007 6232 6456 6681 6905 7130 7354 7578 194 7802 8026 8249 8473 8696 8920 9143 9366 9589 9812 224 195 290035 0257 0480 0702 0925 1147 1369 1591 1813 2034 196 2256 2478 2699 2920 3141 3363 3584 3804 4025 4246 197 4466 4687 4907 5127 5347 5567 5787 6007 6226 6446 198 6665 6884 7104 7323 7542 7761 7979 8198 8416 8635 199 8853 9071 9289 9507 9725 9943 .161 .378 .595 .813 OFNUMBERS. 5 N. 1 2 3 4 5 6 7 8 9 200 301030 1247 1464 1681 1898 2114 2331 2547 2764 2980 201 3196 3412 3628 3844 4059 4275 4491 4706 4921 5136 202 5351 5566 5781 5996 6211 6425 6639 6854 7038 7282 203 7496 7710 7924 8137 8351 8564 8778 8991 9204 9417 204 9630 9843 ..56 .268 .481 .693 .906 1118 1330 1542 212 205 311754 1966 2177 2389 2600 2812 3023 3234 3445 3656 206 3867 4078 4289 4499 4710 4920 5130 5340 5551 5760 207 5970 6180 6390 6599 6809 7018 7227 7436 7646 7854 208 8083 8272 8481 8689 8898 9106 9314 9522 9730 9938 209 320146 0354 0562 0769 0977 1184 1391 1598 1805 2012 207 210 2219 2426 2633 2839 3046 3252 3458 3685 3871 4077 211 4282 4488 4694 4899 5105 5310 5516 5721 5926 6131 212 6336 6541 6745 6950 7155 7359 7563 7787 7972 8176 213 8380 8583 8787 8991 9194 9398 9u()l 9805 ...8 .211 214 330414 (hi 17 0819 1022 1225 1427 163J 1832 2034 2236 202 215 2438 2640 2842 3044 3246 3447 3649 3850 4051 4253 216 4454 4655 4S56 5057 5257 5458 5658 5859 6059 6260 217 6460 6660 6860 7060 7260 7459 7659 7858 8(158 8257 218 8456 8656 8855 9054 9253 9451 9650 9849 . .47 .246 219 340444 0642 0841 1039 1237 1436 1632 1830 2028 2225 198 2-20 2423 2620 2817 3014 3212 3409 3C06 3S02 3999 4196 221 4392 4589 4785 4981 5178 5374 5570 5766 6982 6157 222 6353 6549 6744 6939 7135 7330 7525 7720 7915 8110 223 8305 8500 8694 8889 9083 9278 9472 96(56 9860 . .54 224 350248 0442 0636 0329 1023 1216 1410 1603 1796 1989 193 225 2183 2375 2568 2761 2954 3147 3339 3532 3724 3916 226 4108 4301 4493 4685 4876 5088 5260 6452 6643 5834 227 6026 6217 6408 6599 6790 6981 7172 7363 7554 7744 228 7935 8125 8316 8506 8696 8886 9076 9266 9456 9646 229 9835 ..25 .215 .404 .593 .783 .972 1161 1350 1539 190 230 361728 1917 2105 2294 2482 2671 2859 3048 3236 3424 231 3612 3800 3988 4176 4363 4551 4739 4926 5113 5301 232 5488 5675 5862 6049 6236 6423 6610 6796 H983 7169 233 7356 7542 7729 7915 8101 8287 8473 8659 8845 9030 234 9216 9401 9587 9772 9958 .143 .328 .613 .698 .883 185 235 371068 1253 1437 1622 1806 1991 2175 2360 2544 2728 236 2912 3096 3280 3464 3647 3831 4015 4198 4382 4565 237 4748 4932 5115 5298 5481 6664 5846 6029 6212 6394 238 6577 6759 6942 7124 7306 7488 7670 7852 8034 8216 239 8398 8580 8761 8943 9124 9306 9487 9668 9849 . .30 182 240 380211 0392 0573 0754 0934 1115 1296 1476 1656 1837 241 2017 2197 2377 2557 2737 2917 3097 3277 3456 i 3636 242 3815 3995 4174 4353 4533 4712 4891 5070 5249 6428 243 5606 5785 5964 6142 6321 6499 6677 6856 7034 7212 244 7390 7568 7746 7923 8101 8279 8456 8634 8811 8989 178 245 9166 9343 9520 9898 9875 . .51 .228 .405 .582 .759 246 390935 1112 1288 1464 1641 1817 1993 2169 2345 2521 247 2697 2873 3048 3224 3400 3575 3751 3926 4101 : 4277 248 4452 4627 4802 4977 5152 5326 5501 5676 5850 6025 249 6199 6374 6548 6722 6896 7071 7245 7419 7592 ; 7766 6 LOGARITHMS N. 1 2 3 4 5 6 7 8 9 250 397940 8114 8287 8481 8634 8808 8981 9154 9328 9501 251 9674 9847 ..20 .192 .365 .538 .711 .883 1056 1228 252 401401 1573 1745 1917 2089 2261 2433 2605 2777 2949 253 3121 3292 3464 3635 3807 3978 4149 4320 4492 4663 254 4834 5005 5176 5346 5517 5688 5858 6029 6199 6370 171 255 6540 6710 6881 7051 7221 7391 7561 7731 7901 8070 256 8240 8410 8579 8749 8918 9087 9257 9426 9595 9764 257 9933 .102 .271 .440 .609 .777 .946 1114 1283 1451 258 411620 1788 1956 2124 2293 2461 2629 2796 2964 3132 259 3300 3467 3635 3803 3970 4137 4305 4472 4639 4806 260 4973 5140 5307 5474 5641 5808 5974 6141 6308 6474 261 6641 6807 6973 7139 1303 7472 7638 7804 7970 8135 262 8301 h467 8633 8798 8964 9129 9295 9460 9625 9791 263 9956 .121 .286 .451 .616 .781 .945 1110 1275 1439 264 421604 1788 1933 097 2261 2426 2590 2754 2918 3082 265 3246 3410 3574 3737 3901 4065 4228 4392 4555 4718 266 4882 5045 5208 C371 t534 5697 6860 6023 6186 6349 267 6511 6674 6836 6999 7161 7324 7486 7648 7811 7973 268 8135 8297 8459 8621 8783 8944 9106 9268 9429 9591 269 9752 9914 ..75 .236 .398 .559 .720 .881 1042 1203 270 431364 1525 1685 1846 2007 2167 2328 2488 2649 2809 271 2969 3130 3-290 3450 3610 3770 3930 4090 4-249 4409 272 4569 4729 4888 5048 5207 5367 5526 5685 5844 6004 273 6163 6322 G481 6640 6800 6957 7116 7275 7433 7592 274 7751 7909 8067 8226 8384 8642 8701 8859 9017 9175 158 275 9333 9491 9648 9806 H964 .122 .279 .437 .594 .752 276 440909 1066 1224 1381 1538 1695 1852. 2009 2166 2323 277 2480 2637 2793 2950 3108 3263 ^J419 3576 3732 3889 278 4045 4-201 ^357 4513 4669 4825 4981 5137 6293 5449 279 5604 5760 5915 6071 6226 6382 6637 6692 6848 7003 280 7158 7313 7468 7623 7778 7933 8088 8242 8397 8552 281 8708 8861 9015 9170 9324 9478 9633 9787 9941 ..95 282 450249 0403 0557 0711 0865 1018 1172 1326 1479 1633 283 1786 1940 2093 2247 2400 2553 2706 2859 3012 3165 284 3318 3471 3624 3777 3930 4082 4235 4387 4640 4692 2,-5 4845 4997 5150 5302 5454 5606 5758 5910 6062 6214 286 6366 6518 6670 6821 6973 7125 1276 7428 7579 7731 287 7882 8033 8184 8336 8487 8638 8789 8940 9091 9242 288 9392 9543 9654 9845 9995 .146 .296 .447 .597 .748 289 460898 1048 1198 1348 1499 1649 1799 1948 2098 2248 290 2398 2548 2697 2847 2997 3146 3296 3445 3594 3744 291 0893 4042 4191 4340 4490 4639 4788 4936 5085 5234 292 5383 5532 5680 5829 5977 6126 6274 6423 6571 6719 293 G868 7016 7164 7312 7460 7608 7766 7904 8052 8200 294 8347 8495 8643 8790 8938 9085 9233 9380 9527 9675 "* 147 295 9822 9969 .116 .263 .410 .557 .704 .851 .998 1145 296 471292 1438 1585 1732 1878 2025 2171 2318 2464 2610 297 2756 2903 3049 3195 3341 3487 3633 3779 3925 4071 298 4216 4362 4508 4653 4799 4944 5090 5235 5381 5526 299 6671 5816 5962 6107 6252 6397 6642 6687 6832 6976 ! l OF NUMBERS. 7 N. I 2 3 4 5 6 7 8 9 300 477121 7266 7411 7555 7700 7844 7989 8133 8278 8422 301 8566 8711 8855 8999 9143 9287 9481 9575 9719 9863 302 480007 0151 0294 0438 0582 0725 0809 1012 1156 1299 303 1443 1586 1729 1872 2016 2159 2302 2445 2588 2731 304 2874 3016 3159 3302 3445 3587 3730 3872 4015 4167 142 305 4300 4442 4585 4727 4869 5011 6153 5296 5437 5579 306 5721 5863 6005 6147 6289 6430 6572 6714 6855 6997 307 7138 7280 7421 7563 7704 7845 7986 8127 8269 8410 308 8551 8692 8833 8974 9114 b255 9396 9537 9667 9818 309 9959 ..99 .239 .380 .620 .661 .801 .941 1081 1222 310 491362 1502 1642 1782 1922 2062 2201 2341 2481 2621 311 2760 2900 3040 3179 3319 3458 3597 3737 3876 4015 312 4155 4294 4433 4572 4711 4850 4989 5128 5 - 6 7 5406 313 5544 5683 5822 5960 601)9 6238 6376 (515 (il 53 6791 314 6930 7068 7206 7344 7483 7621 7759 7897 8035 8173 315 8311 8448 8586 8724 8862 8999 9137 <_2I5 94 2 9660 316 9687 9824 9962 . .99 .236 .374 .511 .648 .785 .922 317 503059 1196 1333 1470 1607 1744 1880 -017 2154 2291 318 2427 2564 2/00 2837 29/3 3109 o246 3382 3518 3655 319 3791 3927 40J3 4199 4335 4471 4liU7 47-i3 1878 5014 320 5150 5283 5421 5557 5693 5828 5964 6093 6234 6370 3-21 6505 6640 677o 6911 7046 7181 7316 7451 7586 7721 829 7858 79!) I 812J 8260 8oi)5 b530 8u64 8799 8934 9008 323 9203 9337 9471 961)6 9740 9874 ...9 .143 .2/7 .411 3-24 510545 0679 0813 0947 1081 1215 13*9 14&2 1616 1/50 134 3-25 1883 2017 2151 2284 2418 2551 2G84 2818 2951 3084 326 3218 335 1 3484 3617 3750 3883 401 o 4149 4282 4414 327 4548 4681 4813 4946 5079 &211 P34 r 5476 6609 328 5874 6006 6139 62/1 6403 L635 6668 6800 6932 7034 329 7196 7328 7460 7692 7724 7855 7987 8119 8251 8382 330 8514 8646 8777 8909 9040 9171 9303 9434 9566 9697 331 9828 9959 ..90 .221 .353 .484 .616 .745 .876 10. . 332 521138 1269 1400 1530 1661 1792 1922 2053 2183 2 , , 333 2444 2575 2705 2835 2966 3096 3226 3356 3486 334 3746 3876 4006 4136 4266 4396 4526 4656 4785 4915 335 5045 5174 5304 5434 5563 5693 5822 6951 6081 6210 336 6339 6469 6598 6727 6856 6985 7114 7243 7372 7501 337 7630 7759 7888 8016 8145 8274 8402 8531 8660 8788 338 8917 9045 9174 9302 9430 9559 9687 9815 9943 ..72 339 530200 0328 0456 0584 0712 0840 0968 1096 1223 1361 340 1479 1607 1734 1862 1960 2117 2245 2372 2500 2627 341 2754 2882 3009 3136 3264 3391 3518 3645 3772 3899 342 4026 4153 4280 4407 4534 4661 4787 4914 5041 5167 343 5294 6421 5547 6674 6800 5927 6053 6180 6306 6432 344 6558 6685 6811 6937 7060 7189 7315 7441 7567 7693 129 345 7819 7945 8071 8197 8382 8448 8574 8699 8825 8951 346 9076 9202 9327 9452 9578 9703 9829 9954 ..79 .204 347 540329 0455 0580 0705 0830 0955 1080 1205 1330 1454 348 1579 1704 1829 1953 2078 2203 2327 2452 2576 2701 349 2825 2950 3074 3199 3323 3447 3571 3696 3820 3944 8 LOGARITHMS N. 1 2 3 4 5 6 7 8 9 350 544068 4192 4316 4440 4564 4688 4812 4936 5060 5183 351 5307 5431 5555 5578 5805 5925 6049 6172 6296 6419 352 6543 6666 6789 6913 7036 7159 7282 7405 7529 7652 353 7775 7898 8021 8144 8267 8389 8512 8635 8758 8881 354 9003 9126 9249 9371 9494 9616 9739 9861 9984 .196 122 355 550228 0351 0473 0595 0717 0840 0962 1084 1206 1328 356 1450 1572 1694 1816 1938 2060 2181 2303 2425 2547 357 2668 2790 2911 3033 3155 3276 3393 3519 3640 3762 358 3883 4004 4126 4247 4368 4489 4610 4731 4852 4973 359 5094 5215 5346 5457 5578 5699 5820 5940 6061 6182 360 6303 6423 6544 6664 6785 6905 7026 7146 7267 7387 361 7507 7627 7748 7868 7988 8108 8228 8349 8469 8589 362 8709 8829 8948 9068 9188 9308 9428 9548 9667 9787 363 9907 . .26 .146 .265 .385 .504 .624 .743 .863 .982 364 561101 U21 1340 1459 1578 1698 1817 1936 2056 2173 365 2293 2412 2531 2650 2769 2887 3006 3125 3244 3362 366 3481 3600 3718 3837 3955 4074 4192 4311 4429 4548 367 4666 4784 4903 5021 5139 5257 5376 5494 5612 5730 368 5848 5966 6084 6202 6320 6437 6555 6673 6791 6909 3U9 7026 7144 7262 7379 7497 7614 7732 7849 7967 8084 370 8202 8319 8436 8554 8671 8788 8905 9023 9140 9257 371 9374 9491 9608 9725 9882 9959 ..76 .193 .309 .426 372 57U543 0660 0776 0893 1010 1126 1243 1359 1476 1592 373 1709 1825 1942 2058 2174 2291 2407 2522 2639 2755 374 2872 2988 3104 3220 3336 3452 3568 3634 3800 2915 116 375 4031 4147 4263 4379 4494 4610 4726 4841 4957 5072 376 5188 5303 5419 5534 5650 5765 5880 5996 6111 6226 377 63 il 6457 6572 6687 6802 6917 7032 7147 7262 7377 378 7492 7607 7722 7836 7951 8066 8181 8295 8410 8525 379 8639 8754 8868 8983 9097 9212 9326 9441 9555 9669 380 9784 9898 ..12 .126 .241 .355 .469 .583 .697 .811 381 580925 1039 1153 1267 1381 1495 1608 1722 1836 1950 382 20G3 2177 2291 2404 2518 2631 2745 :J858 2972 3085 383 3199 3312 3426 3539 3652 3765 3879 3992 4105 4218 384 4331 4444 4557 4670 4783 4896 5009 f>122 5235 5348 385 5461 5574 5686 5799 5912 6024 6137 0250 6362 6475 386 6587 6700 6812 6925 7037 7149 7262 7374 7486 7599 387 7711 7823 7935 8047 8160 8272 8384 8496 8608 8720 388 8832 8944 9056 9167 9279 9391 9503 9615 9726 9834 389 9950 ..61 .173 .284 .396 .507 .619 .730 .842 .953 390 591065 1176 1287 1399 1510 1621 1732 1843 1955 2066 391 2177 2288 2399 2510 2621 2732 2843 2954 3064 3175 392 3286 3397 3508 3618 3729 3840 3950 4061 4171 4282 393 4393 4503 4614 4724 4834 4945 5055 5165 5276 5386 394 5496 5606 5717 5827 5937 6047 6157 6267 6377 6487 110 395 6597 6707 6817 6927 7037 7146 7256 7366 7476 7586 396 7695 7805 7914 8024 8134 8243 8353 8462 8572 8681 397 8791 8900 9009 9119 9228 9337 9446 5,656 9666 9774 398 9883 9992 .101 .210 .319 .428 .537 .646 .755 .864 399 600973 1082 1191 1299 1408 1517 1625 1734 1843 1961 OFNUMBERS. 9 N. 1 2 3 4 6 6 7 8 9 400 602060 2169 2277 2386 2494 2603 2711 2819 2928 3036 401 3144 3253 3361 3469 3573 3686 3794 3902 4010 4118 402 4226 4334 4442 4550 4658 4766 4874 4982 6089 5197 403 5305 5413 5521 5628 6736 5844 5951 6059 6166 6274 404 6381 6489 6596 6704 6811 6919 7026 7133 7241 7348 108 405 7455 7562 7669 7777 7884 7991 8098 8205 8312 8419 406 8526 8633 8740 8847 8954 9061 9167 9274 9381 9488 407 9594 9701 9808 9914 ..21 .128 .234 .341 .447 .654 408 610660 0767 0873 0979 1086 1192 1298 1405 1511 1617 409 1723 1829 1936 2042 2148 2254 2360 2466 2572 2678 410 2784 2890 2996 3102 3207 3313 3419 3525 3630 3736 411 3842 3947 4053 4159 4264 4370 4475 4581 4686 4792 412 4897 5003 5108 5213 5319 5424 6529 6634 5740 5845 413 5950 6055 6160 6265 6370 6476 6581 6686 6790 6895 414 7000 7105 7210 7315 7420 7526 7629 7734 7839 7943 415 8048 8153 8257 8362 8466 8571 8676 8780 8884 8989 416 9293 9198 9302 9408 9511 9615 9719 9824 9928 ..32 417 620136 0240 0344 0448 0552 0656 0760 0864 0968 1072 418 1176 1280 1384 1488 1592 1695 1799 1903 2007 2110 419 2214 2318 2421 2525 2628 2732 2835 2939 3042 3146 420 3249 3353 3456 3559 3663 3766 3869 3973 4076 4179 421 4282 4385 4488 4591 4695 4798 4901 6004 610? 5210 422 5312 5415 5518 5621 6724 5827 5929 6032 6135 6238 423 6340 6443 6546 6648 6751 6853 6956 7058 7161 7263 424 7366 7468 7571 7673 7775 7878 7980 8082 8185 8287 103 425 8389 8491 8593 8695 8797 8900 9002 9104 9206 9308 426 9410 9512 9613 9715 9817 9919 . .21 .123 .224 .326 427 630428 0530 0631 0733 0835 0936 1038 1139 1241 1342 428 1444 1545 1647 1748 1849 1951 2052 2153 2255 2356 429 2457 2559 2660 2761 2862 2963 3064 3165 3266 3367 430 3468 3569 3670 3771 3872 3973 4074 4175 4276 4376 431 4477 4578 4679 4779 4880 4981 5081 5182 5283 5383 432 5484 5584 5685 5785 6886 6986 6087 6187 6287 6388 433 6488 6588 6688 6789 6889 6989 7089 7189 7290 7390 434 7490 7590 7690 7700 7890 7990 8090 8190 8290 8389 435 8489 8589 8689 8789 8888 8988 9088 9188 9287 9387 436 9486 9586 9686 9785 9885 9984 ..84 .183 .283 .382 437 640481 0581 0680 0779 0879 0978 1077 1177 1276 1375 438 1474 1573 1672 1771 1871 1970 2069 2168 2267 2366 439 2465 2563 2662 2761 2860 2959 3058 3156 3255 3354 440 3453 3551 3650 3749 3847 3946 4044 4143 4242 4340 441 4439 4537 4636 4734 4832 4931 5029 5127 6226 5324 442 5422 5521 5619 5717 6815 6913 6011 6110 6208 6306 443 6404 6502 6600 6698 6796 6894 6992 7039 7187 7285 444 7383 7481 7579 7676 7774 7872 7969 8067 8165 8262 93 445 8360 8458 8555 8653 8750 8848 8945 9043 9140 9237 446 9335 9432 9530 9627 9724 9821 9919 . .16 .113 .210 447 650308 0405 0502 0599 0696 0793 0890 0987 1084 1181 448 1278 1375 1472 1569 1666 1762 1859 1956 2053 2150 449 2246 2343 2440 2530 2633 2730 2826 2923 3019 3116 10 LOGARITHMS N. 1 2 3 4 5 6 7 8 9 450 653213 3309 3405 3502 3598 3695 3791 3888 3984 4080 451 4177 4273 4369 4465 4562 4658 4754 4850 4946 5042 452 5138 5235 5331 5427 5526 5619 5715 5810 5906 6002 453 6098 6194 6290 6386 6482 6577 6673 6769 6864 6960 454 7056 7152 7247 7343 7438 7534 7629 7725 7820 7916 96 455 8011 8107 8202 8298 8393 8488 8584 8679 8774 8370 456 8965 9060 9155 9250 9346 9441 9536 9631 9726 9821 457 9916 . .11 .106 .201 .296 .391 .486 .681 .676 .771 458 660865 0960 1055 1150 1245 1339 1434 1529 1623 1718 459 1813 1907 2002 2096 2191 2286 2380 2475 2569 2663 460 2758 2852 2947 3041 3135 3230 3324 3418 3512 3607 461 3701 3795 3889 3983 4078 4172 4266 4360 4454 4548 462 4642 4736 4830 4924 5018 5112 6206 5299 5393 6487 463 5581 5675 5769 5862 5956 6050 6143 6237 6331 6424 464 6518 6612 6705 6799 6892 6986 7079 7173 7266 7360 465 7453 7546 7640 7733 7826 7920 8013 8106 8199 8293 466 8386 8479 8572 8665 8759 8852 8945 9038 9131 9324 467 9317 9410 9503 9596 9689 9782 9875 9967 ..60 .153 468 670241 0339 0431 0524 0617 0710 0802 0895 0988 1080 469 1173 1265 1358 1451 1543 1636 1728 1821 1913 2005 470 2098 2190 2283 2375 2467 2560 2652 2744 2836 2929 471 3021 3113 3205 3297 3390 3482 3574 3666 3758 3850 472 3942 4034 4126 4218 4310 4402 4494 4586 4677 4769 473 4861 4953 5045 5137 5228 5320 5412 6503 5595 6687 474 6778 5870 5962 6053 6145 6236 6328 6419 6511 6602 91 45 6694 6785 6876 6968 7059 7151 7242 7333 7424 7616 476 7607 7698 7789 7881 7972 8063 8154 8245 8336 8427 477 8518 8609 8700 8791 8882 8972 9064 9155 9246 9337 478 9428 9519 9610 9700 9791 9882 9973 ..63 .154 .245 479 680336 0426 0517 0607 0698 0789 0879 0970 1060 1151 480 1241 1332 1422 1513 1603 1693 1784 1874 1964 2055 481 2145 2235 2326 2416 2506 2696 2686 2777 2867 2957 482 3047 3137 3227 3317 3407 3497 3587 3677 3767 3857 483 3947 4037 4127 4217 4307 4396 4486 4576 4666 4756 484 4854 4935 5025 5114 5204 5294 6383 5473 5563 5652 485 5742 5831 5921 6010 6100 6189 6279 6368 6458 6547 486 6636 6726 6815 6904 6994 7083 7172 7261 7351 7440 487 7529 7618 7707 7796 7886 7975 8064 8153 8242 8331 488 8420 8509 8598 8687 8776 8865 8953 9042 9131 9220 489 9309 9398 9486 9575 9664 9763 9841 9930 ..19 .107 490 690196 0285 0373 0362 0550 0639 0728 0816 0905 0993 491 1081 1170 1258 1347 1435 1524 1612 1700 1789 1877 492 Ib65 2053 2142 2230 2318 2406 2494 2583 2671 2759 493 2847 2935 3023 3111 3199 3287 3375 3463 3651 3639 494 3727 3815 3903 3991 4078 OQ 4166 4254 4342 4430 4517 495 4605 4693 4781 4868 OO 4956 5044 5131 5210 530? 5394 496 5482 55b9 5G57 5744 5832 5919 6007 1>(W4 6182 6269 497 6356 5444 6531 6618 6706 6793 6880 6968 7055 7142 498 7229 7317 7404 7491 7578 7665 7752 7839 7926 8014 499 8101 8188 8275 8362 8449 8636 8622 8709 8796 8883 OF NUMBERS. 11 N. 1 2 3 4 5 6 7 8 9 600 698970 9057 9144 9231 9317 9404 9491 9578 9664 9751 601 9838 9924 ..11 ..98 .184 .271 .358 .444 .631 .617 602 700704 0790 0877 0963 1050 1136 1222 1309 1395 1482 603 1568 1654 1741 1827 1913 1999 2086 2172 2258 2344 604 2431 2517 2603 2689 2775 2861 2947 3033 3119 3205 86 605 3291 3377 3463 3549 3635 3721 3807 3895 3979 4065 606 4151 4236 4322 4408 4494 4679 4665 4751 4837 4922 | 607 5008 5094 6179 6265 5350 6436 5522 5607 6693 6778 608 5864 5949 6035 6120 6206 6291 6376 6462 6547 6632 509 6718 6803 6888 6974 7059 7144 7229 7315 7400 7485 610 7570 7655 7740 7826 7910 7996 8081 8166 8251 8336 611 8421 8506 8591 8676 8761 8846 8931 9015 9100 9185 612 9270 9355 9440 9524 9609 9694 9779 9863 9948 ..33 613 710117 0202 0287 0371 0456 0540 0825 0710 0794 0879 514 0963 1048 1132 1217 1301 1385 1470 1554 1639 1723 615 1807 1892 1976 2030 2144 2229 2313 2397 2481 2566 616 2650 2734 2818 2902 2986 3070 3154 3238 3326 3407 517 3491 3575 3659 3742 3826 3910 3994 4078 4162 4246 518 4330 4414 4497 4581 4665 4749 4833 4916 6000 5084 619 5167 6251 6335 5418 6502 6586 5869 5753 6836 5920 620 6003 6087 6170 6254 6337 6421 6504 6588 6671 6754 621 6838 6921 7004 7088 7171 7254 7338 7421 7504 7587 622 7671 7754 7837 7920 8003 8086 8169 8253 8336 8419 523 8502 8585 8668 8761 8834 8917 9000 9083 9165 9248 624 9331 9414 9497 9580 9663 9745 9828 9911 9994 ..77 82 625 720159 0242 0325 0407 0490 0573 0656 0738 0821 0903 626 0986 1068 1151 1233 1316 1398 1481 1563 1646 1728 627 1811 1893 .976 2058 2140 2222 2305 2387 2469 2552 528 2634 2716 2798 2881 2963 3045 3127 3209 3291 3374 629 3456 3538 3620 3702 3784 3866 3948 4030 4112 4194 630 4276 4358 4440 4522 4604 4685 4767 4849 4931 6013 631 5095 5176 5258 6340 5422 5503 6585 6667 5748 5830 632 5912 5993 6075 6156 6238 6320 6401 6483 6564 6646 633 6727 6809 6890 6972 7053 7134 7216 7297 7379 7460 534 7541 7623 7704 7785 7866 7948 8029 8110 8191 8273 535 8354 8435 8516 8597 8678 8759 8841 8922 9003 9084 536 9165 9246 9327 9403 9489 9570 9651 9732 9813 9893 637 9974 ..55 .136 .217 .298 .378 .459 .440 .621 .702 538 730782 0863 0944 1024 1105 1186 1266 1347 1428 1508 539 1589 1669 1750 1830 1911 1991 2072 2152 2233 2313 : 510 2394 2474 2555 2635 2715 2796 2876 2956 3037 3117 1 541 3197 3278 , 3358 3438 3518 3598 3679 3759 3839 3919 I 54-2 3999 4079 4160 4240 4320 4400 4480 4560 4640 4720 543 4HOO 4880 4960 5040 6120 5200 5279 5359 5439 5519 544 5.99 5679 6759 5838 5918 5998 6078 6157 6237 6317 80 545 6397 6476 6556 6636 6715 6795 6874 6954 7034 7113 546 7193 7272 | 7352 7431 7511 7590 7670 7749 7829 7908 547 7987 8087 8146 8225 8305 8384 8463 8543 8622 8701 548 8781 8860 8939. 9018 9097 9177 9256 9335 9414 949? 549 9673 9651 9731 9810 9889 9968 ..47 .126 .205 .284 12 LOGARITHMS N. 1 2 3 4 5 6 7 8 9 550 740363 0442 0521 0560 0678 0757 0836 0915 0994 1073 551 1152 1230 1309 1388 1467 1546 1624 1703 1782 1860 55-2 1939 2018 2096 2175 2254 2332 2411 2489 2568 2646 653 2725 2804 2882 2961 3039 3118 3196 3276 3353 3431 554 3510 3558 3667 3745 3823 3902 3980 4058 4136 4215 79 555 4293 4371 4449 4528 4606 4684 4762 4840 4919 4997 556 6076 5153 6231 6309 5387 6465 6543 5621 5699 5777 557 6855 5933 6011 6089 6167 6245 6323 6401 6479 6656 658 6634 6712 6790 6868 6945 7023 7101 7179 7256 7334 659 7412 7489 7567 7645 7722 7800 7878 7956 8033 8110 660 8188 8266 8343 8421 8498 8576 8653 8731 8808 8885 661 8963 9040 9118 9195 9272 9350 9427 9604 9582 9659 662 9736 9814 9891 9968 ..46 .123 .200 .277 .364 .431 663 750508 0586 0663 0740 0817 0894 0971 1048 1125 1202 664 1279 1356 1433 1510 1587 1664 1741 1818 1895 1972 665 2048 2125 2202 2279 2356 2433 2509 2586 2663 2740 566 2816 2893 2970 3047 3123 3200 3277 3353 3430 3506 667 3582 3660 3736 3813 3889 3966 4042 4119 4195 4272 668 4348 4425 4501 4578 4664 4730 4807 4883 4960 6036 569 6112 6189 5265 5341 5417 5494 6570 6646 6722 5799 670 5876 6951 6027 6103 6180 6256 6332 6408 6484 6560 571 6636 6712 6788 6864 6940 7016 7092 7168 7244 7320 672 7396 7472 7548 7624 7700 7775 7851 7927 8003 8079 673 8155 8230 8306 8382 8458 8533 8609 8686 8761 8836 574 8912 8988 9068 9139 9214 9290 9366 9441 9617 9592 74 575 9638 9743 9819 9894 9970 ..45 .121 .196 .272 .347 676 760422 0498 0573 0649 0724 0799 0875 0950 1025 1101 677 1176 1251 1326 1402 1477 1552 1627 1702 1778 1853 678 1923 2003 2078 2153 2228 2303 2378 2453 2529 2604 679 2679 2754 2829 2904 2978 3053 3128 2203 3278 3353 680 3428 3503 3578 3653 3727 3802 3877 3952 4027 4101 581 4176 4251 4326 4400 4475 4550 4624 4699 4774 4848 682 4923 4998 5072 6147 6221 5296 5370 5446 6520 5594 583 6669 5743 6818 6892 6966 6041 6116 6190 6264 6338 584 6413 6487 6562 6636 6710 6785 6859 6933 7007 7082 685 7156 7230 7304 7379 7453 7527 7601 7675 7749 7823 586 7898 7972 8046 8120 8194 8268 8342 8416 8490 8564 587 8638 8712 8786 8860 8934 9008 9082 9156 9230 9303 588 9377 9451 9525 9599 9673 9746 9820 9894 9968 ..42 589 770115 0189 0263 0336 0410 0484 0557 0631 0705 0778 590 0852 0926 0999 1073 1146 1220 1293 1367 1440 1514 691 1587 1661 1734 1808 1881 1955 2028 2102 2175 2248 592 2322 2395 2468 3542 2615 2688 2762 2835 2908 2981 593 3055 3128 3201 3274 3348 3421 3494 3567 3640 3713 594 3786 3860 3933 4006 4079 4152 4225 4298 4371 4444 73 595 4517 4590 i663 4736 4809 4882 4955 6028 5100 d!73 596 5246 531U 5392 5465 6538 5610 5683 5756 6829 5902 597 5974 604 i 6120 6193 6265 6338 6411 6483 6556 6629 598 6701 6774 6846 6919 6992 7064 7137 7209 7282 7354 599 7427 7499 7572 7644 7717 7789 7862 7934 8006 8079 OFNUMBERS. 13 N, 1 2 3 4 6 6 7 8 9 600 778151 8224 8296 83G8 8441 8513 8585 8658 8730 8802 601 8874 8947 9019 9091 9163 92HB 9308 9380 9452 9524 602 9596 6669 9741 9813 9885 9957 ..29 .101 .173 .245 603 780317 0389 0461 0533 0605 0677 0749 0821 0893 0965 604 1037 1109 1181 1253 1324 1396 1468 1540 1612 1684 72 605 1755 1827 1899 1971 2042 2114 2186 2258 2329 2401 606 2473 2544 2616 2688 2759 2831 2902 2974 3046 3117 607 3189 3260 3332 3403 3475 3546 3618 3689 3761 3832 1 608 3904 3975 4046 4118 4189 4261 4332 4403 4475 4546 609 4617 4689 4760 4831 4902 4974 6045 6116 6187 6259 610 6330 5401 6472 6543 5615 5686 5757 5828 6899 5970 611 6041 6112 6183 6254 6325 6396 6467 6538 6609 6680 612 6751 6822 6893 6964 7035 7106 7177 7248 7319 7390 613 7460 7531 7602 7673 7744 7815 7885 79^6 8027 8098 614 8168 8239 8310 8381 8451 8522 8593 8663 8734 8804 615 8875 8946 9016 9087 9157 9228 9299 9369 9440 9510 616 9581 9651 9722 9792 9863 9933 ...4 ..74 .144 .215 617 790285 0356 0426 0496 0567 0637 0707 0778 0848 0918 618 0988 1059 1129 1199 1269 1340 1410 1480 1550 1620 619 1691 1761 1831 1901 1971 2041 2111 2181 2252 2322 620 2392 2462 2532 2602 2672 2742 2812 Q882 2952 3022 621 3092 3162 3231 3301 3371 3441 3511 3581 3651 3721 622 3790 3860 3930 4000 4070 4139 4209 4279 4349 4418 623 4488 4558 4627 4697 4767 4836 4903 4976 5045 6115 624 5185 6254 6324 5393 5463 5532 6602 5672 6741 5811 G9 625 5880 6949 6019 6088 6158 6227 G297 6366 6436 6505 626 6574 6644 6713 6782 6852 6921 6990 7060 7129 7198 627 7268 7337 7406 7475 7545 7614 7683 7752 7821 7890 628 7960 8029 8098 8167 8236 8305 8374 8443 8513 8582 629 8651 8720 8789 8858 8927 8996 9065 6134 9203 9272 630 9341 9409 9478 9547 9610 9685 9754 9823 9892 9961 631 800026 0098 0167 0236 0305 0373 0442 0511 0580 0048 632 0717 0786 0854 0923 0992 1061 1129 1198 1266 1335 633 1404 1472 1541 1609 1678 1747 1815 1884 1952 / 2021 634 2089 2158 2226 2295 2363 2432 2600 2568 2637 2705 635 2774 2842 2910 2979 3047 3116 3184 3252 3321 3389 636 3457 3525 3594 3662 3730 3798 3867 3935 4003 4071 637 4139 4208 4276 4354 4412 4480 4548 4616 4685 4753 638 4821 4889 4957 5025 5093 5161 5229 5297 5365 5433 639 5501 5669 6637 5705 5773 5841 5908 6976 6044 6112 640 6180 6248 6316 6384 6461 6519 6587 6655 6723 6790 641 6858 6926 6994 7061 7129 7157 7264 7332 7400 7467 642 7535 7603 7670 7738 7806 7873 7941 8008 8076 8143 643 8211 8279 8346 8414 8481 8549 8616 8684 8751 8818 644 8886 8953 9021 9088 9156 9223 9290 9358 9425 9492 645 9560 9627 9694 9762 9829 9896 9964 ..31 ..98 .165 646 810233 0300 0367 0434 0501 0596 0636 0703 0770 0837 647 0904 0971 1039 1106 1173 1240 1307 1374 1441 1508 648 1575 1642 1709 1776 1843 1910 1977 2044 2111 2178 649 2245 2312 2379 2445 2612 2579 2646 2713 2780 2847 14 LOGARITHMS N. I 2 3 4 6 6 7 8 9 650 812913 2980 3047 3114 3181 3 247 3314 3381 3448 3514 651 3581 3648 3714 3781 3848 3914 3981 4048 4114 4181 652 4248 4314 4381 4447 4514 4581 4647 4714 4780 4847 653 4913 4980 5046 5113 6179 6246 5312 5378 6445 5511 654 6578 5644 6711 6777 6843 6910 5976 6042 6109 6175 67 655 6241 6308 6374 6440 6506 6573 6639 6705 6771 6838 656 6904 6970 7036 7102 7169 7233 7301 7367 7433 7499 657 7565 7631 7698 7764 7830 7896 7962 8028 8094 8160 658 8226 8292 8358 8424 8490 8556 8622 8688 8754 8820 659 8885 8951 9017 9083 9149 9215 9281 9346 9412 9478 660 9544 9610 9676 9741 9807 9873 9939 ...4 ..70 .136 661 820201 0267 0333 0399 0464 0530 0595 0661 0727 0792 662 0858 0924 0989 1055 1120 1186 1251 1317 1382 1448 663 1514 1579 1645 1710 1775 1841 1906 1972 2037 2103 664 2168 2233 2299 2364 2430 2495 2560 2626 2691 2756 665 2822 2887 2952 3018 3083 3148 3213 3279 3344 3409 666 3474 3539 3605 3670 3735 3800 3865 3930 3996 4061 667 4126 4191 4256 4321 4386 4451 4516 4581 4646 4711 668 4776 4841 4906 4971 6036 5101 5166 5231 5296 6361 669 5426 5491 6556 5621 5686 6751 5815 6880 5945 6010 670 6075 6140 6204 6269 6334 6399 6464 6528 6593 6658 671 6723 6787 6852 6917 6981 7046 7111 7176 7240 7305 672 7369 7434 7499 7563 7628 7692 7757 7821 7886 7951 673 8015 8080 8144 8209 8273 8338 8402 8467 8531 8595 674 8660 8724 8789 8853 8918 8982 9046 9111 9175 9239 65 675 9304 9368 9432 9497 9561 9625 9690 9754 9818 9882 676 9947 ..11 ..75 .139 .204 .268 .332 .396 .460 .525 6W 830589 OG53 0717 0781 0845 0909 0973 1037 1102 1166 678 1230 1294 1358 1422 1486 1550 1614 1678 1742 1806 679 1870 1934 1998 2062 2126 2189 2253 2317 2381 2445 1 680 2509 2573 2637 2700 2764 2828 2892 2956 3020 3083 681 3147 3211 3275 3338 3402 3466 3530 3593 3657 3721 682 3784 3848 3912 3975 4039 4103 4166 4230 4294 4357 683 4421 4484 4548 4611 4675 4739 4802 4866 4929 4993 684 5056 5120 5183 6247 5310 6373 5437 5500 5564 6627 685 5691 5754 6817 5881 5944 6007 6071 6134 6197 6261 686 6324 6387 6451 6514 6577 6641 6704 6767 6830 6894 687 6957 7020 7083 7146 7210 7273 7336 7399 7462 7526 688 7588 7652 7715 7778 7841 7904 7967 8030 8093 8156 689 8219 8282 8345 8408 8471 8534 8597 8660 8723 8786 690 8849 8912 8975 9038 9109 9164 9227 9289 9352 9415 691 9478 9541 9604 9667 9729 9792 9855 9918 9981 ..43 692 840106 0169 0232 0294 0357 0420 0482 0545 0608 0671 693 0733 0796 0859 0921 0984 1046 1109 1172 1234 1297 694 1359 1422 1485 1547 1610 1672 1735 1797 1860 1922 62 695 1985 2047 2110 2172 2235 2297 2360 2422 2484 2547 696 2G09 2672 2734 2796 2859 2921 2983 3046 3108 3170 697 3233 3295 3357 3420 3482 3544 3606 3669 3731 3793 698 3855 3918 3980 4042 4104 4166 4229 4291 4353 4415 699 4477 4539 4601 4604 4726 4788 4850 4912 4974 503(5 OF NUMBERS. 15 N. 1 2 3 4 5 6 7 8 9 700 845098 5160 5222 5-284 5346 5408 5470 5532 6594 5656 701 5718 5780 5842 5904 5906 6J-J8 6090 6151 6213 6275 702 6337 6399 6461 6523 6585 6646 6708 6770 6832 6894 703 6955 7017 7079 7141 7202 7264 7326 7388 7449 7511 704 7573 7634 7676 7758 7819 7831 7943 8004 8066 8128 62 705 8189 8251 8312 8374 8435 8497 8559 8620 8682 8743 706 8805 8866 8928 8989 9051 9112 9174 9235 9297 9358 707 9419 9481 9542 9604 9665 9726 9788 9849 9911 9972 703 850033 0095 0156 0217 0279 0340 0401 0462 0524 0585 709 0546 0707 0769 0830 0891 0952 1014 1075 1136 1197 710 1258 1320 1381 1442 1503 1564 1625 1686 1747 1809 711 1870 1931 1992 2053 2114 2175 2236 2297 2368 2419 712 2480 2541 2602 2663 2724 2785 2846 2907 2968 30-29 713 3090 3150 3211 3272 3333 3394 3455 3516 3577 3637 714 3698 3759 3820 3881 3941 4002 4063 4124 4185 4245 715 4306 4367 4428 4488 4549 4610 4670 4731 4792 4852 716 4913 4974 5034 5095 5156 5216 5277 5337 6398 5459 717 5519 5580 5640 5701 5761 5822 5882 5943 6003 6064 718 6124 6185 6245 6306 6366 6427 6487 6548 6608 6668 719 6729 6789 6850 6910 6970 7031 7091 7152 7212 7272 720 7332 7393 7453 7513 7574 7634 7694 7765 7815 7875 721 7935 7995 8056 8116 8176 8236 8297 8357 8417 8477 722 8537 8597 8657 8718 8778 8838 8898 8958 9018 9078 723 9138 9198 9258 9318 9379 943 9499 9559 9619 9679 724 9739 9799 9859 9918 9978 ..38 ..98 .158 .218 .2<8 60 725 860338 0398 0458 0518 0578 0637 0697 0757 0817 0877 726 0937 0996 1056 1116 1176 1236 1295 1355 1416 1475 727 1534 1594 1654 1714 1773 1833 1893 1952 2012 2072 728 2131 2191 2251 2310 2370 2430 2489 2549 2608 2668 729 2728 2787 2847 2906 2966 3025 3085 3144 3204 3263 730 3323 3382 3442 3501 3561 3620 3680 3739 3799 3858 731 3917 3977 4036 4096 4155 4214 4274 4333 4392 4452 732 4511 4570 4630 4689 4148 4808 4867 4926 4985 6045 733 5104 5163 5222 5282 5341 6400 6459 6519 5578 6637 734 5696 5765 5814 5874 5933 6992 6051 6110 6169 6228 735 6287 6346 6405 6465 6524 6583 6642 6701 6760 6819 736 6878 6937 6996 7055 7114 7173 7232 7291 7350 7409 737 7467 7526 7585 7644 7703 7762 7821 7880 7939 7998 738 8056 8115 8174 8233 8292 8350 8409 8468 8527 8586 739 8644 8703 8762 8821 8879 8938 8997 9056 9114 9173 740 9232 9290 9349 9408 9466 9525 9584 9642 9701 9760 741 9818 9877 9935 9994 . .53 .111 .170 .228 .287 .345 742 870404 0462 0521 0579 0638 0696 0755 0813 0872 0930 743 0989 1047 1108 1164 1223 1281 1339 1398 1456 1515 744 1573 1631 1690 1748 1806 1866 1923 1981 2040 2098 59 745 2156 2215 2273 2331 2389 2448 2506 2664 2622 2681 746 2739 2797 2855 2913 2972 3030 3088 3146 3204 3262 747 3321 3379 3437 3495 3553 3611 3669 3727 3785 3844 748 3902 3960 4018 4076 4134 4192 4250 4308 4360 4424 749 4482 4540 4598 4656 4714 4772 4830 4888 4945 5003 16 LOGARITHMS N. 1 2 3 4 5 6 7 8 9 750 875061 5119 6177 6235 5293 6351 5409 6466 5524 5582 751 6640 6698 6766 6813 6871 6929 5987 6045 6102 6160 752 6218 6276 6333 6391 6449 6507 6564 6622 6680 6737 753 6795 6853 6910 6968 7026 7083 7141 7199 7256 7314 754 7371 7429 7487 7544 7602 7659 7717 7774 7832 7889 57 755 7947 8004 8062 8119 8177 8234 8292 8349 8407 8464 756 8522 8579 8637 8694 8752 8809 8866 8924 8S81 9039 757 9096 9153 9211 9268 9325 9383 9440 9497 9555 9612 758 9669 9726 9784 9841 9898 9956 ..13 ..70 .127 .185 759 880242 0299 0356 0413 0471 0528 0580 0642 0699 0766 760 0814 0371 0928 0985 1042 1099 1156 1213 1271 1328 761 1385 1442 1499 1556 1613 1670 1727 1784 1841 1898 762 1955 2012 2069 2126 2183 2240 2297 2354 2411 2468 763 2525 2581 2638 2695 2752 2809 2866 2923 2980 3037 764 3093 3150 3207 3264 3321 3377 3434 3491 3548 3605 765 3661 3718 3775 3832 3888 3945 4002 4059 4115 4172 766 4229 4285 4342 4399 4455 4512 4569 4625 4682 4739 767 4795 4852 4909 4965 6022 6078 5135 5192 6248 6305 768 5361 5418 6474 6531 5587 5644 5700 5757 6813 6870 769 5926 5983 6039 6096 6152 6209 6265 6321 6378 6434 770 6491 6547 6604 6660 6716 6773 6829 6885 6942 "6998 771 7054 7111 7167 7233 7280 7336 7392 7449 7605 7561 772 7617 7674 7730 7786 7842 7898 7955 8011 8067 8123 773 8179 8236 8292* 8348 8404 8460 8516 8573 8629 8655 774 8741 8797 8853 8909 8965 9021 9077 9134 9190 9246 56 775 9302 9358 9414 9470 9526 9582 9638 9694 9750 9806 776 9862 9918 0974 ..30 ..86 .141 .197 .253 .309 .365 777 890421 0477 0533 0589 0645 0700 0756 0812 0868 0924 778 0980 1035 1091 1147 1203 1259 1314 1370 1426 1482 779 1537 1593 1649 1705 1760 1816 1872 1928 1983 2039 780 2095 2150 2206 2262 2317 2373 2429 2484 2540 2695 781 2651 2707 2762 2818 2873 2929 2985 3040 3096 3151 782 3207 3262 3318 3373 3429 3484 3540 3595 3651 3706 783 3762 3817 3873 3928 3984 4039 4094 4150 4205 4261 784 4316 4371 4427 4482 4538 4593 4648 4704 4759 4814 785 4870 4925 4980 6036 5091 6146 6201 6257 5312 6367 786 5423 6478 6533 5588 6644 5699 5754 6809 5864 5920 787 5975 6030 6085 6140 6195 6251 6306 6361 6416 6471 788 6526 6581 6636 6692 6747 6802 6857 6912 6967 7022 789 7077 7132 7187 7242 7297 7352 7407 7462 7517 7572 790 7627 7683 7737 7792 7847 7902 7957 8012 8067 8122 791 8176 8231 8286 8341 8396 8451 8503 8561 8615 8670 792 8725 8780 8835 8890 8944 8999 9054 9109 9164 9218 793 9273 9328 9383 9437 9492 9547 9602 9656 9711 9766 794 9821 9875 9930 9985 ..39 ..94 .149 .203 .258 .312 55 795 900367 0422 0476 0531 0586 0640 0695 0749 0804 0859 796 0913 0968 1022 1077 1131 1186 1240 1295 1349 1404 797 1458 1513 1567 1622 1676 1736 1785 1840 1S94 1948 798 2003 2057 2112 2166 2221 2275 2329 2384 2438 2492 799 2547 2601 2655 2710 2764 2818 2873 2927 2981 3036 OF NUMBERS. 17 N. 1 2 3 4 5 6 7 8 9 800 903090 3144 3199 3253 3307 3361 3416 3470 3524 3578 801 3633 3687 3741 3795 3849 3904 3958 4012 4066 4120 802 4174 4229 4283 4337 4391 4445 4499 4553 4607 4661 803 4716 4770 4824 4878 4932 4986 5040 5094 5148 6202 804 5256 5310 5364 5418 5472 5526 5580 5634 5688 5742 64 805 5796 5850 5904 5958 6012 6066 6119 6173 6227 6281 803 6335 6389 6443 6497 6551 6604 6658 6712 6766 6820 807 6874 6927 6981 7035 7089 7143 7196 7250 7304 7358 808 7411 7465 7519 7573 7626 7680 7734 7787 7841 7895 809 7949 8002 8056 8110 8163 8217 8270 8324 8378 8431 810 8485 8539 8592 8646 8899 8753 8807 8860 8914 8967 811 9021 9074 9128 9181 9235 9289 9342 9396 9449 9503 812 9556 9610 9663 9716 9770 9823 9877 9930 9984 ..37 813 910091 0144 0197 0251 0304 0358 0411 0464 0518 0571 814 0624 0678 0731 0784 0838 0891 0944 0998 1051 1104 815 1158 1211 1264 1317 1371 1424 1477 1530 1584 1637 816 1690 1743 1797 1850 1903 1956 2009 2083 2115 2169 817 2222 2275 2323 2381 2435 2488 2541 2594 2645 2700 818 2753 2808 2859 2913 2966 3019 3072 3125 3178 3231 819 3284 3337 3390 3443 3496 3549 3602 3655 3708 3761 820 3814 3867 3920 3973 4026 4079 4132 4184 4237 4290 821 4343 4396 4449 4502 4555 4608 4660 4713 4766 4819 822 4872 4925 4977 5030 5033 5136 5189 5241 5594 5347 823 5400 5453 5505 5558 5611 5664 5716 5769 5822 5875 824 5927 5980 6033 6085 6138 6191 6243 6296 6349 6401 825 6454 6507 6559 6612 6664 6717 6770 6822 6875 6927 826 6980 7033 7085 7138 7190 7243 7295 7348 7400 7453 827 750S 7558 7611 7663 7716 7768 7820 7873 7925 7978 828 8030 8083 8185 8188 8240 8293 8345 8397 8450 8502 829 8555 8607 8659 8712 8764 8816 8869 8921 8973 9026 830 9078 9130 9183 9235 9287 9340 9392 9444 9496 9549 831 9601 9653 9706 9758 9810 9862 9914 9967 ..19 ..71 832 920123 0176 0228 0280 0332 0384 0436 0489 0541 0593 833 0645 0697 0749 0801 0853 0906 0958 1010 1082 1114 834 1166 1218 1270 1322 1374 1426 1478 1530 1582 1634 835 1686 1738 1790 1842 1894 1946 1998 2050 2102 2154 836 2206 2258 2310 2382 2414 2466 2518 2570 2622 2674 837 2725 2777 2829 2881 2933 2985 3037 3089 3140 3192 838 3244 3296 3348 3399 3451 3503 3555 3607 3658 3710 839 3762 3814 3865 3917 3969 4021 4072 4124 4147 4228 840 4279 4331 4383 4434 4486 4538 4589 4641 4693 4744 841 4796 4848 4899 4951 5003 5054 5103 5157 5209 5261 842 5312 5364 5415 5467 5518 5570 5621 5673 5725 5776 843 5828 5874 5931 5982 6034 6085 6137 6188 6240 6291 844 6342 6394 6445 6497 6548 6600 6651 6702 6754 6805 52 845 6857 6908 6959 7011 7082 7114 7185 7216 7268 7319 846 7370 7422 7473 7524 7576 7627 7678 7730 7783 7832 847 7883 7935 7986 8037 8088 8140 8191 8242 8293 8345 848 . 8396 8447 8498 8549 8601 8652 8703 8754 8805 8857 849 8908 8959 9010 9081 9112 9163 9216 9266 9317 9368 18 LOGARITHMS N. I 2 3 4 5 6 7 8 9 850 929419 9473 9521 9572 9623 9674 9725 9776 9827 9879 851 9930 9981 ..32 ..83 .134 .185 .236 .287 .338 .389 852 930440 0491 0542 0592 0843 0694 0745 0796 0847 0898 853 0949 1000 1051 1102 1153 1204 1254 1305 1356 1407 854 1458 1509 1560 1610 1661 1712 1763 1814 1865 1915 51 855 1966 2017 2068 2118 2169 2220 2271 2322 2372 2423 856 2474 2524 2575 2626 2677 2727 2778 2829 2879 2930 857 2981 3031 3082 3133 3183 3234 3285 3335 3386 3437 858 3487 3538 3589 3639 3690 3740 3791 3841 3892 3943 859 3993 4044 4094 4145 4195 4246 4269 4347 4397 4448 860 4498 4549 4599 4850 4700 4751 4801 4852 4902 4953 861 5003 5054 5104 5154 5205 5255 5308 5356 5406 5457 862 5507 5558 5608 5658 5709 5759 5809 5860 5910 5960 863 6011 6061 6111 6162 6212 6262 6313 6363 6413 6463 864 6514 6564 6614 6665 6715 6765 6815 6865 6916 6966 865 7016 7086 7117 7167 7217 7267 7317 7367 7418 7468 866 7518 7568 7618 7668 7718 7769 7819 7869 7919 7969 867 8019 8069 8119 8169 8219 8269 8320 8370 8420 8470 868 8520 8570 8620 8670 8720 8770 8820 8870 8919 8970 869 9020 9070 9120 9170 9220 9270 9320 9369 9419 9469 870 9519 9569 9616 9669 9719 9769 9819 9869 9918 9968 8/1 940018 0068 0118 0168 0218 0267 0317 0367 0417 0467 872 0516 0566 0516 0666 0/16 0765 0815 0865 0915 0964 873 1014 1064 1114 1163 1213 1263 1313 1362 1412 1462 874 1511 1561 1611 1660 1710 1760 1809 1859 1909 1958 875 2008 2058 2107 2157 2207 2256 2300 2355 2405 2455 876 2504 2554 2603 26o3 2702 2752 2601 2851 2901 2950 877 3000 3049 3099 3148 3198 3247 3297 3346 3396 3445 878 3495 3544 3593 3643 3692 3742 3791 3841 3890 3939 879 3989 4038 4088 4137 4186 4236 4285 4335 4384 4433 880 4483 4532 4581 4631 4680 4729 4779 4828 4877 4927 881 4976 5025 5074 5124 5173 5222 5272 5321 5370 5419 882 5469 5518 5567 5616 5665 5715 5764 5813 6862 5912 883 5961 6010 6059 6108 6157 6207 6256 6305 6354 6403 884 6452 6501 6551 6600 6649 6698 6747 6796 6845 6894 885 6943 6992 7041 7090 7140 7189 7238 7287 7336 7385 886 7434 7483 7532 7581 7630 7679 7728 7777 7826 7875 I 887 7924 7973 8022 8070 8119 8168 8217 8266 8315 8365 888 8413 8462 8511 8560 8609 8657 8706 8755 8804 8853 889 8902 8951 8999 9048 9097 9146 9195 9244 9292 9341 890 9390 9439 9488 9536 9585 9634 9683 9731 9780 9829 891 9878 9926 9975 ..24 ..73 .121 .170 .219 .267 .316 892 950365 0414 0462 0511 0560 0308 0657 0/06 0754 0803 893 0851 0900 0949 0997 1046 1005 1143 1192 1240 1289 894 1338 1386 1435 1483 1532 1580 1629 1677 1726 1775 48 895 1823 1872 1920 1969 2017 2066 2114 2163 2211 2260 896 2308 2356 2405 2453 2502 2550 ^599 2t>4? 5G96 2744 897 2792 2841 2889 2938 2986 3034 30b3 3131 3180 3228 898 3276 3325 3373 34 21 3470 3518 3566 3bl5 3663* 3711 899 3760 3808 3856 3905 3953 4001 4049 4098 4146 4194 OF NUMBERS. 19 N. 1 2 3 4 6 6 7 8 9 900 954243 4291 4339 4387 4435 4484 4532 4580 4628 4677 901 4725 4773 4821 4869 4918 4966 5014 6062 6110 6158 902 5207 5255 5303 5351 5399 5447 5495 5543 6592 5640 903 5688 5736 5784 6832 6880 6928 5976 6024 6072 6120 904 6168 6216 6265 6313 6361 6409 6457 6505 6553 6601 48 905 6649 6697 6745 6793 6840 6888 6936 6984 7032 7080 906 7128 7176 7224 7272 7320 7368 7416 7464 7512 7559 907 7607 7655 7703 7751 7799 7847 7894 7942 7990 8038 908 8086 8134 8181 8229 8277 8325 8373 8421 8468 8516 909 8564 8612 8659 8707 8755 8803 8850 8898 8946 8994 910 9041 9089 9137 9185 9232 9280 9328 9375 9423 9474 911 9518 9566 9614 9661 9709 9757 9804 9852 9900 9947 912 9995 ..42 ..90 .138 .185 .233 .280 .328 .376 .423 913 960471 0518 0566 0613 0661 0709 0756 0804 0851 0899 914 0946 0994 1041 1089 1136 1184 1231 1279 1326 1374 915 1421 1469 1516 1563 1611 1658 1706 1753 1801 1848 916 1895 1943 1990 2038 2085 2132 2180 2227 2275 2322 917 2369 2417 2464 2511 2559 2608 2653 2701 2748 2795 918 2843 2890 2937 2985 3032 3079 3126 3174 3221 3268 919 3316 3363 3410 3457 3504 3552 3599 3646 3693 3741 920 3788 3835 3882 3929 3977 4024 4071 4118 4165 4212 921 4260 4307 4354 4401 4448 4495 4542 4590 4637 4684- 922 4731 4778 4825 4872 4919 4966 5013 5061 6108 5155 923 5202 5249 5296 5343 5390 5437 5484 5531 5578 5625 924 5672 5719 5766 5813 5860 5907 6954 6001 6048 6095 925 6142 6189 6236 6283 6329 6376 6423 6470 6517 6564 926 6611 6658 6705 6752 6799 6845 6892 6939 6986 7033 927 7080 7127 7173 7220 7267 7314 7361 7408 7464 7501 928 7548 7595 7642 7688 7735 7782 7829 7875 7922 7969 929 8016 8062 8109 8156 8203 8249 8296 8343 8390 8436 930 8483 8530 8576 8623 8670 8716 8763 8810 8856 8903 931 8950 8996 9043 9090 9136 9183 9229 9276 9323 9369 932 9416 9463 9509 9556 9602 9649 9695 9742 9789 9835 933 9882 9928 9975 ..21 ..68 .114 .161 .207 .254 .300 934 970347 0393 0440 0486 0533 0579 0626 0672 0719 0765 935 0812 0858 0904 0951 0997 1044 1090 1137 1183 1229 936 1276 1322 1369 1415 1461 1508 1554 1601 1647 1693 937 1740 1786 1832 1879 1926 1971 2018 20U4 2110 2157 938 2203 2249 2295 2342 2388 2434 2481 2527 2573 2619 939 2666 2712 2758 2804 2851 2897 2943 2989 3035 3082 940 3128 3174 3220 3266 3313 3359 3405 3451 3497 3543 941 3590 3636 3682 3728 3774 3820 3866 3913 3959 4005 942 4051 4097 4143 4189 4235 4281 4327 4374 4420 4466 943 4512 4558 4604 4650 4696 4742 4788 4834 4880 4926 944 4972 5018 6064 5110 5166 5202 5248 6294 5340 5386 46 945 5432 5478 5524 5570 5616 5662 5707 5753 5799 5845 946 5891 6937 5983 6029 6075 6121 6167 6212 6268 6304 947 6350 6396 6442 6488 6533 6579 6925 6671 6717 6763 948 6808 6854 6900 t3946 6992 7037 7083 7129 7175- 7220 949 7266 7312 7358 7403 7449 7495 7541 7586 7632 7678 = L i 20 LOGARITHMS i N. 1 2 3 4 6 6 7 8 9 950 977724 7769 7815 78G1 7906 7952 7998 8043 8089 8135 951 8181 8226 8272 8317 h363 8409 8454 8500 8546 8591 952 8637 8683 8728 8774 8819 8SG5 8911 8956 9002 9047 953 9093 9138 9184 9230 9275 9321 9366 9412 9457 9503 954 9548 9594 9639 9685 9730 9776 9821 9867 9912 9958 46 955 980003 0049 0094 0140 0185 0231 0276 0322 0367 0412 956 0458 0503 0549 0594 0340 0685 0730 0776 0821 0867 957 0312 0957 1003 1048 1093 1139 1184 1229 1276 1320 958 1366 1411 1456 1501 1547 1592 1637 1683 1728 1773 959 1819 1864 1909 1964 2000 2045 2090 2135 2181 2226 960 2271 2316 2362 2407 2452 2497 2543 2588 2633 2678 961 2723 2769 2814 2859 2904 2949 2994 3040 3085 3130 982 3175 3220 3265 3310 3356 3401 3446 3491 3536 3581 963 3626 3671 3716 3762 3807 3852 3897 3942 3987 4032 964 4077 4122 4167 4212 4257 43 j2 4347 4392 4437 4482 965 4527 4572 4617 4662 4707 4752 4797 4842 4887 4932 966 4977 5022 5067 5112 5157 5202 5247 6292 5337 5382 967 6426 5471 5516 5561 6606 5651 5699 5741 5786 5830 968 5875 5920 5965 6010 6055 6100 6144 6189 6234 6279 969 6324 6369 6413 6458 6503 6548 6593 6637 6682 6727 970 6772 6817 6861 6906 6951 6996 7040 7035 7130 7175 971 7219 7264 7309 7353 7398 7443 7488 7532 7577 7622 972 7666 7711 7756 7800 7845 7890 7934 7979 8024 8068 973 8113 8157 8202 8247 8291 8336 8381 8425 8470 8514 974 8559 8604 8648 8693 8737 8782 8826 8871 8916 8960 975 9005 9049 9093 9138 9183 9227 9272 9316 9361 9405 976 9450 9494 9539 9583 9628 9672 9717 9761 9806 9850 977 9895 9939 9983 ..28 ..72 .117 .161 .206 .250 .294 978 990339 0383 0428 0472 0516 0561 0605 0650 0694 0738 979 0783 0827 0871 0916 0960 1004 1049 1093 1137 1182 980 1226 1270 1315 1359 1403 1448 1492 1536 1580 1625 981 1669 1713 1768 1802 1846 1890 1935 1979 2023 2067 982 2111 2156 2200 2244 2288 2333 2377 2421 2465 2509 983 2554 2598 2642 2686 2730 2774 2819 2863 2907 2951 984 2995 3039 3083 3127 3172 3216 3260 3304 3348 3392 985 3436 3480 3524 3568 3613 3657 3701 3745 3789 3833 : 986 3877 3921 3965 4009 4053 4097 4141 4185 4229 4273 987 4317 4361 4405 4449 4493 4537 4581 4625 4669 4713 988 4757 4801 4845 4886 4933 4977 5021 5065 5108 6152 989 5196 5240 5284 6328 6372 5416 6460 5504 6547 6591 990 5635 6679 6723 6767 6811 5854 6898 6942 5986 6030 : 991 6074 6117 6161 6205 6249 6293 6337 6380 6424 6468 992 6512 6555 6599 6643 6687 6731 6774 6818 6862 6906 993 6949 6993 7037 7080 7124 7168 7212 7255 7299 7343 994 7386 7430 7474 7517 7561 7605 7648 7692 7736 7779 44 995 7823 7867 7910 7954 7998 8041 8085 8129 8172 8216 996 8259 8303 8347 8390 8434 8477 8521 8564 8608 8652 997 8695 8739 8792 8826 8869 8913 8956 9000 9043 9087 998 9131 9174 9218 9261 9305 9348 9392 9435 9479 9522 999 9565 9609 9652 9696 9739 9783 9826 9870 9913 9957 TABLE II. Log. Fines and Tangenls. (0) Natural Sines. 21 Sine. D 10 Cosine. D. 10" Tang. D.10 Coiang. N.sine N. cos. 0.000030 10.000000 0.000000 Infinite. 00000 100000 60 6.463726 oooooo 6.463726 13.536274 00029 100000 59 a 764753 oooooo 764756 235244 00058 103001 58 3 940347 oooooo 940847 059153 00037 103000 57 4 7.0ii578o oooooo 7.065786 12.934214 001 16 100000 56 5 162696 oooooo 162696 837304 00145 100300 55 6 241877 9.999999 241878 758122 00175 100000 54 7 308824 999999 308825 691175 00204 100000 53 8 366816 999999 366817 633183 00233 100000 52 9 417968 999999 417970 582030 00262 100030 51 10 463725 999998 463727 636273 00291 100030 50 11 7.505118 9.999998 7.505t20 12.494880 00320 99999 49 12 542903 999997 542909 457091 00349 9999 J 48 13 577668 999997 577672 422328 00378 99999 47 14 609853 999996 609857 390143 00407 99999 46 15 639816 999996 639820 360180 00436 99999 45 16 667845 999995 667849 332151 00465 99999 44 17 694173 999995 694179 305821 00495 99999 43 1 18 718997 999994 719003 280997 00524 99999 42 ; 19 742477 999993 742484 257516 00553 99998 41 i 20 764754 999993 764761 235239 00582 99998 40 21 7.785943 9.999992 7.785951 12.214049 00611 99998 39 22 806146 999991 806155 193845 00640 99998 38 23 825451 999990 825460 174540 00669 99998 37 24 843934 999989 843944 156056 00698 99998 36 25 861663 999988 861674 138326 00727 99997 35 26 878695 999988 878708 121292 00756 99997 34 27 895085 999987 895099 104901 00785 99997 33 28 910879 999986 910894 0391 06 00814 99997 32 29 926119 999985 926134 073866 00844 99996 31 30 940842 999983 940858 059142 00873 99996 30 31 7.955082 9.999982 7.955100 12.044900 00902 99996 29 32 968870 2298 999981 0.2 968889 2298 031111 00931 999% 28 33 982233 2227 999980 0.2 982253 2227 017747 00960 99995 27 34 995198 2161 999979 0.2 995219 2161 004781 00989 99995 26 35 8.007787 2098 999977 0-2 8.007809 2098 11.992191 01018 99995 25 36 020021 2039 999976 0-2 020045 2039 979955 01047 99995 24 37 031919 1983 999975 0-2 031945 1983 968055 01076 99994 23 38 043501 1930 999973 0-2 043527 1930 956473 01105 99994 22 39 054781 1880 999972 0-2 054809 1880 945191 01134 99994 21 40 41 065776 8.076500 1832 1787 999971 9.999969 2 0-2 065806 8.076531 1833 1787 934194 11.923469 01164 01193 99993 99993 20 19 42 43 086965 097183 1744 1703 999968 999966 2 o;2 086997 097217 1744 1703 913003 902783 01222 01251 99993 99992 18 17 44 107167 1664 999964 0]2 107202 1664 892797 01280 99992 16 45 116926 1626 999963 0^3 116963 1627 883037 01309 99991 15 46 126471 1591 999961 0^3 126510 1591 873490 01338 99991 14 47 135810 1557 999959 0.3 135851 1557 864149 01367 99991 13 48 49 60 61 52 53 54 55 66 67 144953 153907 162681 8.171280 179713 187985 196102 204070 211895 219581 1524 1492 1462 1433 1405 1379 1353 1328 1304 1281 999958 999956 999954 9.999952 999950 999948 999946 999944 999942 999940 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.4 144996 153952 162727 8.171328 179763 188036 196156 204126 211953 219641 1524 1493 1463 1434 1406 1379 1353 1328 1304 1281 855004 846048 837273 11.828672 820237 811964 803844 795874 788047 780359 01396 01425 01454 01483 01513 01542 01571 01600 01629 01658 99990 99990 99989 99989 99989 99988 99988 99987 99987 99986 12 11 10 9 8 7 6 5 4 3 58 227134 1259 999938 0.4 227195 1259 772805! 01687 99986 2 59 60 234557 241855 1237 1216 999936 999934 0.4 0.4 234621 241921 1238 1217 765379 01716 758079 j 01745 99985 99985 1 Cosine. Sine. Coi an ST. Tang. H N. cos. \. pjn- 89 Degrees. 22 Log. Sines and Tangents. (1) Natural Sines. TABLE II. Sine. D.10" Cosine. D.10" Tan D 10" Coiang. i N. sine. N. cos. o 8.241855 9.999934 8.241921 11.758079 01742 J9985 60 1 249033 1196 999932 0.4 249102 197 750898 01774 )9984 59 2 256094 117/ 999929 0-4 256165 .177 1 1 KQ 743835 01803 J99S4 58 3 263042 1158 999927 0.4 0,4 263115 L15o 1 1 1 n 736885 01832 99983 57 4 269881 1140 999925 .4 269956 L14U 730044 01862 99983 56 5 276614 1122 999922 0.4 276691 L122 723309 01891 )9982 55 6 283243 1105 999920 0.4 283323 1105 716677 01920 99982 54 7 289773 1 083 999918 0.4 289856 1089 710144 01949 9<>y8i 53 8 29620 7 10/2 999915 0.4 OA 296292 1073 1 AX? 703708 01978 99980 52 9 10 302541) 308794 1056 1041 999913 999910 .4 0-4 302634 308884 lUo/ 1042 1 10 1 697366 691116 02007 02036 99980 99979 51 50 11 8.314954 1027 9.999907 -4 04 8.315046 lU4/ 1 01 ^ 11-684954 02065 9B979 49 12 321027 OAQ 999905 . 4 04 321122 lUlo AGO 678878 02094 99978 48 13 327016 yyo no^ 999902 4 OA 327114 yyt-j 672886 02123 99977 47 14 332924 yoo Q71 999899 .4 OK 333025 985 070 666975 02152 9U977 46 15 338753 y / 1 QXQ 999897 . o OK 333856 J t M 9XQ 661144 02181 9997(5 46 16 344504 yoy 999894 . O OK 344610 OJ 655390 02211 99976 44 17 350181 Q^A 999891 . O OK 350289 , 649711 02240 99975 l:> 18 355783 Q99 999888 O OK 355895 922 644105 02269 99974 42 19 361315 Q1O 999885 O OK 361430 qi i 638570 02298 99974 41 20 366777 QQQ 999882 .O OK 366895 yi i 8QQ 633105 02327 99973 40 21 8.372171 oyy ooo 9.999879 . O OK 8.372292 QJiJ 888 11-627708 02356 9y972 39 22 377499 OOO 877 999876 .O OK 377622 OOO 87Q 622378 02385 99972 38 23 382762 Oil 999873 O OK 382889 /y 617111 02414 99971 37 24 387962 8"R 999870 ,O OK 388092 857 611908 02443 99970 36 25 393101 84fi 999867 . U A K 393234 847 606766 02472 99969 35 26 398179 007 999864 V O OK 398315 837 601685 02501 99969 34 27 403199 Oo / 897 999861 . O OK 403338 828 596662 02530 99968 33 28 408161 oz / 818 999858 .O OK 408304 818 691696 02560 99967 32 29 413068 Olo 999854 . O OK 413213 QAQ 686787 02589 99966 31 30 417919 800 999851 . O Of; 418068 ouy 800 681932 02618 99966 30 31 8.422717 7O.1 9.999848 O Of; 8.422869 ouu 11.577131 02647 99965 29 32 427462 /yi 782 999844 *O 6 427618 783 672382 02676 99964 28 33 432156 / o/* 774 999841 u . o Of; 432315 567685 02705 99963 27 34 436800 / /4 7fjf> 999838 O Of; 436962 7KK 563038 02734 99963 26 35 441394 /OO 758 999834 O OR 441560 /OO 758 558440 02763 99962 25 36 445941 /oo 7nO 999831 . O OR 446110 750 553890 02792 99961 24 37 450440 /OU 999827 . O Of! 450613 649387 02821 99960 23 38 454893 70K 999823 .O OR 455070 735 544930 02850 99959 22 39 459301 /GO 999820 O Or> 459481 ryno 640519 02879 99959 21 40 463666 7on 999816 .0 Of; 463849 / ^Q 720 636151 02908 99958 20 41 8.467985 719 9.999812 O Of; 8.468172 /^u 71 ^5 11.531828 02938 99957 19 42 472263 / 1-4 999809 O O.fi 472454 / lo 707 627546 02967 99956 18 43 476498 999805 O Of; 476693 /U / 700 623307 02996 99955 17 44 480693 ; 999801 .O Of; 480892 / uu 619108 03025 99954 16 45 484848 i 5 999797 .0 485050 . > , 614950 03054 99953 15 46 4889631 ;n 999793 0-7 Ort 489170 fifiO 610830 03083 99952 14 47 4930401 Sq 999790 1 493250 OoU 506750 03112 99952 13 48 497078 SA, 999786 0.7 Ort 497293 fi 602707 03141 99951 12 49 501080 ggj 999782 / Ory 601298 fifi! 498702 03170 99950 11 50 605045 r _ 999778 . / 07 605267 OO1 655 494733 03199 99949 10 51 8.508974 5 9.99977.. . / 0*7 8.509200 ?KA 11.490800 03228 99948 9 52 512867 i 999769 . / 017 613098 OOU 486902 03257 99947 8 53 616726 J 999765 . / 017 516961 f?00 483039 03286 99946 7 54 520551 gg 999761 . / 07 520790 OoO fjOQ 479210 03316 99945 6 55 524343 % 999757 . / On 524586 OOO CiOT 475414 03345 99944 6 66 528102 % 999753 .7 Ort 528349 O^i / 471651 03374 99943 4 67 531828 jjf* 999748 . / Orr 532080 filfi 467920 03403 99942 3 58 635523 p}. 999744 . 1 0*7 635779 010 f1 1 464221 03432 99941 2 59 639186 ,* 999740 . / 07 539447 Ol 1 fiOfi 460553 03461 99940 1 60 542819 C 999735 . / 543084 ouo 466916 03490 99939 Cosine. Sine. Cot a n ;. Tang. N. cos. N.sine. 88 Degrees. TABLE II. Log. Sines and Tangents. (2) Natural Sines. 23 Sjne. D. 10" Cosine. D. 10 Tang. |D. 10 Cotang. IN. sinc.|N. cos GO 8.542819 9.999735 07 8.543084 11.456916 i 03490 99939 1 546422 bUU 999731 . / 546691 DUZ 4533091|03519 99938 59 .2 549995 595 999726 0.7 550268 593 449732! 103548 9993/ 58 3 553539 591 9997-22 0-7 553817 591 446183! 03577 9 )93i 57 4 557054 !? ) 999717 0-8 557336 or>7 442664 j 03606 99935 56 5 560540 581 999713 0-8 600828 582 439172 03635 99934 55 6 663999 576 999708 0-8 564291 577 435709 03664 99933 54 7 667431 572 999704 0.8 507727 573 432273 03693 99932 53 8 670836 507 999099 0-8 00 671137 568 KP 1 428863 03723 99931 52 9 574214 KKQ 999694 b 00 574520 Oo4 K.KC1 425480 03752 99931 51 10 577506 KZ.A 999689 O OQ 577877 ooy K*- X 422123 03781 99929 5C 11 8.580892 OO4 KKA 9.999685 O 00 8.581208 Ooo K - 1 11.418792 03810:99927 49" 12 684193 OOU 999080 O 00 584514 ool 415486 03839 99920 48 13 687469 546 999675 O 687795 647 412205 ! 03808 99925 47 14 690721 542 COO 999070 0.8 O 591051 643 408949!! 03897 99924 46 15 693948 Ooo 999665 .8 O 694283 539 405717 03926 99923 45 16 597152 OJ4 9996(50 .8 697492 535 402508 03955 99922 44 17 600332 530 999055 0.8 600077 631 399323 03984 99921 43 18 603489 526 999650 0.8 603839 627 396161 04013 99919 42 19 606623 e -i (\ 999645 .8 Of. 608978 523 393022 04042 99918 41 20 609734 oiy ~i r* 999640 .8 0.-. 610094 519 389906 04071 99917 40 21 8.612823 OlO C1 1 9.999635 9 On 8.613189 616 11.386811 04100 99916 39 22 615891 Ol 1 KAQ 999629 9 O n 616262 K 383738 03129 99915 38 23 618937 OUO 999324 9 619313 508 380687 04159 99913 37 24 621962 KA1 999619 0-9 622343 505 377657 04188 99912 36 25 624965 OU1 999614 0-9 On 625352 601 374648 04217 99911 35 26 627948 4ol 999608 9 O n 628340 498 371660 04246 99910 34 27 28 630911 633854 490 AW7 999603 999597 9 0-9 On 631308 634256 496 491 368692 365744 04275 99909 04304(99907 33 32 29 636776 4O / ,40 A 999592 9 637184 488 362816 04333|99900 31 30 639680 4o4 A W1 999586 0.9 On 640093 485 369907 04362)99905 30 31 8.642563 4ol AT7 9.999581 9 On 8.642982 482 11.357018 04391(99904 29 32 33 645428 648274 4 / / 474 471 999575 999570 .9 0.9 9 645853 648704 478 475 354147 351296 0442099902 04449 99901 28 27 34 .651102 T: / 1 999564 On 651537 A* 348463 04478 99900 26 35 30 653911 656702 465 4K2 - 999558 999553 .y 654352 657149 409 466 345648 342851 1 04507 04536 99898 99897 25 24 37 659475 4:0^ 999547 659928 463 340072 04565- 99896 23 38 602230 A~r 999541 662089 460 A -o 1 337311 04594 J9894 22 39 664908 d~i 999535 665433 4o7 A r A 334567 04623 99893 21 40 667689 4" > i 999529 In 668160 454 331840 i 04653 99892 20 41 42 8.670393 673080 448 A A^ 9.999524 999518 .U 1-0 1(\ 8.670870 673563 453 449 11. 329130 i 326437 1 04682 04711 99890 99889 19 18 43 675751 44O A AC) 999512 U 676239 446 323761 04740 99888 17 44 45 678405 681043 44,* 440 999506 999500 1 -0 1-0 678900 681544 443 442 321100 318456 04769 04796 99886 99885 16 15 46 683665 4d7 999493 1 .0 684172 438 315828 !l 04827 99883 14 47 686272 434 999487 1.0 6 6784 435 313216,104850 99882 13 48 688863 4OQ 999481 1 .0 1A 689381 433 310619 04885 99881 12 49 691438 497 999475 .U In 691963 430 308037 04914 99879 11 60 693998 *i^> 1 999469 .U 1ft 694529 428 305471 ! 04943 99878 10 51 8.696543 499 9.999463 U 1-1 8.697081 425 11.302919 04972 99876 9 62 699073 % 999456 . 1 699617 423 300383 05001 99875 8 63 701589 417 999450 1 .1 702139 420 297861 05030 99873 7 64 704090 414 999443 1 . 704246 418 295354 05059 99872 6 65 706577 419 999437 1 . 707140 415 292860 1 05088 99870 5 66 709049 A 1 A 999431 1 . 709618 413 290382 05117 99869 4 67 711507 4.1 U 999424 1 . 702083 411 2879171 05146 99807 3 68 59 713952 716383 405 999418 999411 1 1. 714534 716972 408 406 285465 i 05175 283028 05205 99866 99864 2 1 60 718800 403 999404 1 .1 719396 404 280604 05234 99863 Cosine. SnTe! Cotang. Tang. Jj N. cos. N.sine ~ 1 ~ 87 Degrees. Log. Sines and Tangents. (3; Natural Sines. TABLE II. Sine. D. lu Cosine. D. 10 Tang. D. 10 Cotang. |(N. sine N. cos 8.718800 .101 9.999404 8.719396 4(V> 11.280604 05234 99863 60 1 721204 4U1 OQQ 999398 1" 721806 *\J2 oqq 278194 05263 99861 59 2 723595 oyo 999391 724204 oyy 00-7 275796 05292 99860 58 3 725972 396 QO -i 999384 .1 726588 d97 OQK 273412 05321 99858 57 4 728337 oy^i QQO 999378 728959 oyo OQO 271041 05350 99857 56 5 730688 o>y^ 999371 731317 oyo 268683 05379 99855 55 6 733027 390 OQO 999364 .1 733663 391 OQQ 266337 05408 99854 54 7 735354 ooo 999357 735996 ooy 264004 05437 99852 53 8 737667 386 999350 .2 738317 387 261683 05466 99851 52 9 739969 384 999343 .2 740326 385 259374 05495 9984S 51 10 742259 382 999336 . 2 742922 383 OO 1 257078 05524 99847 50 11 8.744536 380 9.999329 .2 8.745207 dol 11.254793 05553 9984G 49 12 746802 378 999322 .2 747479 379 252521 05582 99844 48 13 749055 376 999315 .2 749740 377 250260 05611 99842 47 14 751297 374 999308 .2 751989 375 248011 05640 99841 46 15 753528 372 999301 .2 754227 373 245773 05669 99839 45 16 755747 370 999294 .2 756453 371 243547 05698 99838 11 17 757955 368 999286 .2 758668 369 241332 05727 99836 43 18 760151 366 999279 .2 760872 367 239128 05756 99834 43 19 762337 364 999272 .2 763065 365 236935 05785 99833 41 20 764511 362 QM 999265 .2 765246 364 or*O 234754 05814 99831 40 21 8.766675 ool QKH 9.999257 8.767417 OO* r)>f\ 11.232583 05844 99829 39 22 768828 o59 r>KOr 999250 769578 ooU OKQ 230422 05873 99827 38 23 770970 oo/ OKK 999242 771727 OOO OKf* 228273 05902 99826 37 24 773101 o5o 999235 .3 773866 ooo OEX 226134 05931 99824 36 25 775223 353 O"O 999227 3 775995 OOO o-o 224005 05960 99822 35 26 777333 oo2 999220 .8 778114 OOO OE1 221886 05989 99821 34 27 779434 350 999212 3 780222 o51 219778 06018 99819 33 28 781524 348 999205 .3 782320 350 O AQ 217680 06047 99817 32 29 783605 347 999197 .3 784408 d4o 215592 06076 99815 31 30 785675 345 999189 .3 786486 346 213514 06105 99813 30 31 8.787736 343 9.999181 3 8.788554 345 11.211446 06134 99812 29 32 789787 342 999174 .3 790613 343 209387 06163 99810 28 33 791828 340 999166 .3 792662 341 207338 06192 99808 27 34 793859 339 999158 .3 794701 340 205299 06221 99806 26 35 795881 337 999150 .3 796731 338 OO T 203269 06250 99804 25 36 797894 335 999142 3 798752 OO / OOK 201248 06279 99803 24 37 799897 334 999134 3 800763 ooo 199237 06308 99801 23 38 801892 332 999126 3 802765 334 197235 06337 99799 22 39 803876 331 999118 3 804858 332 195242 06366 99797 21 40 805852 329 999110 .3 806742 331 193258 06395 99795 20 41 8.807819 328 9.999102 3 8.808717 329 11.191283 06424 99793 19 42 809777 326 999094 3 810683 328 189317 06453 99792 18 43 811726 325 999086 1.4 812641 326 187359 06482 99790 17 44 813667 323 999077 1.4 814589 325 185411 06511 99788 16 45 815599 322 oon 999069 1.4 816529 323 OOO 183471 06540 99786 15 46 817522 o20 010 999061 1.4 818461 622 181539 06569 99784 14 47 819436 o!9 01 o 999053 1.4 820384 320 179616 06598 99782 13 48 821343 olo 01 c 999044 1*4 822298 319 177702 06627 99780 12 49 823240 olb O1 K 999036 1 .4 824205 318 176795 06656 99778 11 60 825130 olO 999027 1.4 826103 316 173897 06685 99776 10 51 8.827011 313 9.999019 1.4 8.827992 315 11.172008 06714 99774 9 52 828884 312 999010 1.4 829874 314 170126 06743 99772 8 53 830749 311 999002 1.4 831748 312 168252 06773 99770 7 54 832607 309 998993 1.4 833613 311 166387 06802 99768 6 55 834456 308 998984 1.4 835471 310 164529 06831 99766 5 56 836297 307 998976 1.4 837321 308 162679 06860 99764 4 57 838130 306 998967 1.4 839163 307 160837 06889 99762 3 58 839956 304 998958 1.5 840998 306 159002 06918 99760 2 59 841774 303 998950 1.5 842826 304 157175 06947 99758 1 60 843585 302 998941 1.5 844644 303 155356 06976 99766 Cosine. Sine. Cotang. Tang. N. cos. N.sine. i 86 Degrees. TABU-: II. Log. Sines and Tangents. (4) NaUral Sines. 25 > Sine. D. 10" Cosine. D. 10" Tang. D. 10" Cotang. N. sine. N. cos. 8.843585 OAA 9.998941 1 K 8.844644 onr> 11.155356 06976 99756 60 1 845387 OUU OOQ 998932 1 . Ic; 846455 oUw OA1 153545 07005 99754 59 2 847183 Zyy OQO 998923 . O IK 848260 OU1 9QQ 151740 07034 99752 58 3 848971 zyo 9Q7 998914 O IK 850^57 zyy 9Q8 149943 07063 99750 57 4 850751 Zy I 9QK 998905 O IK 851846 zyo 9Q7 148154 07092 99748 56 5 852525 zyo 9QJ. 998896 IK 853628 * 9Q: 146372 07121 99746 55 6 854291 zy* OQQ 998887 IK 855403 zy. > OGK. 144597 07150 99744 54 7 856049 zyo OOO 998878 .0 IK, 857171 zyo OO ( ? 142829 07179 99742 53 8 857801 ZyZ 291 998869 .O 1 K 858932 zyj 292 141068 07208 99740 52 9 859546 9 ; )0 998860 1 - O 1 fi 860686 901 139314 07237 99738 51 10 861283 Z .>u o 998851 i O 1 fi 862433 ZJl O(i() 137567 07266 99736 50 11 8.863014 zoo os 7 9.998841 1 O 1 fi 8.864173 <*<j\j oco 11.135827 07295 99734 49 12 864738 ZO i 286 998832 1 IK 865906 ^oy 000 134094 07324 99731 48 13 866455 zoo oes 998823 D 1 R 867632 Zoo 987 132368 07353 99729 47 14 868165 zoo 98.4. 998813 JL .O .:? 869351 Zo / 9ox 130649 07382 99727 46 15 869868 ZO4 9Q 998804 ,o 871064 ZoO 98 d 128936 07411 99725 45 16 871565 Zoo 989 998795 872770 ZO4 981 127230 07440 99723 44 17 873255 ZOZ 981 998785 874469 Zoo 989 125531 07469 99721 43 18 874938 ZoJ. 279 998776 _ g 876162 Zoz 981 123838 07498 99719 42 19 876615 279 998766 877849 A^Ol 980 122151 07527 99716 41 20 878285 277 998757 f; 879529 ^OU 070 120471 07556 99714 40 21 8.879949 276 9.998747 U 8.881202 z i y 978 11.118798 07585 99712 39 22 881607 97 "i 998738 882869 Z /O 9,77 117131 07614 59710 38 23 883258 <) 974 998728 884530 Z 1 / 97fi 115470 07643 99708 37 24 884903 Z /^* 271 998718 886185 z ^o 971; 113815 07672 99705 36 25 886542 /o 979 998708 887833 Z 4 O 974 112167 07701 99703 35 26 888174 4tA 971 998699 c 889476 Z < 4 97Q 110524 07730 99701 34 27 889801 A / 1 270 998689 D 6 891112 Z io 272 108888 07759 99699 33 28 891421 9fiO 998679 892742 z iz 971 107258 07788 )9696 32 29 893035 ZoJ 268 998669 7 894300 Z/ 1 970 105634 07817 99694 31 30 894643 Of;7 998659 895984 * 1 V 9AQ 104016 07846 99(592 30 31 8.896246 zo i (U5 9.998649 7 8.897596 zoy OfiQ 11.102404 07875 99689 29 32 897842 wOl) 9 /jr. 998639 7 899203 ^-oo 2fi7 100797 07904 J9687 28 33 899432 ZOO O*^ <1 998629 . / r* 900803 ZD 1 Of?; 099197 07933 #686 2~ 34 901017 ZO4 t>i *j 998619 . / 902398 ZOO O^ X 097602 079(52 9683 26 35 902596 iioo O/ O 998609 903987 ZOO OdA 096013 07991 )9680 25 3i3 37 904169 905736 ZOZ 261 Ofjf) 998599 998589 .7 7 7 905570 907147 Zo4 263 262 094430 092853 08020 08049 99678] 24 99676 23 38 907297 . ZOU OXQ 998578 908719 9fi1 091281 08078 99673 22 39 908853 ^Oa 9^8 998568 7 910285 zol 260 089715 08107 99671 21 40 41 910404 8.911949 zoo 257 9n7 998558 9.998548 / .7 7 911846 8.913401 zou 259 OKO 088154 11.086599 08136 08165 996(58 9966(5 20 19 42 913488 ~ ) / 9^fi 998537 . / 7~ 914951 ZOo 0x7 085049 08194 99664 18 43 915022 *oo OKK 998527 916495 zo / 9!Sfi 083505 08223 99661 17 44 916550 <OO OK/4 998516 918034 ZOO OKC 081966 08252 99(559 16 45 918073 ^04 2,%o 998506 919568 zoo 9n^i 080432 08281 99657 15 46 919591 Do 998495 _ 921096 ZOO O" A 078904 08310 99(554 14 47 921103 252 998485 922619 zo4 077381 ! 08339 99652 13 48 922(510 251 998474 .8 9241 36 253 2-^0 075864 08368 99649 12 49 9-J4112 250 998464 .8 925649 02 074351 08397 99647 11 5. ) 925(509 249 o in 998453 .8 g 927156 251 OPCA 072844 08426 99644 10 51 8.927100 f -iJ 9.998442 8.928658 ZJU 11.071342 08455 99642 9 928587 248 o/i re 998431 .8 930155 249 O in 069845 08484 99639 8 53 930068 !*47 O/ift 998421 931647 ^4y O/lrt 068353 08513 99637 7 51 931544 ^i*iO ^AK 998410 O 933134 Z^lo 917 066866 08542 99635 6 55 !)33015 ,w4:O . ) 1 1 998399 _ 9346 1(5 Z-l / " 1C 065384 08571 99632 5 56 934481 Z \~ Q 1Q T98388 _ y 936093 - M > 245 063907 (IS(iOl) 99630 4 57 : ;:;r ..:-. ^UO 998377 937565 O i 1 062435 08629 99627 3 58 937398 fc J43 o 10 998366 .8 939032 z44 9J.J. 060968 08658 99625 2 59 60 938S50 94u296 ^4^ 241 998355 998344 .b 940494 941952 Z44 243 059506 058048 08687 08716 99622 99619 1 (losino. SJneT~ Co tang. Tang. il N. cos. N.sinc. / 85 Degrees. 26 Log. Sines and Tangents. (5) Natural Sines. TABLE 11. Sine. D. 10" Cos-ine. D. 10 Tang. D. 10 Cotang. | jN. sine X. COR 8.940296 9.998344 8.941952 11.058048 08716 9961 60 1 941738 Z4U 993333 1 .9 943404 242 056596 08745 9961 59 2 943174 239 998322 1 .9 944852 241 055148 08774 99614 58 3 944606 998311 1.9 946295 240 053705 08803 99612 57 4 946034 238 998300 1 .9 947734 240 052266 08831 99609 56 5 947456 26 i 998289 1 .9 949168 239 050832 08860 9960* 55 6 948874 236 998277 1.9 950597 238 049403 08889 99804 54 7 950287 00- 998266 1 .9 1 Q 952021 937 047979 08918 99602 53 8 951698 234 998255 i . y 1q 953441 Zo / 046559 08947 99599 52 9 953100 933 998243 y 1 Q 954856 __ 045144 08976 9959( 51 10 954499 Zoo 998232 i . y 1 q 956267 934 043733 09005 99594 50 11 8.955894 9.39 9.998220 i , y 1q 8,957674 93d 11.042326 09034 99591 49 12 957284 . 998209 . y 1 Q 959075 QOO 040925 09063 99588 48 13 958670 930 998197 i . y 1 Q 960473 Zoo 039527 09092 99586 47 14 960052 ZoU 998186 i . y 1r\ 961866 232 038134 109121 99583 46 15 961429 OQ 998174 .9 If-* 963255 ~31 036745 09150 99580 45 16 962801 228 998163 .9 t q 964639 231 035361 09179 99578 44 17 964170 227 998151 A * y 1 Q 966019 92Q 033981 09208 99575 43 18 965534 997 998139 JL , y A 967394 99Q 032606 09237 99572 42 19 966893 ZZ i 998128 Z . \) 2 968766 9OQ 031234 09266 99570 41 ! 20 968249 99" 998116 2 A 970133 997 029867 |09295 99567 40 21 8.969600 994. 9.998104 . u O A 8.971496 ZZ / 11.028504 09324 99564 39 22 970947 ZZ4 998092 9 ft 972855 99fi 027145 09353 )95(>2 38 23 972289 223 998080 Z. U 2 974209 90 K 025791 09382|99559 37 24 973628 998068 z . u 2/1 975560 zzo 024440 09411199556 36 25 974962 999 998056 . U O A 976906 91 023094 0944099553 35 26 976293 99 \ 998044 z , \) 2 A 978248 993 021752 09469(99561 34 27 28 977619 978941 220 998032 998020 V 2.0 2 A 979586 980921 222 0204141 0949899548 019079 09527199545 33 32 29 980259 91 Q 998008 . U 9 O 982251 901 1 7749 I 09556 99542 31 30 31 32 981573 8.982883 984189 ziy 218 218 217 997996 9.997984 997972 Z. U 2.0 2.0 2 983577 8.984S99 986217 220 220 016423! 09585 11.015101 J09614 013783:109642 99540 99537 99534 30 29 28 I 33 985491 **i i 997959 o n 987532 f) 1 U 012468J 09671)99531 27 34 988789 Ol Q 997947 ^ . U O t\ 988842 011158 09700(99528 26 35 988083 O1 ^ 997935 O 1 990149 01 7 009851 09729 99526 25 36 37 989374 990660 *> 10 214 01 \ 997922 997910 * i 2.1 21 991451 992750 - ! i 216 008549 007250 06758 99523 0978799520 24 23 38 991943 z!4 21 3 997897 .1 2 1 994045 216 91 K 005955 |09816J99517 22 39 993222 Z1O 91 o 997885 Z.I i 995337 Z1O 004663 09845 99514 21 40 994497 Zl^ OlO 997872 Z.I 9 1 996624 O1 A 003376 0987499511 20 41 42 43 44 45 S. 995 768 997036 998299 999560 ). 0008 16 o to to to to t E co o i i > t J. 997860 997847 997835 997822 997809 o to to to to t 8.997908 999188 9.000465 001738 003007 o to to to to t - I tO CO CO ,4 11.002092 000812 0.999535 998262 996993 0990399508 09932199508 09961 99503 09990,99500 10019(99497 19 18 17 16 15 46 002069 997797 24 004272 **!.*. 995728 10048199494 14 47 003318 9/vo 997784 .1 21 005534 210 994466 10077|99491 13 48 004563 ZUo 9/17 997771 . 1 O 1 006792 Q 993208 1 10108 i 99488 12 49 005805 ZU 997758 Z . 1 2-1 008047 8 991953 1013599485 11 50 OJ7044 9 r 997745 . 1 21 009298 208 990702 1 10164:99482 10 51 9.008278 9/)K 9.997732 . 1 2] 9.010546 9O7 0.989454 10192 99479 9 52 009510 zuo 205 997719 . 1 2 1 011790 207 988210 10221199476 8 53 010737 997706 013031 686969 10250:99473 7 54 011962 204 203 997693 2.1 2 2 014268 206 985732 10279^99470 6 55 013182 203 997680 2 2 015502 _ 984498 10308,99467 5 56 014400 202 997667 9 9 016732 904 983268 ! 10337J99464 4 57 015613 9( jo 997654 9 9 017959 ZUTC 983041 1036699461 3 58 016824 zuz om 997U41 Z . Z 29 019183 .-!q 980817 |j 10395 99458 2 59 018031 9", 997628 . Z 20 020403 Q 9;9597 11)424199455 1 60 019235 " 997614 . ~f 021620 9783801 10453199452 Cosine, j Sine. Cotang. Tang. i N. cos.JN.^ine. ~ 84 D^rees. TABLE 11. Log. Sines and Tangents. (6 C ) Natural Sines. 27 Sine. D. 10" Cosine. ]D. 10" Tang. iD. 10" Co tang. N. sine. N. cos. j 9.019235 Of\f\ 9.997614 o Q 9.021620 909 10.978380 | 10453 J9452 60 1 020435 200 997601 J*J 022834 Z\j 909 977166 ; 10482 99449 59 2 021632 199 1 QQ 997588! J J 024044 Z\jZ 201 975956 ; 105 11 9944(> 58 3 022825 iyy 1 QS 997574 ! * * 025251 201 974749 ; 10540 99443 57 4 024016 iyo 1 OS 997561 ^ Q 026455 *>\jl 200 973545, 1056H 99440 56 5 025203 lc/o 997547 ** 027655 1 QQ 972345| 1059 < 99437 55 6 026386 997534 --J 028852 iyy 199 971148 10626 99434 54 7 027567 i or* 997520; J J 030046 1 Q8 969954 1 0055 99431 53 8 028744 lUb 1Qfi 997507 ;; 031237 iyo 198 968763 10684 J9428 52 9 029918 iyo 1 d" 997493! * 2 032425 1Q7 967575 10713 99424 51 10 031089 iyo 997480 * -o 033609 iy i 1 0*7 966391] 10742 99421 50 11 9.032257 195 1 Q4 9.997466 2-3 20 9.034791 iy / 196 10.965209 j 10771 99418 49 12 033421 j y*i 1 Qd 997452 o 20 035969 196 964031 10800 99415 48 13 14 034582 035741 L9*m 193 1 Q9 997439 997425 o 2.3 20 037144 038316 195 195 962856 10829 961684 1 10858 99412 99409 47 46 15 036896 i&*> 1 Q2 997411 o 20 039485 194 960515 10887 99406 45 16 038048 iy^ 997397 . o 20 040651 1 Q4. 959349 10916 99402 44 17 039197 . 1 01 997383 o 20 041813 iy4 i no 958187 10945 99399 43 18 040342 iyi 1QO 997369 J 2. 042973 iyo 193 957027 10973 99396 42 19 041485 lc/U 1 Qft 997355 O 20 044130 192 955870 i 11002 99393 41 20 21 042625 9.043762 iy*J 189 1HQ 997341 9.997327 o 2.3 2 A 045284 9.046434 liy-w 192 191 954716 10.953566 11031 11060 99390 99386 40 39 22 23 044895 046026 loy 180 1 QQ 997313 997299 4 2-4 2 A 047582 048727 i^/i 191 190 952418 951273 11089 11118 99383 38 J9380 1 37 24 047154 loo 187 997285 4 2 A 049869 190 950131 11147 99377 36 ! 25 048279 lo / 1 1>7 997271 4 2 A 051008 189 948992 11176 99374 35 26 049400 io / 1 8fi 997257 4 2 A 052144 10^7 18Q 947856 11205 99370 34 27 050519 loo I8fi 997242 .4 2 A 053277 lOi7 188 946723 11234 99367 33 28 051635 loo 1 QK 997228 .4 2 A 054407 loo 188 945593 11263 99364 32 29 052749 loo i wx 997214 4 2 A 055535 187 944465 11291 99360 31 30 053859 loo 1 ,4. 997199 4 2 A 056659 lo / 187 943341 11320 99357 30 31 9.054966 Io4 1 0,4 9.997185 4 2 A 9.057781 lo / i Q 10.942219 11349 99354 29 32 056071 1O4 10 4 997170 4 2 A 058900 1OO 186 941100 11378 99351 28 33 057172 1O4 i eo 997156 4 2 A 060016 185 939984 11407 99347 27 34 058271 loo 1 WO. 997141 4 2 A 061130 lS r j 938870 11436 99344 26 35 059367 loo 1 ftO 997127 4 2,4 OG2240 lOu 1 HT 937760 11465 99341 25 36 060460 LoZ 1 CQ 997112 .4 24 083348 lou 184 936652 11494 99337 24 37 061551 1O.4 181 997098 . 4 2x 064453 184 935547 11523 99334 23 38 062639 lol 181 997083 4 2e 065556 183 934444 11552 99331 22 39 063724 lol 1 80 997068 O 2s 066655 183 933345 11580 99327 21 40 064806 lou 180 997053 O 2K 067752 182 932248 11609 99324 20 41 9.085885 loU 1 7Q 9.997039 O 2c 9.068846 182 10-931154 11638 99320 19 42 066962 1 1 y 997024 O 2, K 069038 181 930062 11667 99317 18 43 068036 1 TO 997009 O 071027 181 928973 11696 99314 17 44 069107 l i J 1TQ 996994 2 .5 072113 lol 1 81 927887 11725 99310 16 45 070176 to 178 996979 2 -5 2K 073197 lol 180 926803 11754 99307 IB 46 071242 1 fo 996964 2e 074278 180 925722 11783 99303 14 47 072306 177 996949 D 2c 075356 179 924644 11812 99300 13 48 073366 1 1 / 1 tfi 996934 . D O K 076432 179 923568 11840 99297 12 49 074424 I/O ITfi 996919 ;i O 2K 077505 178 922495 11869 99293 11 50 075480 i O 1 ^7^ 996904 .0 2pr 078576 178 921424 1189& 99290 10 61 il. 073533 1 to 1 "7K 9.996889 O 2f 9.079644 178 10-920356 11927 99286 9 6-2 077583 1 tO Irrpr 996874 .O 2c 080710 177 919290 11956 99283 8 53 078631 to 996858 . O 2c 081773 177 918227 11985 99279 7 54 079676 996843 . O 2K 082833 176 917167 12014 99276 6 55 080719 1 7*} 996828 O 2c 083891 176 916109 12043 99272 6 56 081759 1 IO 996812 . O 2 084947 175 915053 12071 99269 4 57 082797 i 79 996797 . O 2J 086000 175 914000 12100 99265 3 58 083832 1 1 .Z 996782 t o 087050 17K 912950 12129 99262 2 59 084864 172 1 79 996766 2.6 2/? 088098 / O 174 911902 12158 99258 1 60 085894 1 iz 996751 .0 089144 910856 12187 99255 Cosine. Sine. Cotang. Tang. N. co, N.sine. / 83 Degrees. Log. Sines and Tangents. (7) Natural Sines. TABLE II. Sine. D. 10 Cosine. D. ](/ Tang. L>. JO Cotang. :N. sine N. COH. 1 9.085894 OS6!)22 171 9.996751 996735 2.6 2r; 9.089144 090187 174 1 7^? 10.9108561! 12187 909813 12216 99255 99251 60 59 2 OS7947 1 7O 996720 . O o r: 091228 I/O 908772 12245 99248 58 3 088970 & v 17O 996704 O 2CL 092266 907734 12274 99244 57 4 089990 1 / U 1 70 996688 .O 2 a 093302 906698 12302 99240 56 5 01)1008 1 / U 1f>Q 996673 . O 2fj 094336 1 70 905664 12331 99237 55 6 092024 ioy 1 fiQ 996657 . o 2c 095367 1 i A 904633 12360 99233 54 7 093037 ioy i ^w 906641 o 8 096395 1 71 903605 12389 99230 53 8 094047 lOo 1fiP. 996625 A> . O 2 a 097422 1/1 902578 12418 99226 52 9 095056 lOo 1 00 996610 O 2fj 098446 901554 12447 99222 51 10 096082 lOo 996594 . o 2r> 099468 900532 12476 99219 50 11 9.097055 167 i7 9.996578 .O 2 7 ). 100487 170 16Q 10.899513 12504 99215 49 12 098036 10 1 996562 101504 loy 898496 12533 99211 48 13 099065 166 lfif> 996546 2.7 2 7 102519 169 IfiQ 897481 12562 99208 47 14 1000G2 100 18.fi 996530 27 103532 ioy tCQ 896468 12591 99204 46 15 101056 1OO IfJK 996514 . / 27 104542 -I DO 1 fiS 895458 12620 99200 45 16 102048 1OO ICC 996498 4 27 105550 lOo 1fi8 894450 12649 99197 44 17 103037 1OO 184 996482 . i o 7 106556 lOo 1R7 893444 12678 99193 43 18 104025 1O4 184. 996465 < 27 107559 ID / 892441 12706 99189 42 19 105010 1O4: 184. 996449 . / 9 7 108560 Ififi 891440 12735 99186 41 20 105992 lO*i 1 CO 996433 < 1 27 109559 1OO i ca 890441 12764 99182 40 21 9.106973 loo t (>q 9.996417 . / 27 9.110556 100 1fifi 10.889444 12793 99178 39 22 107951 loo IfJO 996400 . / 217 111551 1OO ifin 888449 12822 99175 38 23 108927 loo ion 996384 . / 2rl 112543 loo ifin 887457 12851 99171 37 24 109901 10,4 182 996368 , / 27 113533 100 IKK 886467 12880 99167 36 25 110873 1O^ 1 fiO 996351 / 2rj 114521 100 1J< 885479 12908 99163 35 26 111842 lo^ 1 M 996335 . i 21 115507 1O4 IRA 884493 12937 99160 34 27 112809 lol 996318 . 7 116491 104 883509 12966 99156 33 28 113774 161 IP A 996302 2.7 20 117472 164 IH -i 882528 12995 99152 32 29 114737 1OU 1 MO 996285 .0 20 118452 AOo 1f;0 881548 13024 99148 31 30 115698 1OU 996269 .0 20 119429 1OO 1 /;) 880571 1 13053 99144 30 31 9.116656 160 9.996252 .0 9.120404 mOSt 1 iO 10.879596 13081 99141 29 32 33 117613 118567 159 159 996235 996219 2.8 2.8 121377 122348 lOxi 162 i /--I 878623 | 877652 13110 13139 99137 99133 28 27 34 119519 159 996202 2.8 123317 lol i PI 876683 13168 99129 26 35 120469 158 996185 2.8 124284 lul 1 at 875716 13197 99125 25 36 121417 158 996168 2.8 125249 lol 874751 13226 J9122 24 37 122362 158 996151 2.8 126211 160 873789 13254 J9H8 23 38 123306 157 996134 2.8 127172 160 872828 13283 J9H4 22 39 124248 157 1^7 996117 2.8 20 128130 160 1 KQ 871870 13312 J9110 21 40 125187 O t 1 n.8 996100 .0 2Q 129087 .toy 1 KQ 870913 13341 J9106 20 41 9.126125 loo 9.996083 .O 9.130041 1OJ i rfi 10.869959 18370 J9102 19 42 127060 156 996066 2.9 130994 loU 1 KQ 869006 13399 J9098 18 43 127993 156 996049 2.9 131944 loo 1 Q 868056 13427 J9094 17 44 128y25 155 996032 2.9 132893 loo 1 KQ 867107 13456 )9091 16 ; 45 129854 155 996015 2.9 133839 loo 1 K^7 866161 13485 J9087 15 46 130781 154 995998 2.9 134784 Io7 I cry 865216 13514 J9083 14 47 131706 154 995980 2.9 135726 Io7 1 XT 864274 13543 J9079 13 48 132630 154 i cq 995963 2.9 O Q 136367 -Io7 156 863333 13572 J9075 12 49 133551 100 995946 *z , y 137605 1 KC 1 862395* 13600 J9071 11 50 134470 153 1 -0 995928 2,9 O Q 138542 156 156 861458 i 13629 J9067 10 51 52 9 . 135387 136303 lOo 152 1 KO 9.995911 995894 A . y 2.9 9.139476 140409 155 1EC 10. 860524! 113658 - 859591 1 113687 99063 99059 9 8 53 137216 15^ 1 nO 995876 2. 9 o o 141340 UO 1 ^ 858660 ! 137 16 )9055 7 54 55 56 138128 139037 139944 IO4 152 151 1 ni 995859 995841 995823 M . U 2.9 2.9 o q 142269 143196 144121 100 154 154 154 857731 13744 856804 113773 855879 1 113802 )9051 J9047 99043 6 5 4 57 140850 1O1 i r.i 995806 M . y 2 9 145044 153 854956 13831 J9039 3 58 141754 I Ol i no 995,88 (1 145966 15U 854034 13860 )9035 o 59 142655 1OU 995771 /^ . y 1468S5 1 KO 853115 138x8 J >9031 1 60 143555 150 995753 2.9 147803 loo 852197 113917 JO(W7 Cosine. Sine. Cotang. Tang. N. C-OP. \.siiio. T" 8:2 Degrees. TABLE II. Log, Sines and Tangents. (8) Natural Sines, 29 Sine. I). 10" Cosine. 1). li/ Tang. D. 10" Co tang. N. sine. N. cos. o 9.143555 9.995753 9.147803 10.852197 13917 99027 60 1 144453 150 995735 3.0 148718 153 851282! 13946 99023 59 2 145349 149 995717 3.0 149632 152 850368 113975 99019 58 3 146243 149 995699 3.0 3D 150544 152 1 "O 849456] 140fV4 99015 67 4 147136 149 995681 .0 151454 1O.4 848546 14033 99011 56 5 148026 148 995664 3.0 152363 151 847637 14061 99006 55 6 148915 148 995646 3.0 153269 151 846731 14090 99002 54 7 149802 148 995628 3.0 154174 151 845826 14119 98998 53 8 9 150686 151569 147 147 995610 995591 3.0 3.0 155077 155978 150 150 844923 844022 14148 14177 98994 98990 52 51 10 152451 147 995573 3.0 156877 150 843123 14205 98986 50 11 9.153330 147 9.995555 3.0 9.157775 150 10.842225 14234 98982 49 12 154208 146 995537 3.0 3f\ 158671 149 841329 14263 98978 48 13 155083 146 995519 .0 3 /\ 159565 149 1 ylQ 840435 14292 98973 47 14 155957 146 995501 .u 160457 i4y 1 4Q 839543 14320 98969 46 15 156830 145 1 * 995482 3. 3 161347 14o 1 .18 838653 14349 98965 45 16 157700 145 995464 162236 14O 1 -Itt 837764 14378 98961 44 1? 158569 145 995446 . 163123 14o 1 J.8 836877 1440/ 98957 43 18 159435 144 995427 . 164008 1-iO 1 A7 835992 1443G 98953 42 19 160301 144 995409 . 164892 14< 1 d.7 835108 14464 98948 41 20 161164 144 995390 . 165774 l*i / 1 J.7 834226 14493 98944 40 21 9.162025 144 9.995372 . 9.166654 IT- / 1 Ad 10-833346 14522 98940 39 22 162885 143 995353 . 31 167532 14o 146 832468 14551 98936 38 23 163743 143 995334 . i 31 168409 14O 1 Af! 831591 14580 98931 37 24 164600 143 995316 . X 31 169284 1 4O 830716 14608 98927 36 25 165454 142 995297 . l 170157 145 829843 14637 98923 35 26 166307 142 995278 3. 1 171029 145 828971 14666 98919 34 27 167159 142 995260 3. 1 171899 145 828101 14695 98914 33 28 168008 142 995241 3.1 39 172767 145 1 A A 827233 14723 98910 32 29 168856 141 995222 V ri* 3f) 173634 144 826366 14752 98906 31 30 169702 141 995203 .4 39 174499 144 1 A A 825501 14781 98902 30 31 9.170547 141 9.995184 *m 30 9.175362 144 1AA 10.824638 14810 98897 29 3-2 171389 140 995165 ,A 3O 176224 J.44 1 A1 823776 14838 98893 28 33 172230 140 i An 995146 .4 q o 177084 14o 143 822916 14867 98889 27 34 173070 141) i in 995127 O . A q o 177942 1 40 822058 14896 98884 26 35 173908 14U i on 995108 O . Al 39 178799 1-rO 142 821201 14925 98880 25 36 174744 io\y 1 OO 995089 >& 30 179655 i^^f 149 820345 14954 98876 24 37 175578 i oy 1 Q(i 995070 . w 39 180508 ii& 1 4 2 819492 14982 98871 23 38 176411 loy 1 OO 91*5051 <> Q O 181360 14:^ i <o 818640 15011 98867 22 39 177242 loy 1 OQ 995032 S*o 182211 f. Km i 41 817789 15040 98863 21 40 178072 loo 1 OU 995013 0* 39 183059 141 141 816941 15069 98858 20 41 9.178900 loo 9.994993 . *> q 9 9.183907 141 m 10-816093 15097 98854 19 42 179726 994974 Ov 39 184752 1 A1 815248 15126 988491 18 43 180551 137 994955 . 39 185597 141 1 40 814403 15155 !)8845 17 44 181374 137 994935 ^ 39 186439 14U 140 813561 15184 98841 16 45 182196 1 Q"7 994916 - 3q 187280 140 812720 15212 98836 15 46 183016 lo / 1 1fl 994896 49 3q 188120 140 811880 15241 98832 14 47 183834 loo 1 Qfi 994877 O 3q 188958 139 811042 15270 98827 13 48 184651 loo 1 Qi 994857 . O 3q 189794 1QU 810206 15299 98823 12 49 185466 loo 1 Q! 994838 o 3q 190629 1OJ7 1 3Q 809371 15327 98818 11 50 186280 lob 1 OK 994818 o 3q 191462 loy 1 34 808538 15356 98814 10 St 9 . 187092 loo 1 OK 9.994798 . O 3q 9.192294 10^7 1 18 10-807706 15385 98809 9 52 187903 loo 1 O^ 994?79 O 3q 193124 loo Iqo 806876 15414 98805 8 53 188712 loo 994759 O 193953 oo 1 oQ 806047 15442 98800 7 54 189519 135 994739 3.3 3q 194780 loo 138 805220 15471 98796 6 55 190325 994719 . o 195606 804394 15500 98791 5 56 191130 134 994700 J-* 196430 1 O.7 803570 15529 98787 4 57 191933 1 Q/1 994680 *; 197253 lo < 802747 15557 98782 3 58 192734 Io4 9946601 J J 198074 801926 15586 98778 2 59 193534 133 994640! o o 198894 137 801106 15615 98773 1 60 194332 133 994620 | d 199713 136 800287 15643 98769 Cosine. Sine. Cotang. Tang- N. cos. N.sine. i 81 Degrees. 30 Log. sines and Tangents. (9) Natural Sines. TABLE II. Sine. D. 10" Cosine. D. 10" Tang. D. 10 Cotang. N. sine. N. cos.| o 9.194332 9.994620 9.199713 10.800287 15643 98769 60 1 195129 133 994600 3.3 200529 136 799471 15672 98764 59 2 195925 133 1 QO 994580 3.3 201345 136 136 798655 15701 98760 58 3 196719 lo^ 1QO 994560 O , O 3 4 202159 135 797841 15730 98755 57 4 197511 LOZ 1 V) 994540 3 4 202971 135 797029 15758 98751 56 5 198302 lo> 1 *}9 994519 3 4 203782 135 796218 15787 98746 55 6 199091 lO. 1 O1 994499 3 4 204592 135 795408 15816 98741 54 7 199879 I J i 1 O 1 994479 3 4 205400 794600 15845 98737 53 8 200366 lot 99 1459 3 4 206207 793793 15873 98732 52 9 201451 994438 3 4 207013 792987 15902 98728 51 10 202234 1 QH 994418 q 4 207817 792183 15931 98723 50 11 9.203017 loU 1 Qli 9.994397 o , ^ 3 A 9.208619 133 10.791381 15959 98718 49 12 203797 loU 1 f ll\ 994377 . 4: 34 209420 133 790580 15988 98714 48 13 204577 .loU 1 QA 994357 . *x 34 210220 i qq 789780 16017 98709 47 14 205354 loU 1 OQ 994336 ^r 34 211018 loo 133 788982 16046 98704 46 15 206131 i^y 1 OG 994316 . * 3 A 211815 133 788185 16074 98700 45 16 203906 l-6;J 1 on 994295 . * 34 212611 1 ^52 787389 16103 98695 44 17 207679 uy i on 994274 ^4 3X 213405 lo 132 786595 16132 98690 43 18 208452 -Izy/ 1 OQ 994254 o 3K 214198 i qo 785802 16160 98686 42 19 209222 izo 1 OQ 994233 o 3K 214989 iOAl i qo 735011 16189 98681 41 20 209992 14O 994212 O 3K 215780 LiKt 784220 16218 98676 40 21 9.210760 1 OQ 9.994191 O 3K 9.216568 1 ^1 10.783432 16246 98671 39 22 211526 l^O 994171 O 3K 217356 lol i q-i 782644 16275 98667 38 23 212291 127 994150 O Se 218142 -lo.L 781858 16304 98662 37 24 213055 127 994129 .O 3c 218926 i qn 781074 16333 98657 36 25 213818 127 994108 Q 3E 219710 loU i qn 780290 16361 98652 35 26 27 214579 215338 127 127 994087 994066 .O 3.5 3K 220492 221272 1OU 130 i qo 779508 778728 1639098648 16419198643 34 33 28 216097 126 994045 .O 3C 222052 AoU 1 qn 777948 16447 98638 32 29 216854 126 994024 .O 3e 222830 loU 10Q 777170 16476 98633 31 30 217609 126 1 OI 994003 .O 5 223606 *j& 129 776394 16505 98629 30 31 9.218363 1.4O 1 OX 9.993981 o . o K 9.224382 129 10.775618 16533 98624 29 32 219116 Izo 1 OK 993960 O . c> q K 225156 129 774844 16562 98619 28 33 219868 l^O 1 OX 993939 O . U 35 225929 129 774071 16591 98614 27 34 220618 1*0 1 OX 993918 . o 3K. 226700 128 773300 16620 98609. 26 35 221367 ItiO 1 O" 993896 . O 3ft 227471 128 772529 16648 986.04 25 36 222115 iVO 1 O/< 993875 o 3c 228239 128 771761 16677 98600. 24 37 222861 1Z4 993854 . O 3 229007 770993 16706 98595 23 38 223606 124 1 f)A 993832 .0 3(\ 229773 127 770227 16734 98590 22 39 224349 Iz4 993811 .0 3f; 230539 127 769461 16763 98585 21 40 225092 124 1 OQ 993789 .D 3c 231302 768698 16792 98580 20 41 9.225833 l!2o 1 OO 9.993768 .D 3f; 9.232065 197 10.767935 16820 98575 19 42 226573 Izd 993746 .O 3f! 232826 1.4 i 197 767174 16849 98570 18 43 227311 123 1 OQ 993725 .O q R 233586 1 ~ i 126 766414 16878 98565 17 | 44 45 228048 228784 l^O 123 993703 993681 O U 3.6 3R 234345 235103 126 126 765655 764897 16906 98561 16935 98556 16 15 46 229518 993660 D 3fJ 235859 126 764141 16964 98551 14 r 47 230252 993638 . D 3 236614 1 Oft 763386 16992 98546 13 48 230984 122 993616 .0 3^; 237368 AjMJ 1 OR 762632 17021 98541 12 49 231714 122 993594 .D 317 238120 1^0 1 OK 761880 17050 98536 11 50 232444 122 993572 . / 317 238872 l^o 1 OK 761128 17078 98531 10 51 9.233172 121 9.993550 / 3rr 9.239622 1^0 1 OK 10.760378 17107 98526 9 52 233899 121 994528 1 37 240371 l-^O 125 759629 17136 98521 8 53 234625 101 993506 i 37 241118 758882 17164 98516 7 54 235349 1*1 993484 i 3rf 241865 758135 17193 98511 6 55 236073 120 993462 1 3rj 242610 1 OA 7573GO 17222 98506 5 56 236795 120 1 oA 9J3440 - I 37 243354 Iz-l 756646 17250 98501 4 57 237515 izU 993418 I 37 244097 755903 17279 98496 3 58 238235 993396 . ( 244839 1OQ 755161 17308 98491 2 59 238953 120 1 1 (i 993374 3.7 3rr 245579 !o 754421 17336 98486 1 60 239370 iiy 993351 . / 246319 753681 17365 98481 Cosine. Sine. Cotang. Tang. IN. cos. N.sine. ~ T ~ 80 Degrees. TABLE II. Log. Sines and Tangents. (10) Natural Sines. 31 / Sine. D. 10" Cosine. D. 10" Tang. D. 10" Cotang. N.sine. N. cos. 9.239670 nq 9.993351 3ij 9.246319 10.753681 17365 98481 60 1 240386 iiy 993329 . / 3n 247057 1 OQ 752943 17393 98476 59 2 241101 119 1 1 Q 993307 .7 3>7 247794 12o 752206 17422 98471 58 3 241814 i iy 993285 , / 248530 751470 17451 98466 57 4 242526 119 993262 3.7 249264 122 750736 17479 98461 56 5 243237 118 993240 3.7 249998 122 750002 17508 98455 55 6 243947 118 993217 3.7 250730 122 749270 17537 98450 54 7 244656 118 993195 3.8 251461 122 748539 17565 98445 63 8 245363 118 993172 3.8 , 252191 122 747809 17594 98440 52 9 246069 118 993149 3.8 i 252920 121 747030 17623 98435 51 10 246775 117 993127 3.8 30 253648 121 746352 17651 98430 50 11 9.247478 117 9.993104 .0 9.254374 121 10.745626 17680 98425 49 12 248181 117 993081 3.8 30 255100 121 744900 17708 98420 48 13 248883 117 993059 . 30 255824 121 744176 17737 98414 47 14 249583 117 993036 .O 30 256547 120 743453 17766 98409 46 15 250282 116 993013 .O 30 257269 120 OA 742731 17794 98404 45 16 25W980 116 992990 .0 257990 \20 742010 7823 98399 44 17 251677 116 992967 3.8 30 258710 120 741290 17852 98394 43 18 252373 116 992944 .0 259429 120 740571 17880 98389 42 19 253067 116 992921 3.8 30 260146 120 739854 17909 98383 41 20 253761 116 992898 .0 260863 119 739137 17937 98378 40 21 9.254453 115 9.992875 3.8 30 9.261578 119 10.738422 17966 98373 39 22 255144 115 992852 .0 30 262292 119 737708 17995 98368 38 23 255834 115 992829 .0 3fk 263005 119 736995 18023 98362 37 24 256523 115 992806 . y 3r\ 263717 119 736283 18052 98357 36 25 257211 115 992783 . y 3q 264428 1 1S 735572 18081 98352 35 26 257898 992759 . y 3q 265138 llo 1 1S 734862 18109 98347 34 27 258583 : 992736 . y 3q 265847 J lo 1 18 734153 18138 98341 33 28 259268 992713 . y 3q 266555 llo 733445 18166 98336 32 29 259951 992690 . y 3q 267261 1 18 732739 18195 98331 31 30 1 31 260S33 9.261314 113 992666 9.992643 . y 3.9 267967 9.268671 1 1O 117 732033 10.731329 18224 98325 18252,98320 30 29 3-2 261994 113 992619 3.9 3q 269375 117 730625 18281198315 28 33 262673 992596 . y 3q 270077 729923 18309|98310 27 34 263351 992572 . y Q q 270779 729221 18338198304 2o 35 36 264027 264703 113 992549 992525 o. y 3.9 271479 272178 116 728521 727822 18367198299 18395198294 25 24 37 265377 112 110 992501 3.9 30. 272876 116 727124 1842419828^ 23 38 2436051 11^ 992478 . y A A 273573 116 1 1 a 726427 18452^98283 22 39 266723 992454 ^ . u 4. n 274269 I lo I | 726731 1.848198277 31 40 41 267395 9.268065 112 992430 9.992406 ^ . u 4.0 4f\ 274964 9.275658 l lo 116 725036 10.724342 18509:98272 18538 J 98267 ^0 19 42 268734 . 992382 . 4f\ 276351 115 723649 18567198261 18 43 269402 992359 .0 4f\ 277043 115 722957 18595 98256 I/ 44 270069 Ill 992335 .V 4f\ 277734 115 722266 18624 ! 9825U 16 45 270735 111 992311 .0 278424 115 721576 1865298245 15 46 271400 m - 992287 4.0 4 A, 279113 115 1 | e 720887 18681 98240 14 47 272064 . 1 1 n 992263 . U 4f\ 279801 lit) 720199 1871098234 13 48 272726 11U 110 992239 . O 4(\ 280488 : 719512 1873898229 12 49 50 51 52 273388 274049 9.274708 275367 llw 110 110 110 1 10 992214 992190 9.992166 992142 . 4.0 4.0 4.0 4 281174 281858 9.282542 283225 : 114 114 114 718826 718142 10.717458 716775 18767 98223 18795:98218 1882498212 1885298207 11 10 9 8 53 276024 i J-v i no 992117 ^ . \J 283907 716093 1888198201 7 54 276681 iuy 1 f \Q 992093 . 284588 715412 18910 ! 98196 6 55 277337 iuy 1 0Q 992069 . 4 285268 i 714732 18938198190 5 56 277991 lUi7 1 OQ 992044 4 285947 714053 1896798185 4 57 278644 J.U*7 1 0Q 992020 4 286624 713376 1899598179 3 58 59 279297 279948 ll/*7 109 1 AQ 991996 991971 4. 4 287301 287977 113 712699 712023 1902498174 19052 98168 2 1 60 280599 iuo 991947 288652 711348 19081 98163 Cosine. Sine. Co tang. Tang. N. eos. N.nne. 79 Degrees. 32 Log. Sines and Tangents. (11) Natural Sines. TABLE II. i Sine. 1 [>. 10" Cosine. U. 10" Tally;. ID. iu ixitang. N. sine. N. cos. J. 280599 "1 AQ 9.991947 41 9.288652 no 10.711348 19081 98163 60 1 281248 lUo 991922 . l 289326 z 710674 19109 98157 59 2 281897 108 991897 4.1 289999 710031 19138 98152 68 3 282544 108 991873 4. 1 290671 709329 19167 98146 57 4 283190 108 991848 4.1 291342 708058 19195 98140 56 6 28383f> 108 1 A 7 991823 4. 1 41 292013 707987 19224 98135 55 6 284480 10 / 1 n7 991799 . i 41 292682 707318 19252 J8129 54 7 285124 10 / 1 A 7 991774 . i A 9 293350 706650 19281 J8124 53 8 285766 ID/ 991749 4. -^ , o 294017 705983 19309 98118 52 9 286408 107 991724 4.2 . o 294684 111 705316 19338 J8112 51 10 287048 107 991699 4.2 295349 111 704651 19366 98107 50 1 1 ). 287687 107 3. 991674 4.2 9.296013 111 TO. 703987 19395 98101 49 12 288326 106 991649 4.2 40 296677 111 1 1 703323 19423 98096 48 13 288964 106 991624 . 297339 1 1U 702661 19452 98090 47 14 289600 106 991599 4.2 40 298001 110 110 701999 19481 98084 46 15 290236 106 991574 .* . o 298662 1 1U 1 1 A 701338 19509 98079 45 16 290870 106 991549 4.2 299322 110 700678 19538 98073 44 17 291504 106 991524 4.2 299980 110 700020 19566 )8067 43 18 292137 105 991498 4.2 40 300638 110 1 OQ 699362 19595 )8061 42 19 292768 105 991473 .* 301295 iuy 698705 19623 98056 41 20 293399 105 991448 4.2 A O 301951 109 1 AO 698049 19652 98050 40 21 9.294029 105 9.991422 4.^ 40 9.302607 iuy 1 OQ 10-697393 19680 98044 39 22 294658 105 1 AT* 991397 .* 40 303261 iuy 1 AO 696739 19709 98039 38 23 295286 ll/O 1 (\A 991372 .<& 40 303914 iuy 1 AO 696086 19737 98033 37 24 295913 104 1 A 4 991346 .*> 40 304567 1LM 109 695433 19766 98027 36 25 296539 104 1 A/1 991321 .0 40 S05218 lU^ 108 694782 19794 J8021 35 26 297164 104 1 A/I 991295 .0 40 305869 J.UO 1 AQ 694131 19823 38016 34 27 297788 104 1 A/I 991270 .0 40 306519 1U<^ 1 08 . 693481 19851 98010 33 28 298412 104 1 f\ A 991244 .0 40 307168 luo 108 692832 19880 98004 32 29 299034 104 1 A4 991218 .0 40 307815 iUO 1 AQ 692185 19908 97998 31 30 299655 104 991193 .0 40 308463 1UO 1 08 691537 19937 97992 30 31 9.300276 103 9.991167 .0 40 9.309109 1UO 1 07 10-690891 19965 97987 29 32 300895 103 991141 .0 40 309754 1U * 1 07 690246 199&4 97981 28 33 301514 103 991115 .O 40 310398 1U / 1 r\7 689602 20022 97975 27 34 302132 103 991090 ;O 40 311042 llf I 1 O" 688958 2X)051 97969 26 35 302748 103 991064 .0 40 311685 1U< 1 f>7 688315 20079 97963 25 36 303364 103 1 AO 991038 .O 40 312327 1U< 107 687673 20108 97958 24 37 303979 WZ 1 no 991012 . o 40 312967 lUi 1 07 687033 20136 97952 23 38 304593 l(}Z 1 AO 990986 * o A Q 313608 i \ft 1 0fi 686392 20165 97946 22 39 305207 l\) 1 AO 990960 *1, O A Q 314247 AUO 106 685753 20193 97940 21 40 305819 \\K 1 AO 990934 TC O 4 A 814885 1UO 106 685115 20222 97934 20 41 9.306430 lUSe 1 AO 9 990908 . ^fc 4 A 9.315523 1UD 10I-! 10-684477 20250 97928 19 42 307041 lift 1 09 990882 ^* 4 4 316159 J.UO 106 683841 20279 97922 18 43 307650 k\Ke 990855 4 A 316795 1 AM 683205 2030? 97916 17 44 308259 101 990829 % * 4 A 317430 1UO 1 Aii 682570 20336 97910 16 45 308867 101 1 O1 990803 ^ 4 A 318064 1UO 1 O r ) 681936 20364 97905 15 46 309474 1 Vrl 1 A1 990777 . *x A A 318697 1UO 105 681303 20393 97899 14 47 310080 1U1 1 A1 990750 ! * A A 319329 i n^ 680671 20421 97893 13 48 310685 1O1 1 A1 990724 T: * ^t A A 319961 1UO 1 0^ 680039 20450 97887 12 49 311289 1UI 990697 4. ^ 320592 JIUO 1 AX 679408 20478 97881 11 50 311893 100 990671 4.4 321222 lUo 1 f\^ 678778 2050? 97875 10 51 9.312495 100 i on 9.990644 4.4 A 4 9.321851 lOo 105 10-678149 20535 97869 9 52 313097 1UU i An 990618 rr . TC 4- 4 322479 677521 20563 97863 8 63 313698 1UU i on 990591 4t . ^ A 4 323106 676894 20592 97857 7 54 314297 1UU 1 1 M t 990565 . ^t A A 323733 104 676267 20620 97851 6 55 314897 1UU i t\i\ 990538 Q . Tt 4 A 324358 lU^i 1 04 675642 20649 97845 5 56 315495 1UU 1 AA 990511 TC 4 K 324983 1UT: 1 04 675017 2007? 97839 4 57 316092 1UU QU 990485 ^t O d. r> 325607 AUTC 674393 20706 97833 3 56 316689 yy QU 990458 4 k O A ft 326231 673769 20734 97827 2 59 317284 yy ou 990431 4 . O A & 326853 1 O-t 673147 20763 97821 1 60 317879 yy 990404 *. O 327475 lU-t 672525 20791 97815 ~CosiiieT~ Sine. Co tang. Tang. || N. cos N.rfne 78 Degrees. TABLE II. Log. Sines and Tangents. (12) Natural Sines. 33 Sine. D. 10" Cosine. D. 10" Tang. D. 10" Cotang. jN . sine. i>, CO*. 9.317879 on n 9.990404 4e 9.327474 1 no 10.672526 20791 97815 60 1 318473 yy . u 98.8 990378 .0 4 K, 328095 LUJ 1 0*^ 671905 20820 97809 59 2 319.W6 f kQ *7 990351 4. O 328715 LU<j 671285 20848 97803 58 3 319658 9o . 7 98.6 990324 4.5 4K 329334 103 1 ? 6 1 0-Jou 20877 97797 57 4 320249 no A 990297 : . O 329953 i j.> 670U47 20905 97791 56 5 320840 9o.4 98 3 990270 4.5 4K 330570 103 1 QQ 669430 20933 97784 55 6 321430 98 2 990243 . o 4K 331187 iUo i no 668813 20962 97778 54 7 322019 oVo 990215 O 4K 331803 lUo 1 09 668197 20990 97772 53 8 322607 Jo . U 07 q 990188 O 4K 332418 1U4 i no 667582 2101997766 52 9 323194 J I . y Q7 7 990161 . O 4K 333033 1U-* 1 09 666967 2104797760 51 10 323780 y i . i Q7 (\ 990134 . O 4K 333646 iinfl 1 09 666354 2107697754 50 11 9.324366 J i . O 07 5 9.990107 . O 4fi 9.334259 L(J& i no 10.665741 211049774S 49 12 324950 J i . O Q7 1 990079 :. O 4 334871 1 \j 1 AO 665129 2113297742 48 13 325534 y / . o Q7 9 990052 . o 4f! 335482 IU-4 i no 664518 2116197735 47 14 326117 y / . z O7 t\ 990025 . O 4(\ 336093 LU-6 1 fkO 663907 2118997729 46 15 326700 y i \J Q.IA Q 989997 . o 4c 336702 iUDI 663298 21218 97723 45 16 327281 yo . y 96.8 989970 . o 4.6 337311 1 01 662689 2124697717 44 17 327862 Qfi fi 989942 A C 337919 iivl 1 ni 662081 2127597711 43 18 328442 yo . o 989915 4 . D 4r 338527 lUi 661473 21303 97705 42 19 329021 96. 5 989387 .O 4 339133 101 660867 2133197698 41 20 329599 96.4 9K <2 989860 .0 A fi 339739 101 660261 2136097692 40 21 9.330176 o . -^ 9.989832 4 . O 4" ! 9.340344 10.659656 21388 97686 39 22 330753 96. 1 989804 .O 340948 101 659052 2141797680 38 23 331329 96. 9~ Q 989777 4. 6 4fj 341552 101 i nn 658448 2144597673 37 24 331903 O . u O~ ry 989749 . o 4ry 342155 1UU 657845 2147497667 36 25 332478 yo. / QX & 989721 . 1 A 7 342757 100 i an 657243 2150297661 35 26 333051 yo . o QX 4 989693 TT. 1 A 7 343358 1UU 1 00 656642 2153097655 34 27 333624 yO 4: 989665 4. i 4rf 343958 \.\J\) 656042 21559 97648 33 28 334195 95.3 QX O 989637 .7 4rr 344558 100 1 C\(\ 655442 21587:97642 32 29 334766 yo. z QX A 989509 . 1 47 345157 1UU i i\i\ 654843 21616 ! 97636 31 30 335337 yj . \) Ql Q 989582 . t 47 345755 1UU 1 (\ \ 654245 21644 97630 30 i 31 9.335906 y-. y Q 1 W 9.989553 * i A 7 9.346353 1U*J OQ A 10.653647 2167297623 29 32 336475 y- . o Q 1 M 989525 4.1 4. 7 346949 \J J H QO ^ 653051 2170197617 28 33 337043 y^t . o Q/1 X. 989497 4 . / A 7 347545 yy c: GO O 652455 2172997611 27 34 337610 y4 . o Q.1 /I 989469 4. 1 A 7 348141 yy . >i GO 1 651859 2175897604 26 35 338176 y-i . 4 Q 1 Q 989441 4 . / 47 348 ?35 yy . i GO f 651265 21786:97598 25 36 338742 y . o uA i 989413 . / 47 349329 yy . v 98. 650671 2181497592 24 37 339306 y4. i 989384 . * 4rr 349922 9.j 17 650078 21843 ! 97685 23 38 339871 94. no (i 989356 . / 4rj 350514 o . * O-2 ^i 649486 21871 97579 22 39 340434 yo.y qq 7 989328 :. / 47 351106 yo . u QS fi 648894 21899 97573 21 40 340996 yo. t QO C- 989300 . / 47 361697 C7O . O UW Q 648303 21928 97566 20 41 9.341558 yo . o nQ Pi 9.989271 . / 47 9.352287 yo . o *IP 9 10.647713 21956 97560 19 42 342119 yo . O nQ A 989243 . / A 7 352876 i7O . A QL? I 647124 21985 97553 18 43 44 342679 343239 \jo . 4 93.2 f\Q 1 989214 989186 4 . / 4.7 4r* 353465 354053 yo . i 98.0 07 o 646535 645947 22013 97547 22041197541 17 16 45 343797 9<J- 1 989157 . / 4.7 354640 y / . y ()- 7 645360 22070 97534 15 46 344355 93 . rkO Q 989128 . / A ft 355227 %M * I 07 644773 22098 97528 14 47 48 344912 345469 9-^.y 92.7 Q9 fi 989100 989071 4.o 4.8 4C 355813 356398 y / . o 97.5 C)7 ,d 644187 | 22126 643^02 22155 97521 97515 13 12 49 346024 y^ . o fiO K 989042 . O A Q 356982 y < . T 07 q 643018 22183 97508 11 50 346579 y^. o nO /I 989014 4 . O 4 357566 y * . o Q7 1 642434 22212 97502 10 51 52 9.347134 347687 y/* . 4 92.2 /v~> -i 9.988985 988956 . O 4.8 4U 9.358149 358731 y / . i 97.0 OH Q 10.641851 22240 641269) 122268 97496 97489 9 8 53 348240 92.1 988927 . O 40 359313 yo . y or; Q 640687 22297 97483 7 54 34879-2 92. 988898 . o 4Q 359893 yu . o QO n 640107 3232597470 6 55 56 57 349343 349893 350443 91 .9 yl.7 91.6 988869 988840 988811 .0 4.8 4.8 40 360474 361053 361632 y o , / 96.6 96.5 Q; r 639526 638947 638368 22353 | 22382 22410 97470 97463 97457 5 4 3 68 350992 91.5 988782 .y 4n 362210 yo.o f\f\ o 637790 22438 97450 2 59 351540 91 .4 988753 .9 362787 yo.z ot; 1 637213 22467 97444 1 60 352088 91.3 988724 4.9 363364 yt>. i 636636 22495 97437 Cosine. Sine. Cotang. Tang. N. cos. N.sine. 77 Degrees. 34 Log. Sines and Tangents. (13) Natural Sines. TABLE 11. ~i Sine. D. 10" Cosine. D. 10" Tang. D. 10" Cotang. i N.sine_ N. cos. 1 9.352088 352635 91.1 9.988724 988695 4.9 4f\ 9.363364 363940 96.0 (\K. O 10.636636 636060 22495 22523 97437 97430 60 59 2 353181 91 .0 988666 .y 4Q 364515 y>.y f^K. O 635485 22552 97424 58 3 353726 90.9 988636 .y 4(\ 365090 95.0 r\K. >7 634910 22580 97417 67 4 354271 90.8 988607 .y 365664 95.7 634336 22608 97411 66 5 354815 90.7 988578 4.9 366237 95.5 633763 22637 97404 65 6 355358 90.5 988548 4.9 366810 95.4 633190 22665 97398 54 7 355901 90.4 988519 4.9 40 367382 95.3 632618 22693 97391 53 8 356443 90.3 988489 .y 367953 95.2 632047 22722 97384 52 9 356984 90.2 988460 4.9 40 368524 95.1 631476 22750 97378 61 10 357524 90. 1 988430 .y 4Q 369094 95.0 630906 2277897371 60 11 9.358064 89.9 9.988401 .y 4Q 9.369663 94.9 f\A Q 10.630337 2280797365 49 12 358603 89.8 988371 .y 4Q 370232 94.8 629768 2283597358 48 13 14 359141 359678 89.7 89.6 988342 988312 ,y 4.9 5 A, 370799 371367 94.6 94.6 629201 628633 22863197351 22892197345 47 46 15 360215 89.6 988282 . U 5/\ 371933 94.4 628067 22920 973(38 45 16 360752 89.3 988252 .U Bf\ 372499 94.3 627501 22948 97331 44 17 361287 89.2 on 1 988223 .U 5 A 373064 94.2 f\A 1 626936 22977 97325 43 18 361822 89. 1 988193 . U 5f\ 373629 94.1 626371 23005 97318 42 19 362356 89.0 QQ Q 988163 ,u 5n 374193 94.0 riQ O 625807 23033 97311 41 20 362889 88. y QQ Q 988133 . U 5 A 374756 9o.y r\O C 625244 23062 97304 40 21 9.363422 88. O QQ rj 9.988103 .U 5 A 9.375319 9o.o 10.624681 23090 97298 39 22 363954 88. / 988073 .U 6 A. 376881 93.7 624119 23118 97291 38 23 364485 88.5 QQ A 988043 .U 5 A 376442 93.5 rvO A 623558 23146 975284 37 24 365016 00.4 QQ Q 988013 .U 5 A 377003 93.4 622997 23176 97278 36 25 365546 OO.O 987983 . U 6f\ 377563 93.3 622437 23203 97271 35 26 366075 88.2 987953 .0 5f\ 378122 93.2 621878 23231 97264 34 27 366604 88.1 QQ A 987922 .0 5 A 378681 93.1 AQ A. 621319 23260 97257 33 28 367131 OO.U 07 Q 987892 U 5 A 379239 9o. U nO O 620761 23288 97251 32 29 367659 o / .y o^y >7 987862 . U 5{\ 379797 9-^.y 620203 23316 97244 3] 30 368185 87.7 OT C. 987832 .0 51 380364 92.8 619646 23345 97237 30 31 9.368711 87. o 9.987801 .1 5-1 9.380910 92.7 10.619090 23373 97230 29 32 369236 87.5 987771 .1 6-f 381466 92.6 618534 23401 97223 28 33 369761 87.4 987740 .1 54 382020 y2 5 617980 23429 97217 27 34 370285 87.3 987710 .1 61 382575 92.4 617425 23458 97210 26 35 370808 87.2 81 i 987679 . 1 51 383129 92.3 616871 23486 97203 25 36 371330 7.1 987649 . 1 5-1 383682 92.2 616318 23514 J7 96 24 37 371852 87.0 987618 .1 51 384234 93.1 615766 23542 97189 23 38 372373 86.9 987588 . 1 61 384786 92.0 615214 23571 97182 22 39 372894 86.7 987557 . 1 6] 385337 91.9 0^4663 23599 97176 21 40 373414 86.6 987526 . 1 51 385888 91.8 614112 23627 97169 20 41 9.373933 86.5 9.987496 . 1 51 9.386438 91.7 10.613562 23656 97162 19 42 374452 86.4 987465 . 1 51 386987 91.5 613013 23684 97155 18 43 374970 86.3 987434 . 1 51 387536 91.4 612464 23712 97148 17 44 375487 86.2 oa 1 987403 . 1 59 388084 91.3 611916 23740 97141 16 45 376003 OO. 1 QG 987372 . 5 2 388631 91 .2 01 1 611369 23769 97134 15 46 376519 OO. U or Q 987341 K*9 389178 yi . i 01 n 610822 23797 97127 14 47 377035 oo y 0** Q 987310 *-*.-* 60 389724 yi . u 610276 23825 97120 13 48 377549 00.0 987279 . Z 6c\ 390270 90.9 609730 23853 97113 12 49 378063 85.7 OK a 987248 .z 6C\ 390815 90.8 609185 23882 97106 11 50 378577 85 .0 987217 ,2> 5n 391360 90.7 60S640 23910 97100 10 51 9.379089 85.4 OK O 9.987186 .2 50 9.391903 90.6 10.608097 23938 J7093 9 52 379601 bo.o OK O 987155 .2 392447 90.5 607553 23966 970H6 8 53 380113 85.^ 987124 5.2 392989 90.4 607011 23995 97079 7 54 380624 85. 1 987092 6.2 393531 90.3 606469 24023 97072 6 55 381134 85.0 987061 6.2 394073 90.2 605927 24051 97066 6 56 381643 84.9 987030 6.2 394614 90.1 605386 24079 9-, 053 4 57 382152 84.8 986998 6.2 395154 90.0 604846 24108 97051 3 58 382661 84.7 84. fi 986967 5.2 59 395694 89.9 QO Q 604306 241 3b 97044 2 59 383168 o4 * D R4. ^ 986936 . & 59 396233 oy ,o QO *7 603767 24164 97037 1 60 383675 o< Q 986904 . w 396771 oy . i 603229 24192 97030 Cosine. Sine. Cotang. Tang. > N. cos. N.sine. " 76 Degrees. TABLE II. Log. Sines and Tangents. (14) Natural Sines. 35 Sine. D. 10 Cosine. D. 10" Tang. D. 10 Co tang. N. sine N. cos. 9.383675 QA A 9.986904 59 9.396771 10.603229 24192 97030 60 1 2 384182 384687 o4. T 84.3 QI 9 986873 986841 . 5.3 50 397309 397846 89^6 QQ C 602691 602154 1 24220 ! 24249 97023 1 59 97015 58 3 385192 o-* . A 81 1 986809 . o 50 398383 oy . o 901617 24277 97008 57 4 385697 Ox. 1 810 986778 o 50 398919- n o 601081 24305 97001 56 5 386201 o . vJ 986746 > o 399455 OJ.O 600545 24333 96994 55 6 386704 83 < 9 986714 5 .3 399990 89.2 600010 2436296987 54 7 387207 83 . 8 986683 5 .3 400524 89.1 699476 2439096980 53 8 387709 83 . 7 Qq f! 986651 6.3 50 401058 89.0 QQ Q 698942 2441896973 52 9 388210 oo . O QO K 986619 . O 50 401591 oo. y QQ Q 698409 24446 96966 51 10 388711 oo . O 986587 . O 402124 OO.O 597876 24474 96959 50 11 9.389211 83. 4 Qo o 9.986555 6. 3 9.402656 88.7 10.597344 24503 96952 49 12 389711 oo.o 83 2 986523 5.3 5 3 403187 88.6 QQ K 596813 24531 96945 48 13 390210 83 1 986491 5 3 403718 OO . O QQ 4 596282 24559 96937 47 14 390708 986459 404249 OO . T: 596751 24587 96930 46 15 391206 83. 986427 5.3 404778 88.3 595222 24615 96923 45 16 391703 SZ.O 986395 5.3 405308 88.2 694692 24644 96916 44 17 392199 32.7 986363 6.3 405836 88.1 594164 24672 96909 43 18 392695 32. 6 986331 5.4 406364 88.0 693636 24700 96902 42 19 393191 32.5 986299 5.4 406892 87.9 693108 24728 96894 41 20 393685 an q 986266 6.4 5 A 407419 87.8 87 7 692581 24756 96887 40 21 9.394179 JA O 9.986234 . 4 9.407945 a 1 . i 10.592055 24784 96880 39 22 394673 32.2 986202 5.4 408471 87.6 591529 24813 96873 38 23 395166 32. 1 986169 6.4 408997 87.5 691003 24841 96866 37 24 395658 32.0 Q1 O 986-137 5.4 409521 87.4 690479 24869 96858 36 25 396150 oi.y Q1 Q 986104 5.4 410045 87.4 589955 24897 96851 35 26 396641 31.0 986072 5.4 410569 87.3 589431 24925 96844 34 27 397132 31.7 Ol ft 986039 5.4 411092 87.2 688908 24954 96837 33 28 397621 31.7 986007 6.4 411615 87.1 688385 24982 96829 32 29 398111 31.6 985974 5.4 412137 87.0 687863 25010 96822 31 30 398600 31.6 985942 5.4 412658 86.9 587342 25038 96815 30 31 9.399088 81.4 21 q 9.985909 6.4 6K. 9.413179 86.8 O? "*1 10.586821 25066 J6807 29 32 399575 51 . o 985876 . o 413699 bO 686301 25094 96800 28 33 400062 31 .2 01 1 985843 6.5 6K 414219 86.6 Oi K 585781 25122 J6793 27 34 400549 51.1 81 fl 985811 . o K. K 414738 oo. 585262 26151 J6786 26 35 401035 51 . U 985778 o. o 415257 36 . 4 684743 25179 96778 25 36 401520 30.9 985745 5.5 5 p. 415775 36.4 O/ O 584225 25207 J6771 24 37 402005 ?n 7 985712 .0 6K 416293 00.0 583707 25235 96764 23 38 402489 30.7 985679 . O 416810 b 6.2 683190 25263 96756 22 39 402972 30.6 985646 5.6 417326 86.1 582674 25291 96749 21 40 403455 80.6 en A 985613 5.5 5K 417842 86.0 582158 25320 96742 20 41 42 9.403938 404420 5U.4 80.3 20 o 9.985580 985547 . O 6.6 6C 9. 418358 4188/3 35!8 0.581642 681127 25348 25376 96734 96727 19 18 43 404901 5U A 985514 . O 3K 41938 / Bo . oer d 680613 25404 96719 17 44 45 405382 405862 80.0 170 Q 985480 985447 6.6 419901 420415 So.b 85.6 Q" f^ 580099 579585 25432 25460 J6712 96705 16 15 46 406341 /y . y 7Q 8 985414 5{? 42092 / 8O.O QK A 579073 25488 96697 14 47 406820 /y .o fit\ ri 985380 .0 6f> 421440 OO.4 8 1 - Q 678560 25516 96690 13 48 407299 IV. i "7O fl 985347 .0 6? 421952 O.o QK O 678048 25545 96682 12 49 407777 /y.o rf(\ er 985314 . o 5> 422463 85. -* 8-> i 577537 25573 96675 11 50 408254 /y .0 986280 .0 5 422974 O. 1 8" n 677026 25601 96667 10 61 .408731 79.4 3.985247 .O 3.423484 o.u 0.576616 25629 96660 9 62 53 409207 409682 /9.4 79.3 985213 985180 6 . 6 6.6 5 C 423993 424503 84.9 84.8 8J. 8 676007 675497 | 25657 25686 96653 96645 8 7 64 410157 Q 1 985146 . o 5(i 425011 o4.o 674989 25713 J6638 8 55 410632 r n 985113 . o 425519 84. 7 674481 25741 J6630 5 66 411106 O W Q 985079 H 426027 QA Pi 673973 25766 96623 4 67 411579 /o . y 8Q 985045 5f! 426534 o4.O QA A 673466 1 25798 96615 3 68 412052 . O 70 7 985011 .O 427041 o4.4 QA Q 672959 25826 96608 2 59 412524 O . / rrQ 984978 5? 427547 84 ,o Q -1 Q 672453 25854 96600 1 60 412996 /O.D 984944 .O 428052 o4.o 671948 25882 96593 Cosine. Sine. Co tang. Tang. N. cos. N.pine. i 75 Degrees. 36 Log. Sines and Tangents. (15) Natural Sines. TABLE II. Sine. I). 10" Cosine. D. 10" Tang. D. 10" Cotaug. N. sine. N. cos. 9.412996 9.984944 5Hf 9.428052 C A O 10.571948 25882 96593 60 1 413467 78.5 984910 .7 428557 o4.ii Q A 1 571443 25910 96585 59 o 413938 78.4 984876 5.7 429062 o4. 1 O A A 5709381,25938 96578 58 3 414408 78.3 170 Q 984842 5.7 5w 429566 o4.U QO q 570434 25966 96570 57 4 414878 /o . o i- C C 984808 . / 517 430070 oo .y CO C 569930 25994 96562 56 6 415347 t*i .* 984774 . / 430573 oo o QO Q 669427 26022 96555 55 6 415815 78.1 984740 5.7 431075 oo .0 QO *7 668925 26050 96547 54 7 416283 78.0 984706 5.7 431577 oo . i oo r; 568423 26079 96540 53 8 416751 77.9 984672 5.7 432079 OO . O oo r 667921 26107 96532 52 9 417217 77.8 984637 5.7 432580 oo .0 OQ A 567420 26135 96524 51 10 417684 77.7 984603 5.7 433080 O0.4 OO Q 566920 26163 96617 50 11 9.418150 77.6 9.984569 5.7 9.433580 OO .0 10.566420 26191 96509 49 12 418615 77.5 984535 5.7 434080 83 .2 665920 26219 96602 48 13 419079 77.4 984500 6.7 434579 83.2 00 1 565421 26247 96494 47 14 419544 77.3 riry o 984466 6.7 435078 oo . 1 QO A 664922 26275 96486 46 15 420007 77 .0 984432 6.7 435576 oo . U 664424 26303 96479 45 16 420470 77.2 984397 5.8 50 436073 82.9 QO ft 663927 26331 96471 44 17 420933 77. 1 984363 .0 436670 O^ O 663430 26359 96463 43 18 421395 77.0 984328 6.8 437067 82.8 QO *7 562933 26387 96456 42 19 421857 76 .9 984294 5.8 50 437563 OA . 1 OO ft 662437 26415 96448 41 20 422318 76.8 rir- ri 984259 .0 50 438059 oZ.o QO 5 661941 26443 96440 40 21 9.422778 1(3 . i 9.984224 .O 9.438664 o-w t> QO A 10.561446 26471 96433 39 22 423238 76.7 n{* fy 984190 5.8 5Q 439048 O/i.4 QO . 660952 26500 96425 38 23 423697 7b.o 984155 .0 439543 O/w . O QO O. 660457 26528 96417 37 24 424156 76.5 i*lC A 984120 5.8 50 440036 O^.o 00 O 559964 26556 96410 36 25 424615 76.4 984085 .0 440529 O-& . * QO 1 559471 26584 96402 35 26 425073 76.3 984050 5.8 441022 cw.l Q/- A 658978 26612 96394 34 27 425530 76.2 984015 5.8 441514 o2 . U Qi q 658486 26640 96386 33 28 425987 76.1 983981 5.8 50 442006 ol .y 01 Q 657994 26668 96379 32 29 426443 76.0 983946 .0 50 442497 01 .y 01 Q 657503 26696 96371 31 30 426899 76.0 983911 .0 442988 ol .0 Q1 1 657012 26724 96363 30 31 9.427364 75.9 9.983876 5.8 9.443479 ol . i Q1 (^ 10.556521 26752 96355 29 32 427809 75.8 983840 5.8 443968 ol .t> O-l ? 556032 26780 96347 28 33 428263 75.7 983805 5.9 444458 ol .u Q1 K 655542 26808 96340 27 34 428717 75.6 983770 5.9 444947 ol .0 Q1 A 555053 26836 96332 26 35 429170 75.5 nK. A 983735 5.9 5 A 445435 ol .4 01 Q 554565 26864 96324 25 36 429623 75 .4 983700 .y 446923 O J. . O Q1 O 654077 26892 96316 24 37 430075 75.3 983664 5.9 446411 ol .< Q1 O 653589 26920 96308 23 38 430527 75.2 983629 5.9 446898 ol .^ Q1 1 653102 26948 96301 22 39 430978 75.2 983594 6.9 447384 ol . 1 Q1 A 652616 26976 96293 21 40 431429 75.1 983558 5.9 447870 ol .U QA A 552130 27004 96285 20 41 9.431879 75.0 9.983523 5.9 9.448356 yo.y 10.551644 27032 96277 19 42 432329 74.9 983487 6.9 448841 80.9 C>f\ O 651159 27060 96269 18 43 432778 74.9 n A Q 983452 6.9 6t\ 449326 ou.O QA 7 550674 27088 96261 17 44 433226 74. o 1 4 n 983416 . y 5 A 449810 oU . i OA 550190 27116 96253 16 45 433675 74.7 n A r* 983381 .y 50 460294 ou . o QA f] 549706 27144 96246 15 46 434122 74 b 983345 .y 5 A 450777 OU . O QA K 649223 27172 96238 14 47 434569 74.5 983309 .y 5(-\ 451260 OU . O QA A 648740 27200 96230 13 48 49 435016 435462 74.4 74.4 "7/1 o. 983273 983238 .y 6.0 6 A 451743 452225 OU . ^ 80.3 QA 548257 547775 27228 27256 96222 96214 12 11 60 435908 /4 o *7 A O 983202 u f> n 452706 Ov . ^ QA O 547294 27284 96206 10 51 9.436353 /4 . 9.983166 O . U 9.453187 OU . * QA I 10.546813 27312 96198 9 52 436798 74.1 rt A A 983130 6. 6 A 453668 OU. A QA A 646332 27340 96190 8 53 437242 /4. U 983094 . U 454148 OU . U I-7Q A 545852 27368 96182 7 54 437686 74-0 ryo A 983058 6.0 6 A 454628 /y .y 7Q Q 645372 27396 96174 6 55 438129 to y 983022 . U 455107 /y . y (7Q Q 644893 27424 96166 5 56 438572 73.8 982986 6.0 455686 7y .o r<A *7 644414 27452 96158 4 57 439014 73.7 982950 6.0 456064 /y . rr(\ R 643936 27480 96150 3 58 439456 73.6 982914 6.0 6f\ 456542 ?y .0 7Q fi 643458 27508 96142 2 59 439897 73 .6 982878 . 457019 lV v u net x 642981 27536 96134 1 60 440338 73.5 982842 6.0 457496 7y .0 542504 27564 96126 Cosine. Sine. Cotang. Tang. N. cos. |N. sine. / 74 Degrees. TABLE II. Log. Sines and Tangents. (16) Natural Sines. 37 Sine. D. 10" Cosine. D. 10" Tang. D. 10" Cotang. N. sine. N. cos.j 9.4t0338 70 A 9.982842 6O 9.457496 *7Q A 10.542504 27564 96126 60 ] 440778 i O 4 ryo o 982805 . U 457973 /y .4 542027 27592 96118 59 2 441218 /O . O 982769 6.0 458449 79.3 541551 27620 96110 58 3 441658 73 .2 982733 6. . 458925 79 .3 541075 27648 96102 57 4 442096 73 1 982696 6. 459400 79.2 540600 27676 96094 56 5 442535 73. 1 70 n 982660 6. 459875 79.1 7Q n 540125 27704 96086 55 6 442973 to . u 982624 . 460349 / y . u 539651 27731 96078 54 7 443410 72.9 f*-i 982587 6.1 460823 ^9.0 639177 27759 96070 53 8 443847 72.0 982551 6.1 461297 8.9 538703 27787 96062 52 9 444284 72.7 982514 6.1 461770 8.8 538230 27815 96054 51 10 444720 72.7 982477 6.1 462242 8.9 537758 27843 96046 50 11 9.445155 72.6 9.982441 6.1 9 462714 8.7 10.537286 27871 96037 49 12 445590 72.5 982404 6. 1 463186 8-6 536814 27899 96029 48 13 446025 72.4 79 T 982367 6. 1 61 463658 8-5 78 ^i 536342 27927 96021 47 14 446459 t Z . o 7Q o 982331 . 61 464129 /o O 170 A 535871 27955 96013 46 15 446893 f jltO 79 9 982294 . i 61 464599 IQ .4 170 O 535401 27983 96005 45 16 447326 i Z ^ 79 1 982257 . i 61 465069 /O ." *7Q Q 534931 28011 95997 44 17 447759 |3* A co n 982220 . R 9 465539 to <-* /7O 9 534461 28039 95989 43 18 448191 i Z . \J *7o n 982183 b.^ fi 9 466008 /O.xi TO 1 533992 28067 95981 42 19 448623 i ^ . \J n-t Q 982146 D.- 6O 466476 /o . i 533524 28095 95972 41 20 449054 71 .y rf-t fi 982109 .* 6O 466945 78-0 533055 28123 95964 40 21 9.449485 71 .0 ^1 *7 9 982072 .^ Go 9 467413 78.0 10.532587 28150 95956 39 22 449915 71. < rrt fl 982035 ,-<5 467880 77-9 n-fr O 532120 28178 95948 38 23 450345 71 .0 981998 6.2 60 468347 77-8 531653 28206 95940 37 24 450775 71.6 ~ , K 981961 .& r O 468814 77-8 531186 28234 95931 36 25 451204 71 .0 981924 6.* r> O 469280 77-7 530720 28262 95923 35 36 451632 71.4 71 *} 981886 6.^ 69 469746 77-6 530254 28290 95915 34 27 45*060 71 .<> r^-l O 981849 .4 69 470211 77-5 529789 28318 95907 33 28 452488 71 ."J *l 9 981812 .<* 69 470676 77-5 529324 28346 95898 32 29 452915 71 *7 1 1 981774 .s 60 471141 77-4 528859 28374 95890 31 30 453342 71.1 * i (\ 981737 .A 60 471605 77.3 628395 28402 95882 30 31 9.453768 71 .v frl A 9.981699 .^ 60 9 472068 77-3 10.527932 28429 95874 29 32 454194 /I .U -f A O 981662 .0 60 472532 77.2 527468 28457 95865 28 33 454619 7u. y r~n o 981625 .0 60 472995 77.1 627005 28485 95857 27 34 455044 7U.o rrA r7 981587 .0 60 473457 77.1 526543 28513 95849 26 35 455469 7U.7 981549 .0 473919 77.0 626081 28541 96841 25 36 455893 70.7 981512 6.3 474381 76.9 625619 28569 95832 24 37 456316 70.6 981474 6.3 474842 76.9 625158 28597 95824 23 38 456739 70.5 981436 -6.3 475303 76.8 524697 28625 96816 22 39 457162 70.4 r A A 981399 6.3 60 475763 76.7 524237 28652 95807 21 40 457584 7U.4 i-rA. Q 981361 .0 60 476223 76.7 623777 28680 95799 20 41 9.458006 70. 6 i~l\ O 9.981323 .0 60 9 476683 76.6 10 623317 28708 95791 19 42 458427 /U.xJ -M 1 981285 .0 60 477142 76.5 622858 28736 95782 18 43 458848 7U. 1 981247 .0 477601 76.5 622399 28764 95774 17 44 459268 70. 1 981209 6.3 478059 76.4 521941 28792 95766 16 45 459688 70. 981171 6.3 478517 76.3 621483 28820 95757 15 46 460108 69.9 r*O ft 981133 6.3 6 A 478975 76.3 r-yr r\ 621025 28847 95749 14 47 460527 by . o />Q O 981095 .4 6 A 479432 76.2 520568 28875 95740 13 48 460946 by . o /Q "7 981057 .4 6 A 479889 76.1 620111 28903 95732 12 49 461364 by. < 981019 .4 480345 76.1 619655 28931 95724 11 50 461782 69 .6 rCI Pi 980981 6.4 6 A 480801 76.0 519199 28959 95715 10 51 9.462199 by o ?n K 9.980942 .4 9 481257 75.9 10.518743 28987 95707 9 52 462616 by . o 980904 6.4 481712 75.9 618288 29015 95698 8 53 463032 69.4 ,.(\ 980866 6.4 482167 75.8 517833 29042 95690 7 54 463448 by . o 980827 6.4 482621 75.7 617379 29070 95681 6 55 463864 69.3 980789 6.4 483075 75.7 516925 29098 95673 5 56 464279 69.2 980750 6.4 483529 75.6 516471 29126 95664 4 57 464694 69. 1 /Q A 980712 6.4 483982 75.5 616018 29154 95656 3 58 465108 by . u f*o n 980673 6,4 484435 75.5 615565 29182 95647 2 59 465522 by. u r-Q O 980635 6.4 484887 75.4 515113 2920!- 95639 1 60 465935 bo. y 980596 6.4 485339 75.3 514661 29247 95630 Cosine. Sine. Cotang. Tang. N. cos. N.sine. 73 Degrees. Log. Sines and Tangents. (17) Natural Sines. TABLE II. Sine. D. 10 Cosine. D. 10 Tang. D. 10 Cotang. N. sine N.cos 9.465935 ftO o 9.980596 6 A 9.485339 7ft "i 10.514661 [ 29237 95630 60 1 466348 Do . o 980558 . 4 6 A 485791 /O . d 514209 I 29265 95622 59 2 466761 DO. 8 980519 . 4 486242 75. 2 513758 i 29293 95613 58 3 467173 68.7 nQ r> 980480 6.5 6 5 486693 75. 1 613307 29321 95605 57 4 467585 Do . o 980442 6K 487143 WK f\ 612857 29348 95596 56 6 6 7 467996 468407 468817 68.5 68.5 68.4 980403 980364 980325 . O 6.5 6.5 6C 487593 488043 488492 /o. U 74.9 74.9 r?A o 612407 611957 511508 29376 29404 1 29432 95588 95579 95571 55 54 53 8 469227 68.3 980286 . D 6^ 488941 /4.o 511059 29460195562 52 9 469637 68.3 980247 . O 6E 489390 74.7 W A W 610610 29487 95554 51 10 470046 68 . 2 980208 . o 6cr 489838 /4. / T A (Z 610162 29515 95545 50 11 9.470455 68. 1 9.980169 .O 6 5 9.490286 74. 10.509714 29543 95536 49 12 470863 68 . 980130 6K. 490733 w 4 e 609267 29571 95528 48 13 471271 68.0 980091 . o 6K. 491180 74. W.4 A 608820 29599 95519 47 14 471679 67.9 980052 . 6E 491627 74.4 *7 A A 608373 29626 95511 46 15 472086 67.8 980012 .O 6K 492073 74.4 W A 607927 29654 95502 45 16 472492 67.8 979973 . O 6K 492519 74. o n A o 507481 29682 95493 44 17 472898 67.7 />W 979934 . o 6 6 492965 74. d 74 Q. 607035 29710 95485 43 18 473304 O/ . O 979895 6(* 493410 <4 . ^ ^ 1 1 606590 29737 95476 42 19 473710 67.6 979855 . o 6f! 493854 74. 1 606146 29765 95467 41 20 474115 67. 6 979816 . O 6? 494299 A ( 605701 29793 95459 40 21 9.474519 67.4 9.979776 . O eft 9.494743 74*0 10.505257 29821 95450 39 22 474923 67.4 979737 . o 6f5 495186 /4 . v 70 Q 504814 29849- 95441 38 23 475327 67.3 979697 . o 6 495630 /d . y 504370 29876 95433 37 24 475730 67.2 979658 .0 6ft 496073 73. 8 70 7 603927 29904 95424 36 25 476133 67.2 979618 . o eft 496615 /d . / 70 7 603485 29932 95415 35 26 476536 67. 1 979579 . o eft 496957 /d . 1 603043 29960 95407 34 27 476938 67.0 979539 o eft 497399 70 ft 602601 29987 95398 33 28 477340 66.9 979499 . o 6> 497841 /d , O 502159 30015 95389 32 29 477741 66.9 979459 .0 eft 468282 73. 5 WO A 601718 30043 J5380 31 30 478142 66.8 979420 . o eft 498722 7d.4 501278 30071 95372 30 31 9.478542 66.7 9.979380 . o 6ft 9.499163 WO O 10.500837 30098 95363 29 32 478942 66.7 979340 . o 6ft 499603 7d.d 600397 30126 95354 28 33 479342 66.6 979300 o 67 500042 wo 9 499958 30154 95345 27 34 479741 66.5 979260 . / 67 600481 /d .2 wo 1 499519 30182 95337 26 35 36 480140 480539 66.5 66.4 979220 979180 . / 6.7 6rr 500920 501359 7d. 1 73.1 499080 498641 30209 30237 95328 95319 25 24 37 38 480937 481334 66.3 66.3 979140 979100 . / 6.7 6rr 501797 602235 73. 73.0 wO O 498203 497765 30266 30292 95310 J5301 23 22 39 481731 66.2 979059 . / 617 602672 <4. y wO Q 497328 30320 J5293 21 40 482128 66.1 979019 . / 6T 603109 72.0 496891 30348 J5284 20 41 9.482525 66.1 9.978979 . / Gey 9.503546 wo rt 10.496454 30376 95275 19 42 482921 66.0 978939 . / 6ry 603982 "2. i wo >7 496018 30403 95266 18 43 483316 65.9 978898 . i 67 504418 "2.1 495582 30431 95257 17 44 483712 65.9 978858 . / 67 604854 c 495146 30459 95248 16 45 484107 65.8 978817 . / 6*7 605289 /2 . o 494711 30486 95240 15 46 484501 65.7 978777 . / 67 505724 72 . 5 494276 30514 95231 14 47 484895 65.7 978736 . / 6rt 506159 wo A 493841 30542 95222 13 48 485289 65.6 978696 . / 6Q 506593 72. 4 493407 30570 95213 12 49 485682 65.5 978655 . O 6Q 607027 m o 492973 3059? 95204 n 50 486075 65.5 978615 .0 507460 72.2 492540 30625 95196 10 61 9.486467 65.4 9.978574 " S 9.607893 wo 1 0.492107 30653 95186 9 52 486860 66.3 978533 U .0 60 608326 72. 1 491674 30680 95177 8 53 487251 65.3 978493 ,0 6Q 508759 72. 1 491241 30708 95168 7 54 487643 65.2 978452 . bo 609191 72.0 wi O 490809 30736 95159 6 55 488034 65.1 978411 . O 6Q 609622 71 .y wi n 490378 30763 95150 6 66 488424 65.1 978370 .0 6O 610054 71 .y W1 Q 489946 30791 95142 4 57 488814 65.0 978329 .O 6Q 510485 71.0 489515 30819 95133 3 58 489204 65.0 9 78288 . 6Q 610916 71 .8 489084 30846 95124 2 59 489593 34. y 978247 . o 60 511346 71.7 w-l ? 488654 30874 95115 1 60 489982 64.8 978206 .O 511776 71 .0 488224 3090 ^ 95106 Cos[ne7~ Sine. Cotang. Tang. ||N. cos. N.sinc. ~~ r T> Degrees. TABLE II. Log. Sines and Tangents. (18) Natural Sines. 39 _ ._ Sine. D. 10" Cosine. D. 10" Tang. D. 10" Cotang. N. sine N. cos. 9.489982 RA. R 9.978206 60 9.511776 71 R 10.488224 30902 95106 60 1 490371 D4. o C1 Q 978165 . o 60 512206 < 1 .D rT 1 487794!! 30929 95097 59 o 490759 O 1 ! o 978124 o 60 612635 / 1 .D 487365 | i30957 95088 58 3 491147 64.7 978083 .0 613064 71 .5 486936 130985 95079 57 4 491535 64.6 P.A P. 978042 6.9 6Q 613493 71.4 IJ-I A 486507 i 31012 95070 66 6 491922 D4.D 64 5 978001 .y 6 a 613921 /1 .4 71 486079 31040 95061 55 6 492308 64 4 977959 y 60 514349 4 1 . O 71 i 485651 31068 95052 54 7 492695 O4: . * 64 4 977918 . y 6q 614T77 4 1 . O 71 485223 31095 95043 53 8 493081 DHt. * 64 ^ 977877 . y 6q 615204 i x 2p 71 484796 31123 95033 52 9 493466 O^. O R4 2 977835 y 6 9 615631 1 A 71 1 484369 31151 95024 51 10 493851 O^T . Z fi4 1 977794 6q 616057 *. I 71 A 483943 31178 96016 50 11 9.494236 D i. Z 64 1 9.977752 . y 6 a 9.516484 i A . U 71 H 10.483516 31206 95006 49 12 494621 O*x. 1 64 1 977711 . y 6q 516910 / 1 . U 70 Q 483090 31233 94997 48 13 495005 D^. 1 fi.4 977669 . y 6q 617335 iv . y 70 Q 482665 31261 94988 47 14 495388 O^. \j f\\ Q 977628 . y 6q 617761 / u. y 70 ft 482239 31289 94979 46 15 495772 DO . y cq Q 977686 . y 6q 518185 /u . o 70 ft 481816 31316 94970 45 16 496154 DO . y f\\ P. 977644 . y 7 618610 i U . o 70 7 481390 31344 94961 44 17 496537 DO . o CO fi 977503 i , \) 7 619034 /U. 7 70 ft 480966 31372 94952 43 18 496919 Do. / CO 7 977461 i . U 7 519458 /U.O 70 P. 480542 31399 94943 42 19 497301 DO . / CO C. 977419 v 7 n 519882 /U. b *7A K 480118 31427 94933 41 20 497682 Do.D 977377 i . U 7r\ 520305 /U. 5 479695 31454 94924 40 21 9.498064 63.6 CO C 9.977335 .0 7 9.520728 70.5, 70 A 10.479272 3148294915 39 22 23 24 25 26 498444 498825 499204 499584 499963 Do . O 63.4 63.4 63.3 63.2 cq 9 977293 977251 977209 977167 977125 . v 7.0 7.0 7.0 7.0 7 521151 621573 621995 622417 622838 / U 4 70.3 70.3 70 3 70.2 rrf\ r> 478849 478427 478005 477583 477162 31510 ! 94906 3153794897 3156594888 3159394878 3162094869 38 37 36 35 34 27 600342 DO . Z cq i 977083 / . U 7 623259 /U. 2 r-rf\ 1 476741 31648 94860 33 28 500721 Do . 1 CO 1 977041 / . U 7 n 523680 ". 1 r~i\ -t 476320 31675 94851 32 29 501099 Do , 1 976999 i .U 524100 /U.I 475900 31703 94842 31 30 601476 63 . 976957 7.0 624520 70.0 475480 31730^94832 30 31 9.501854 62.9 9.976914 7.0 9.524939 69.9 tO- 475061 31758^94823 29 32 33 34 35 502231 502607 602984 503360 62.9 62.8 62.8 62.7 R9 fi 976872 976830 976787 976745 7.0 7.1 7.1 7.1 7 1 525359 625778 626197 626615 69.9 69.8 69.8 69.7 ?Q i*r 474641 474222 473803 473385 3178694814 31813948U5 3184194795 31868194786 28 27 26 25 36 37 603735 504110 D-* . D 62.6 976702 976660 / , 1 7.1 7-t 627033 527451 oy , / 69.6 472967 472549 31896 94777 3192394768 24 23 38 504485 62.5 976617 . l 71 627868 69.6 472132 3195194758 22 39 604860 62.5 976574 .1 628285 69.5 471715 3197994749 21 40 605234 62.4 976532 7.1 528702 69.5 471298 32006 94740 20 41 9.505608 62.3 ro Q 9.976489 7.1 71 9.629119 69.4 r>(\ ci 10.470881 32034^94730 19 42 505981 O-4 .0 fi9 "2 976446 . i 7 1 629535 t>y . o fiQ . 470465 3206194721 18 43 506354 u - ~ c.o o 976404 i . A 7 1 529950 Dy , o f\Q Q 470050 3208994712 17 44 506727 D.4 . ^ 976361 * 71 630366 Dy . o 469634 32116 94702 16 45 607099 62 . 1 976318 . 1 71 630781 69 .2 469219 32144 94693 15 46 507471 62. 976275 . 1 71 531196 69.1 468804 3217194684 14 47 507843 62.0 976232 . 1 70 531611 69.1 468389 32199 94674 13 48 608214 61 .9 976189 ,Z 7 O 632025 69.0 467975 32227 94665 12 49 508585 61 .9 fil H 976146 .^ 7 2 532439 69.0 fitf q 467561 32250 94656 11 50 508956 Dl . o C1 O 976103 / . ^ 7 2 532853 DO . y fiQ. q 467147 32282 94646 10 51 9.509326 D 1 . o P.1 7 9.97G060 * . * 7 2 9.633266 DO . y fiK ft 10.466734 32309 94637 9 52 509696 Dl . P.1 fi 976017 i . -^ 7 1 533679 Do , o fiR C 466321 32337 94627 8 53 610065 Dl .O 975974 f . ~ 70 534092 Do . o /c * 465908 32364 94618 7 54 610434 61 .6 975930 . Z 70 634504 DO. 7 465496 32392 94609 6 55 610803 61.5 975887 . * 70 534916 68 . 7 465084 32419 94599 5 56 511172 61 .5 975844 .* 635328 68.6 464672 32447 94590 4 57 611540 61 .4 976800 7.2 635739 68.6 464261 32474 94580 3 68 511907 61 .3 975757 7.2 636150 68.5 463850 3250294571 2 59 512275 61 .3 975714 7.2 636561 68.6 463439 3252994561 1 60 512642 61 .2 975670 7.2 536972 68.4 463028 3255 , 94552 Cosine. Sine. Cotang. Tang. N. coR.JN.sine. T 71 Degrees. 40 Log. Sines and Tangents. (19) Natural Sines. TABLE II. Sine. U. 10 Cosine. i). iU Tang. i>. 10" Cotaug. N. sine.|N. cos.| o 9.512642 9.975670 70 9.536972 rjo A 10.463028 3255 94552 60 513IM 61 .2 n i 9756-27 . <J 7 ^ 537382 DO .^ CQ q 462618 3258 94542 59 t 513375 ol . l 975583 / . o 7 ^ 537792 DO . i r>Q * 462208 3261 94533 58 j 513741 ol . 1 975539 / . o 538202 bo . i c-0 9 461798 3263 94523 57 i 514107 61 .( 97549( 79 638611 bo . z fiP. 9 461389 3266 94514 56 5 614472 60 * 975452 7 3 539020 bo .Z 68 1 460980 3269 94504 55 6 514837 cf\ 975408 70 539429 bo . . 460571 3272 94495 54 515202 bU.c fin s 975365 . O 7 3 639837 r i 460163 3274 94485 53 8 615586 bu c 975321 7 3 640245 o 459755 3277 94476 52 c 615930 fin r 975277 7 ^ 640653 R7 t 459347 3280 94466 51 10 516294 bU. / 975233 1 . O 7 3 641061 b i . J 458939 3283 94457 60 11 9.516657 PA P. 9.975189 I . O 9.641468 R7 10.458532 3285 94447 49 12 617020 bU -0 975145 7 ^ 641876 b/ .0 R7 P. 458125 3288 94438 48 13 517382 AX 975101 / . o 70 542281 b/ .0 457719 3291 94428 47 14 517745 60 . 4 975057 o 542688 fi7 *- 457312 3294 94418 46 15 518107 rn r 975013 7q 643094 D / . i R7 f 456906 32969 94409 45 16 618468 A C 974969 . J 7 4 643499 D / . 1 S7 ( 456501 3299 94399 44 17 618829 fin 9 974925 i . *x 7 4 643905 R7 t 456095 33024 94390 43 18 519190 bO-z 974880 i.4 7 A 644310 i . C /-jO" p- 455690 3305 94380 42 19 519551 60- 1 orv -I 974836 .4 7 A 644715 a/ .0 455285 33079 94370 41 20 519911 bU- 1 974792 . 4 7 4 545119 37 / 454881 33106 94361 40 21 9.520271 oU.U 9.974748 9.545524 3 / . T 10.454476 33134 94351 39 22 620631 60.0 59 9 974703 7.4 7 4 645928 37. c R7 c 454072 3316 94342 38 23 520990 K,q q 974659 1 ,*r 7 4 646331 3 / . c R7 Q 453669 33189 94332 37 24 521349 oy . y 974614 /.4 7 4 646735 3 i . fi7 2 453265 33216 94322 36 25 621707 Kf\ Q 974570 1.4 647138 3 / .^ a<*f 1 452862 33244 94313 35 26 622066 a9.o 974525 7. 4 647640 67.1 /.-y -I 452460 33271 94303 34* 27 522424 69.7 974481 7.4 647943 67.1 an A 452057 33298 94293 33 28 622781 59.6 974436 7.4 548345 37.1 451655 33326 94284 32 29 623138 59-6 974391 7.4 7 4 648747 ifi <H 451253 33353 94274 31 30 523495 c 974347 i . 4 7er 549149 bb . y r;/j Q 450851 33381 94264 30 31 9.523852 a9.5 ^Q A 9.974302 .6 7 fi 9.649550 bb.y Rfi 8 10.450450 33408 94264 29 32 524208 oy .4 974257 / . 649951 )b . o r*r> Q 450049 33436 94245 28 33 34 524564 624920 59.4 69.3 974212 974167 7. 5 7.5 650352 650752 bb.o 66.7 /?/-. ry 449648 449248 33463 33490 94235 94225 27 26 35 625275 59.3 974122 7.6 651152 bb. i /> y-> 448848 33518 J4215 25 36 625630 59.2 974077 7.5 651552 )6.b 448448 33545 94206 24 37 625984 59-1 974032 7.6 651952 56.6 448048 33573 94196 23 38 626339 59-1 973987 7.5 652351 56.5 447649 33600 4186 22 39 626693 39-0 973942 7.6 652750 66.5 T> K. 447250 33627 4176 21 40 627046 59 co n 973897 7.5 553149 >b.o 446851 33655 4167 20 41 9.527400 X5-9 9.973852 7.5 9.563548 66.4 0.446452 33682 4157 19 42 627753 >8-9 CO Q 973807 7.5 653946 66.4 446054 33710 4147 18 43 628105 >o.o 973761 7.5 654344 66.3 445656 33737 4137 17 44 528458 )8-8 973716 7.5 654741 66.3 445259 33764 4127 16 45 1 46 528810 529161 S8-7 58 7 ~O f 973671 973625 7.6 7.6 555139 655536 66.2 66.2 444861 444464 33792 33819 4118 4108 5 14 47 529513 oo -b 973580 7.6 666933 66.1 444067 ! 33846 4098 3 48 529864 )8-6 973535 7.6 656329 66.1 443671 1 33874 4083 2 49 530215 >8-5 973489 7.6 556725 66.0 443275 1 33901 4078 1 50 530565 S8-5 973444 7.6 557121 66.0 442879 ! 33929 4068 51 52 .530915 531265 58-4 58-4 -Q q .973398 973352 7.6 7.6 7(? .557517 557913 65.9 65.9 fi^ Q 0.442483 3395b 442087 il 33983 4058 4049 9 8 53 531614 3o O 973307 .b 558308 30 . y 441692 34011 4039 7 54 55 531963 532312 58-2 58-2 973261 973215 7.6 7.6 558702 659097 65.8 65.8 441298 ! i 34038 440903 34065 4029 4019 5 56 532661 68. 1 973169 7.6 559491 65.7 440509 34093 4009 4 57 533009 58.1 973124 7.6 559885 65.7 440115 34120 3999 3 58 53335 / 58.0 7.6 560279 65.6 439721 ! 34147 3989 2 59 533704 !>8 973032 7.6 560673 35.6 439327! 134175 3979 1 60 534052 )7.9 972986 7.7 561066 66.5 438934 34202 3969 o Cosine. Sine. Co tang. Tang. N. cos. .sine. 70 Degrees. TABLE II. Log. Sines and Tangents. (20) Natural Sines. 41 Sine. D. 10 Cosine. D. 10" Tang. D. 10" Cotanir. N. sine. N. cos. 9.534052 57 8 9.972986 7 7 9.561066 65 5 10.438934! 34202 93969 60 1 534399 972940 i i 661459 438541| 134229 93959 59 2 534745 57 . 7 972894 7.7 661851 65.4 438149 34257 93949 68 8 535092 57.7 o7 " / 972848 7.7 7 7 662244 65.4 ex. q 437756 i 34284 93939 57 4 535438 972802 / . / 562636 OO o 437364 i 34311 93929 56 5 635783 57.6 972755 7.7 663028 65.3 436972 134339 93919 55 6 536129 57.6 972709 7.7 7*7 563419 65.3 f!K C) 436581 ^34366 93909 64 7 8 536474 536818 5 / . 5 57.4 57 4 972663 972617 . / 7.7 7 7 563811 564202 OO . A 65.2 65 1 436189 34393 4357981,34421 93899 93889 53 C2 9 537163 972570 / . i 564592 Oc) . i 435408 34448 93879 51 10 537507 57. 3 972524 7.7 564983 65.1 435017 134475 93869 50 11 9.537851 57. 3 9.972478 7.7 9.565373 65.0 10.434627 34503 93859 49 12 538194 57.2 972431 7.7 565763 65.0 434237 34530 93849 48 13 538538 67. 2 x,7 i 972385 7.8 7 8 566153 64. 9 64 9 433847 34557 93839 47 14 538880 O / . 1 57 1 972338 1 . O 7 8 666542 64 9 433458 34584 93829 46 15 539223 57 n 972291 i . O 7 8 666932 433068 34612 93819 45 16 539565 / . u 972245 / .0 70 567320 M Q 432680 34639 !>3S09 44 17 539907 57.0 x/: q 972198 .0 7Q 567709 o4.o 64 7 432291 34666 93799 43 18 540249 oo . y x<: n 972151 . O 7 8 568098 OTT . / 431902 34694 93789 42 19 540590 OO . J K? O 972105 1 . O 7Q 568486 4 f 431514 34721 93779 41 20 540931 Oo . o ftfi 8 972058 . O 70 568873 64 . 6 f}4 6 431127 34748 93769 40 21 9.541272 OO O 9.972011 . o 7 8 9.569261 f!A X 10.430739 34775 93759 39 22 541613 f n 971964 / .0 70 569648 O4 . 430352 3480393748 38 23 541953 06.7 971917 .0 570035 64.5 429965 34830 93738 37 24 642293 56 . 6 XI J 971870 7.8 70 670422 64.5 429578 34857 93728 36 25 542632 ob . b 971823 .0 570809 64 . 4 429191 34884 93718 35 > 26 542971 56.5 Xf X 971776 7.8 7 8 571195 64.4 428805 34912 93708 34 27 28 543310 543649 oo . o 56.4 56 *4 971729 971682 / . o 7.9 7 9 571581 571967 64.3 64 2 428419 428033 34939 93698 34966 193688 33 32 29 543987 56. 3 971635 7 g 572352 o^ . ^ 64 2 427648 34993 93677 31 30 544325 56 3 971588 7 Q 572738 R4 1 427262 35021 93667 30 31 9.544663 9.971540 i . y 7 Q 9.573123 o^ . * 10.426877 35048 93657 29 32 645000 OO. ~i 971493 . y 573507 O4. 1 426493 35075 93647 28 33 545338 56 . 2 xr; 1 971446 7.9 7 A 573892 64. 1 ( 1 | V 426108 35102 93637 27 34 545674 OO . 1 56 . 1 971398 . y 7 9 574276 o* . u 64 425724 35130 93626 26 35 54d01 1 X> A 971351 574660 425340 35157 93616 25 36 546347 OO . U 56 971303 7 Q 575044 63 .9 63 9 424956 35184 93606 24 37 546683 971256 i . y 575427 424573 35211 93596 23 38 *547019 55.9 971208 7.9 575810 63.9 424190 35239 93585 22 39 547354 55.9 x,x o 971161 7.9 7 Q 676193 63.8 423807 35266 .93575 21 40 547689 Oo . o 971113 576576 ?Q r~ 423424 35293 93565 20 41 9.548024 55 . 8 9.971066 7.9 8f\ 9.576958 00.7 10.423041 35320 93555 19 42 548359 55.7 xx 7 971018 .0 8 A 577341 63.7 422659 35347 93544 18 43 548693 oo t 55 . 6 970970 U 8 A 677723 R3R 422277 35375 93534 17 44 549027 5o . o 970922 . U 8 A 678104 rVfi 421896 35402 93524 16 45 549360 XX X. 970874 . U 8/\ 678486 OO.O 421514 35429 93514 15 46 649693 oo . 55.5 970827 .0 8n 578867 63.5 6*} Fi 421133 35456 93503 14 47 650026 55 4 970779 . u 8 679248 oo . o 420752 35484 93493 13 48 550359 KX A 970731 o . u 8 A. 679629 /0 A 420371 35511 93483 12 49 650692 OO.4 xx q 970683 . U 8 A 680009 bo . 4 419991 35538 93472 11 50 551024 oo . o x* q 970635 . U 8 A 580389 poo 419611 35565 93462 10 51 J. 551 356 Oo . o XX O 9.970586 . U 8 A 9.580769 rq Q 10.419231 35592 93452 9 52 53 551687 552018 OO . > 55.2 55 2 970538 970490 . V 8.0 8 A 581149 681528 63^2 418851 418472 35619 35647 93441 93431 8 7 54 552349 970442 . \J 681907 bo. 2 418093 35674 93420 6 55 552680 55. 1 XX 1 970394 8.0 8 A 582286 63.2 417714 35701 93410 5 56 553010 oo . i 55.0 970345 u 8 1 582665 63 . 1 417335 35728 93400 4 57 553341 970297 583043 bo. l 416957 35765 93.S89 3 58 553670 55.0 970249 8. 1 683422 63.0 416578 35782 93379 2 59 554000 54. 9 970200 8. 1 583800 63.0 416200 35810 93368 1 60 554329 54.9 970152 8. 1 584177 62.9 415823 35837 93358 Cosine. Sine. Cotang. Tang. N. cos. X.sino. 69 Degrees. 42 Log. Sines and Tangents. (21) Natural Sines. TABLE II. Sine. D. 10 Cosine. D. 10 Tang. D. 10"i Cotang. jjN.sine. N. cos. - 9.554329 X,1 Q 9.970152 9.584177 71710.415823" 35837 93358 60 1 554658 O4.o X 1 Q 970103 . 584555 O-^ , y 89 Q 415445 35864 93348 59 9 554987 O4. O 970055 . 684932 u^ . y GO Q 415068 35891 93337 58 3 555315 54.7 970006 . 685309 2 . o GO Q 414691 35918 93327 1 57 4 555643 54.7 969957 . 585686 2.0 414^14 35945 933 16 1 5;i 6 5559 71 64.6 969909 8. 686062 62.7 89 7 413938 35973 93306 ! 55 6 7 556299 556626 54. 6 54.5 Mer 969860 969811 . 8. 586439 586815 O^& . / 62.7 r>o i-j 413561 413185 36000 36027 93295 i 54 93286 1 5; 8 556953 . 5 969762 . , 587190 O-i . O f>o ii 412810 36054 93274 By 9 557280 54. 4 969714 . 587566 t)^i . I) /?(> ~ 412434 36081 93264 51 10 11 557606 9.557932 54.4 54.3 969665 9.969616 8. 8O 587941 9.588316 O,* . O 62.5 89 & 412059 10.411684 36108 36135 93253 93-243 s 12 558258 54.3 969567 . * 8O 588691 \)A . O iO /I 411309 3616L 93232 4h 13 658583 54. 3 969518 . & 8O 589066 D^ . 4 f) ,4 410934 36190 93222 47 14 558909 54.2 ~ \ o 969469 .2 8f\ 689440 O* . 4 GO O 410560 36217 93211 4t i 15 559234 o4.!2 969420 . * 8O 589814 4.O /?o o 410186 36244 93201 45 1G 559558 54. 1 p;4. i 969370 .2 Q 9 590188 \)2 . o &9 ^ 409812 36271 93190 4-i 17 559383 O4, 1 pM A 969321 o . A 89 690562 ^ o &9 9 409438 36298 93180 4o 18 560207 t>4 . K A A 969272 . w 8c% 590935 +A 60 o 409065 36325 US 169 4-2 19 560531 o-l.O 969223 .2 8f) 691308 si.y >f> O 408692 36352 93159 41 20 560855 53.9 XQ o 969173 .2 80 591681 \)2.2 fi9 1 408319 36379 93148 40 21 9.561178 Oo . y KQ Q 9.969124 . 2 8Q 9.592054 D^. I 89 1 10.407946 36406 93137 39 22 561501 Oo .O KQ Q 969075 . A 8c\ 692426 D.4. 1 fio f\ 407574 36434 93127 38 23 561824 Oo. O xo n 969025 . 2 8n 692798 Q2 , ?r> 1 1 407202 36461 93116 3< 24 25 562146 562468 Oo . / 53.7 968976 968926 . 2 8.2 8 O 593170 693542 0^ . U 61.9 >-! A 406829 406458 36488 36515 93106 93095 36 35 26 562790 53 .6 968877 .3 8 693914 oi ,y ^1 Q 406086 36542 93084 34 27 563112 53.6 XQ C 968827 .3 8q 694286 Ol ,O fi1 R 405715 36569 93074 33 28 563433 OO.O 968777 . o 8 594656 Dl . o ^1 O 405344 36596 93063 32 29 563755 53.5 968728 .3 8 595027 ol .0 , 1 ri 404973 36623 93052 ol 30 584075 53.5 XQ A 968678 .3 80 695398 Dl.7 fil 7 404602 36650 93042 30 31 9.564396 OO.4 XQ A 9.968628 . o 80 9.695768 Ol . / R1 10.404232 36677 93031 29 32 564716 00.4 xq q 968578 .0 8q 696138 01 . / 61 fi 403862 36704 930-20 28 33 565036 Oo . o cq q 968528 O 8q 596508 Ol . O 61 ft 403492 36731 93010 27 34 565356 Oo . o xq o 968479 .O 8q 596878 Ol . O 61 fi 403122 36758 92999 26 35 565676 Oo , Z XQ o 968429 . O 8q 597247 01 . o fil K 402753 36785 92988 25 36 565995 Oo , Z xq i 968379 .O 80 697616 Ol , o fi1 K 402384 36812 92978 24 37 566314 Oo . 1 XQ 1 968329 . O 80 697985 Ol . o fil f^ 402015 36839 92967 23 38 666632 Oo . 1 XQ 1 968278 . O 80 598354 Ol .0 fi1 A 401646 36867 92956 22 39 566951 oo . 1 53 968228 .O R A 598722 Ol . 4 61 4 401278 1 36894 92945 21 40 667269 XQ A 968178 o . 4 8 A 599091 fil Q 400909 36921 92935 20 41 9.567587 oo . U XQ q 9.968128 . 4 8 A 9.599459 Ol . o fil ^ 10.400541 i 36948 92926 19 42 567904 o<* . y xo n 968078 . 4 84 699827 Ol . o ft1 o 400173 36975 92913 18 43 568222 02 . y XO O 968027 .4 8 A 600194 ol . d f!1 9 399806 37002 92902 17 44 568539 *jz , o XQ 967977 . 4 8 A 600562 Ol . .4 fil 9 399438 37029 92892 16 45 568856 O& . o XO 967927 .4 8 A 600929 Ol . ^ fil 1 399071 37056 92881 15 46 669172 <3Z . o xo fj 967876 .4 8 A 601296 Ol . 1 fil 1 398704 37083 92870 14 47 569488 0* . 1 xo 7 967826 .4 8 A 601662 Ol . 1 fil 1 398338 37110 92859 13 48 49 669804 670120 O-^. / 52.6 XQ ft 967775 967725 .4 8.4 84. 602029 602395 Ol . 1 61.0 61 397971 137137 397605 37164 92849 J2838 12 11 50 570435 O-^ . D XO X 967674 T: 8 A 602761 ?i A 397239 j 37191 J2827 10 51 9.670751 oz .0 xo x 9.967624 .4 c 4 9.603127 Ol . u 60 Q 10.396873 37218 J2816 9 52 571066 oz , o 967573 o . 4 8. 603493 ou . y 396507 37245 J2805 8 53 54 671380 571695 52.4 52.4 Xo o 967522 967471 .4 8.6 8pr 603858 604223 60.9 60.9 f>{\ 396142 37272 3957771137299 92794 92784 7 6 55 572009 O J.J ^Q q 967421 . O 8K 604588 DU.O 80 ft 395412 37326 92773 5 56 572323 OZ . O 967370 . 8x 604953 OU .0 IA O 1 395047 37353 92762 4 57 572636 52.3 xo o 967319 . O 8K 605317 oU.7 80 7 394683 j| 37380 J2751 3 58 572950 O^. Z 967268 . 8e- 605682 DU . / r>f\ rt 394318; 37407 32740 2 59 673263 52. Q 967217 . O ( 06046 oU. 7 R{\ A 393954 137434 42729 1 60 573575 52.1 967166 8 . 6 606410 bU.o 393590 I 37461 92718 Cosine. Sine. Cotang. Tang. N. cos. N.pinc. ~ T ~ 68 Degrees. TABLE IT. Log. Sines and Tangents. (22) Natural Sines. 43 Sine. D. 10" Cosine. D. 10" Tang. D. 10" Cotaug. iN.sinc. N. cos. 9.573575 52 1 9.967166 8K 9.606410 60 6 10. 393590!] 37461 92718 60 1 573888 CO A 967115 . O 8fL 606773 393227||37488 92707 59 2 574200 O^.U 967064 .O 8C 607137 fiOK 3928631137515 92697 58 3 674512 ci q 967013 o 8 re 607500 fin 6 392500 i 37542 92686 57 4 574824 Ol . J 966961 O 607863 ou . o 392137 37569 92675 5(5 6 675136 51 .9 966910 8.5 608225 60.4 391775 37595 92664 55 6 575447 51 .9 51 8 966859 8.5 8K 608588 60.4 fiO 4 391412 37622 92653 54 7 575758 ~ 1 966808 . o 608950 ou . ^ 391050 37649 92642 53 8 576069 o 1 . o ei 7 966756 8.5 8ft 609312 60.3 fin ^ 390688 37676 92631 52 9 576379 Ol . 1 ci 7 96(5705 . o 8ft 609674 OU. o fin *3 390326 37703 92620 51 10 676689 01 . C 1 fi 966653 . o 8ft 610036 OU. o fifl 1 389964 37730 92609 | 50 11 9.576999 Ol . O ci ft 9.966602 . o 8ft 9.610397 OU. A fin i 10.389603 37757 92598 49 12 577309 Ol . O C 1 ft 966550 . o Q ft 610759 OU. A fiO 2 389241 37784 92587 48 13 577618 Ol . O ei c 966499 8ft 611120 fin 1 388880 37811 92576 47 1 14 577927 Ol . O 966447 .0 8ft 611480 ou . i 388520 37838 92565 46 15 578236 51 .4 966395 .0 Q ft 611841 fin i 388159 37865 92554 45 16 578545 51 4 966344 O . O 8ft 612201 OU . 1 fin n 387799 37892 92543 44 17 578853 el Q 966292 . o 612561 ou . u 387439 37919 92532 43 18 679162 51.0 966240 8.6 612921 60.0 387079 37946 92521 42 19 579470 51.3 966188 8.6 613281 60.0 386719 37973 92510 41 20 679777 51 . 3 966136 8.6 613641 59.9 386359 37999 92499 40 21 9.580085 51.2 CI O 9.966085 8.6 87 9.614000 59.9 -q Q 10-386000 38026 92488 39 22 580392 Ol *> CI 1 966033 . / 87 614359 oy . o F.Q H 385641 3805392477 38 23 580699 Ol . 1 Ci 1 965981 . / 87 614718 oy . o P.Q P. 385282 38080192466 37 24 681005 Ol . 1 CI 1 965928 . i 87 615077 oy . o 384923 38107 92455 36 25 681312 01.1 CI A 965876 . / 87 615435 K.Q 7 384565 38134 92444 35 26 581618 01 . U fil 965824 . / 8/7 615793 oy. i P.Q 7 384207 38161 92432 34 27 681924 Ioi . u f A f\ 965772 . / 616151 oy . i 383849 38188 92421 33 28 582229 I * 965720 8. 7 87 616509 59.6 383491 38215 92410 32 29 30 31 682535 582840 9.583145 50.9 50.8 50 8 965668 965615 9.965563 . / 8.7 8.7 87 616867 617224 9.617582 59^6 59.5 383133 382776 10-382418 38241 92399 38268192388 38295 92377 31 30 29 32 683449 50 7 965511 . / 87 617939 en c 382061 38322192366 28 33 583754 CM 7 965458 . / 8r-f 618295 381705 38349J92355 27 34 584058 ou . / 9d540d 1 618652 59.4 381348 38376 92343 26 35 584361 50. 6 9G5353 8.7 619008 59 . 4 380992 38403 92332 25 36 584665 50.6 965301 8.8 619364 59. 4 380636 38430 92321 24 37 584968 50.6 CI) C 965248 8.8 80 619721 59.3 380279 38456 92310 23 38 585272 50 5 965195 . o 80 620076 59. 3 cq <\ 379924 38483 92299 22 39 585574 965143 . o 80 620432 oy . o 379568 3851092287 21 40 585877 nO 4 965090 .0 80 620787 CO O 379213 38537 92276 20 41 9.586179 9.965037 .0 80 9.621142 oy . A 59 2 10-378858 38564 92265 19 42 686482 CA Q 964984 .0 80 621497 en 1 378503 38591 92254 18 43 586783 CA Q 964931 .0 80 621852 oy . i CA 1 378148 38617 92243 17 44 45 46 587085 587386 587688 OU. o 50.2 50.2 CA 1 964879 964826 964773 .0 8.8 8.8 80 622207 622561 622915 59 .0 59.0 CO A 377793 377439 377085 3864492231 3867192220 3869892209 16 15 14 47 587989 OU. 1 CA I 964719 O 80 623269 59. U cQ O 376731 38725 92198 13 48 688289 OU. 1 CA 1 964666 .O 8Q 623623 oo. y CQ O 376377 3875292186 12 49 588590 OU. 1 CA A 964613 .9 8J-* 623976 5o . y 376024 3877892175 11 50 588890 OU. U en A 964560 .9 8q 624330 eO Q 375670 38805 92164 10 51 9.589190 OU . U 9.964507 . y 8q 9.624683 Oo . O CQ C 10.375317 38832 92152 9 52 589489 Q Q 964454 . y 8f\ 625036 OQ .0 CQ Q 374964 3885992141 8 53 589789 4-9 9 964400 .9 8Q 625388 5o.o ^o 17 374612 3888692130 7 54 590088 964347 . y 8Q 625741 CO 7 374259 3891292119 6 55 590387 49 8 964294 . y 8Q 626093 Oo. / CO rj 373907 3893992107 6 56 57 590686 590984 49!7 49 7 964240 964187 .y 8.9 8Q 626445 626797 OO. / 58.6 CQ ! 373555 373203 38966 192096 38993192085 4 3 ; 58 59 60 591282 591580 591878 49.7 49.6 964133 964080 964026 . y 8.9 8.9 627149 627501 627852 OO. D 58.6 58.5 372851 372499 372148 39020 ! 92073 3904692062 39073|92050 2 1 Cosine. Sine. Cotang. Tang. N. cosJN.sine. 67 Degrees. Log. Sinos and Tangents. (23) Natural Sines. TABLE II. Sin;;. 1). lu Cosine. U. 1U Tang. D. Ju Cotang. ; N. sine N. cos 9.591878 49 6 9.964026 8 Q 9.627852 58 5 10.372148 39073 92050 60 1 592176 49 a 963972 o . J 8Q 628203 58 371797 39100 92039 59 r 592473 49 5 963919 . y 8 9 628554 Oo . K 371446 39127 92028 58 3 592770 963865 o . J 9ft 628905 Oo . co 371095 39153 92016 57 4 593067 49. 5 4Q 4 963811 .0 9 A 629255 5o. KU , 370745 i 39180 92005 56 5 593363 4:y . 4 4-9 4 963757 . U Q 629606 Oo . CO 370394 39207 91994 55 6 593659 ^J . 4 49 1 963704 J , U Q f) 629956 OO . t KO 370044! 39234 91982 54 7 5939o5 4J . o 49 ^ 963650 y . u Q 630306 OO . t CO 369694 39260 91971 53 ! 8 594251 4y . o AQ *} 963596 y . u 9 A 630656 Oo . i CO 3693441139287 91959 52 9 594547 4 J O 4Q 9 963542 . u 9 A 631005 Oo . t, CO 368995 ! 39314 91948 51 10 594842 4y . z 963488 . U 9/1 631355 Oo . KQ 368645 39341 919^6 50 11 9.595137 49 .2 49. 1 9.963434 . U 9 9.631704 Oo . 58. 10.368296 ;39367 91925 49 1-2 595432 49 1 963379 9.0 632053 58.1 367947 j 39394 91914 48 13 595727 4-.J 1 AC\ 1 963325 9 A 632401 CO 1 367599 39421 91902 47 14 598021 4y i 4Q 963271 . u Q (I 632750 Oo . 1 58.1 367250 39448 91891 46 15 596315 4y . U 49 963217 7 V q A 633098 58. C 366902 39474 91879 45 Hi 596609 4c* U 48 Q 963163 i7 . V 9 633447 K r 366553 39501 91868 44 17 596903 4:0 . y 48 Q 963108 y . u 9 1 633795 OO . \j KQ /; 366205 39528 91856 43 18 597196 4o . y 48 Q 963054 9*1 634143 Oo . \j C7 Q 365857 39555 91845 42 19 597490 40 . y 48 8 962999 9*1 634490 O * . o fi7 Q 365510 39581 91833 41 20 597783 4o . O 48 8 962945 . i 91 634838 Of .0 K7 Q 365162 39608 91822 40 21 9.598075 4O . O 48 7 9.962890 . i 91 9.635185 o t . y K7 8 10.364815 39635 91810 39 22 598368 4o . / 48 7 962836 . i 91 635532 O / . c K7 8 364468 39661 91799 38 23 598660 4o . / 40 7 962781 . i 9 1 635879 ( .0 57 8 364121 39688 91787 37 24 598952 <o . / 48 fi 962727 91 636226 O * . o 57 7 363774 39715 91775 36 25 599244 4O . O 48 fi 962672 9 1 636572 o / . / K7 7 363428 39741 91764 35 26 599536 4o .O 48 fi 962617 91 636919 Di . / K7 7 363081 39768 91752 34 27 599827 4o . O 48 f 962563 . i 91 637265 Di . K7 7 362735 39795 91741 33 28 600118 4:0 . O 48 fi 962508 . i 91 637611 O / . / K7 fj 362389 39822 91729 32 29 600409 4:0 . O /4Q >4 962453 . i 91 637956 * . D fi7 R 362044 39848 J1718 31 30 600700 4:0 . 4 48 4 962398 . i 90 638302 O4 .D FL7 R 361698 39875 91706 30 3t 9.600990 4O .4 y*Q y* 9.962343 4 90 9.638647 O/ .O K7 Fi 10.361353 39902 J1694 29 32 601280 4o . 4 48 3 962288 . & 9 2 638992 O/ ,O K7 f> 361008 39928 J1683 28 33 601570 48 3 962233 92 639337 . O K7 fi 360663 39955 )1671 27 34 35 601860 602150 48^2 48 2 962178 962123 9. 3 Q 639682 640027 D . O 57.4 ^7 4 360318 359973 39982 40008 91660 91648 26 25 36 602439 4o -^ 48 9 962067 J . ^ Q 640371 3 < . 4 "7 ^ 359629 40035 91636 24 37 602728 4o . Z J.8 1 962012 \J . A 9O 640716 3 . 4 7 q 359284 40062 91625 23 38 603017 4o . 1 jQ 1 961957 . -^ 9O 641060 3 / . O K7 ^ 358940 40088 91613 22 39 603305 48 . 1 iQ -| 961902 . 9f) 641404 O/ .0 r-ry o 358596 40115 91601 21 40 603594 4o.l /i8 n 961846 . A 90 641747 5/.3 K7 n 358253 40141 191590 20 41 9.603882 4o . U 48 9.961791 , .* 9 2 9.642091 o/ .is 57 o 10.357909 40168 U578 19 42 604170 4o \J 47 9 961735 g g 642434 J . A i7 9 357566 40195 91566 18 43 604457 4 1 y 47 9 961680 Q 2 642777 5 4 . ^ i7 9 357223 40221 91555 17 44 604745 4 v 47 Q 961624 y . <& Q 643120 < = 7 i 356880 40248 91543 16 45 605032 4 / y 47 8 961569 y . o 9Q 643463 O< . 1 7 i 356537 40275 91531 15 46 605319 4/ o 47-8 961513 . o 9 3 643806 O/ . 1 57 1 356194 40301 J1519 14 47 605606 47 8 961458 90 644148 o / . i FJ7 /i 355852 40328 91508 13 48 605892 4 / o /IT *7 961402 . o 90 644490 o/ .u 7 o 355510 40355 91496 12 49 606179 4 / / /I "7 "7 961346 . o 90 644832 o 1 ,(J K7 n 355168 40381 91484 11 50 605465 4 / / -1*7 1 961290 # o 90 645174 O/ .U K.fi O 354826 40408 )1472 10 51 9.606751 4/ -D 47 6 9.961235 . o 9 3 3.645516 oo.y 16 n 0.354484 40434 91461 9 52 607036 itf *O <-7 r 961179 9 q 645857 oo . y & a 354143 40461 91449 8 53 607322 4/ -O 47 5 961123 . o 9 3 646199 oo.y : Q 353801 40488 91437 7 54 607607 4: / O 47 fi 961067 9 3 646540 oo . y K 353460 40514 91425 6 55 607892 4 / O 47 4 961011 9Q 646881 )O . o ifi 8 353119 40541 H414 5 56 603177 4 / 4: 47 4 960955 . o 9 Q 647222 OO .O PY; o 352778 40567 )1402 4 57 608461 4 / . 4: 47.4 960899 y . *j 9 3 647562 OO ,o 16 7 352438 40594 )1390 3 58 608745 47 3 960843 94 647903 Ou . / Sfi 7 352097 40621 )1378 2 59 609029 960786 648243 JO . / 351757 40647 H366 1 60 609313 47.3 9J0730 9.4 648583 >6.7 351417 40674 31355 Cosine. Sine. Cotang. Tang. N. cos. X.sine. C6 Degrees. TABLE II. Log. Sines and Tangents. (24) Natural Sines. 45 Sine. D. 10 Cosine. D. 10 Tang. D. 10 Cotang. j N. sine N. cos 9.609313 47 9.960730 9 A 9.648583 ^ 10.351417 40674 91355 60 1 609597 4 1 . t 47 1 960874 . * 9 A 648923 06 . ( 351077 40700 91343 59 r 609830 4 / . ^ 9o06I8 . 4: 9 A 649263 Xfi f. 350737 4072, 91331 58 610164 47.* 960561 , rt 64960- oo . o 350398 40753 91319 57 610447 47.2 47 1 960505 9.4 94 649942 56 6 350058 40780 91307 56 5 610729 4 / . J 4 7 1 980448 . 4; 94 650281 06 . 5 XR P. 349719 40806 91295 55 6 611012 4 / . J 47 ( 960392 . 4 o 4 650620 OO . xq e 349380 40833 91283 54 7 611294 4/ . I 47 ( 960335 9 A 650959 oy .0 XR 4 349041 40860 91272 53 8 611576 4 1 . 1 47 f 960279 . *x 94 651297 OO . 4 348703 40886 91260 52 9 611858 4 i . I 960222 . 4 94 651636 Xfi A 348364 140913 91248 51 10 612140 4fi c 960165 . 4 94 651974 OO . H xft o 348026 40939 91236 50 11 9.612421 4O . <J 9.960109 . 4- 9K 9.652312 OO..J 10.347688 1 40966 91224 49 12 612702 4b. ._ 960052 . O 652650 Ob .<d 347350 ! 40992 91212 48 13 612983 46.8 4fi 8 959995 9.5 9X 652988 56. a 347012 41019 91200 47 14 613264 4:0 . C 4fi 7 959938 . o 653326 ftfl 5 346674 41045 J1188 46 15 613545 4-O . i 4fi 7 959882 9x 653663 Kf! O 346337 41072 91176 45 16 613825 4O . < AC n 959825 . O 654000 Ob .2 346000 41098 J1164 44 17 18 614105 614385 4b. / 46.6 959768 959711 9.5 9.5 654337 654174 56.2 56.1 345663H 41 125 ! 91 152 345326 41 151 91 140 43 42 19 20 614665 614944 46. t 46.6 959654 959596 9.5 9.5 655011 655348 56.1 56.1 344989 41178 91128 344652 I 41204 91 lib 41 40 21 9.615223 46 . 5 AG X 9.959539 9.5 9K 9.655684 56.1 10.344316 i 41231 91104 o9 22 615502 40 . C 959482 .O 656020 56 .C 343980 j: 41 257 9 1092 38 23 615781 46.5 959425 9.5 656356 56. (J 343644 4128491080 37 24 616030 46.4 959368 9.5 656692 56.0 343308 14131091068 36 25 26 616338 616616 46.4 46.4 959310 959253 9.5 9.6 657028 657364 55.9 55.9 342972 342636 41337|91056 41363191044 35 34 27 616894 48.3 9591 95 9.6 657699 55.9 342301 4139091032 33 28 617172 46. 3 959138 H ~ 658034 55.9 341966 4141691020 32 29 30 31 32 617450 617727 J. 6 18004 618281 46.2 46.2 46.2 46.1 959081 959023 9.958965 958908 9.6 9.6 9.6 9.6 658369 658704 9.659039 (>59373 55.8 55.8 55.8 55.8 341631 341296 10.3409bl 34(>o27 41443 ( 91 008 4146900996 4149690984 4152290972 31 30 29 28 33 618558 if i 958850 9.6 G59708 55 . 7 340292 4154990960 27 34 618834 4b. 1 958792 9.6 G60042 55.7 339958 41575 90948 26 35 619110 46.0 4H (\ 958734 9.6 660376 55.7 339624 41602 90936 25 36 619386 40 . U 958677 9.6 660710 55 . 7 339290 41628 90924 24 37 619662 46. 958619 9.6 661043 55.6 338957 41655 190911 23 38 619938 45. 9 958561 9.6. 661377 55.6 338623 4168190899 22 39 620213 45. 9 A X O 958503 9.6 661710 35.6 338290 41707 90887 21 40 620488 40 . y \ x o 958445 9.7 662043 35.5 337957 4173490875 20 41 9.620763 4O . o A x Q 9.958387 9.7 J. 662376 55.5 337624 4176090863 19 42 43 621038 621313 4O.O 45.7 A X r* 958329 968271 9.7 9.7 662709 663042 55.5 55.4 337291 336958 41787J90851 41813 90839 18 17 44 621587 4o. 7 A X rj 958213 9.7 663375 55.4 336625 41840^0826 16 45 46 621861 622135 4o.7 45.6 958154 968096 9.7 9.7 663707 664039 55.4 55.4 336293 335961 4186690814 4189290802 15 14 47 622409 r? 958038 9.7 664371 35.3 335629 4191990790 13 48 622682 4o. 6 ,1 X X 957979 9.7 664703 55.3 335297 41945 90778 12 49 622956 4O . A X x 957921 9.7 665035 35.3 334966 4197290766 11 50 51 623229 .623512 4O. 45.5 957863 J. 957804 9.7 9.7 665366 J. 665697 55.3 55.2 334634 0.334303 4199890753 4202490741 10 9 52 53 54 623774 624047 624319 4o 4 45.4 45.4 4** o 957746 957687 957628 9.7 9.8 9.8 666029 666360 666691 55.2 55.2 55.1 333971 333620 333309 4205190729 4207790717 4210490704 8 7 6 65 624591 o.J 957570 9.8 667021 35.1 332979 42130^ (0692 6 56 624863 45.3 957511 9.8 667352 55. 1 332648 42156 ( . )0680 4 57 625135 46.3 957452 9.8 667682 >5.1 332318 42183 i >0668 3 58 625406 15.2 AK. O 957393 9.8 668013 35.0 331987 42209 i )0655 2 59 625677 10 .2 957335 J.8 668343 35. 331657 42235 10643 1 60 625948 15.2 957276 9.8 668672 o5.0 331328 42262 10631 Cosine. Sine. Cotang. Tang. N. cos.lk.sine. 65 Degrees. 4G Log. Sines and Tangents. (25) Natural Sines. TABLE II. Sine. D. 10 Cosine. }D. 10" Tang. D. 10 Cotang. |,N.sine. N. cos. 9. 625948 AK. 1 9.957276 90 9.668673 CK 10.331327 42262 90631 60 1 2 626219 626490 40 . 1 45.1 A X 1 957217 957158 . o 9.8 90 669002 669332 OO . U 54.9 Mq 330998 330868 42288 42315 90613 90606 59 68 3 626760 40 . 1 4C A 957099 . o 9Q 669661 . y 330339 ; 42341 90594 57 4 627030 4O . U 957040 . O 669991 04. y 330009: 42367 90582 56 5 627300 45.0 /I X A 956981 9.8 9Q 670320 54.8 54 8 329680 I 42394 90669 55 6 627570 4O . U 956921 . O 9q 670649 54 S 329351 1 42420 90557 54 7 627840 44 q 956862 . y 9q 670977 329023 42446 90545 53 8 628109 4-t . y AA O 956803 , y 9 9 671306 54 7 328694 42473 90532 52 9 628378 T" 956744 9* 9 671634 04 . / 54 7 328366 42499 90520 51 10 628647 A A Q 956684 9Q 671963 328037 42525 90507 60 11 9.628916 44. o 44 7 9.956625 . y 9 9 9.672291 54. 7 54 7 10.327709 42552 90495 49 12 629185 44 . / 44 7 956566 99 672619 54 6 327381 | 42578 90483 48 13 629453 4^r . 1 44 7 956506 99 672947 54 6 327053 42604 90470 47 14 629721 44 a 956447 9 9 673274 54 326726 42631 90458 46 15 629989 4:4 . D 44 6 956387 99 673602 O4 . U 54 6 326398 42657 90446 45 16 630257 44 a 956327 . ^ 9 9 673929 54 5 326071 42683 90433 44 17 630524 44 . O 44 6 956268 99 674257 54 6 325743 42709 90421 43 18 630792 44 c 956208 10 674584 54 5 325416 42736 90408 42 19 631059 44 , O 956148 10 674910 Ort . O 325090 42762 90396 41 20 631326 44 % I) 44 5 956089 10 675237 54 4 324763 42788 90383 40 21 9.631593 44.4 9 . 956029 10.0 9-675564 54*4 10.324436 42816 90371 39 22 631859 44.4 955969 10.0 675890 54 4 324110 42841 90358 38 23 632125 44.4 955909 676216 54 3 323784 j 42867 90346 37 24 25 632392 632658 44 . 3 44.3 955849 955789 io!o 10 676543 676869 54^3 54 3 323457 42894 323131 II 42920 J0334 90321 36 35 26 632923 44 3 955729 10 677194 54 3 322806 1| 42946 9030y 34 27 633189 44 2 955569 10 677520 54*2 322480 142972 90296 33 28 633454 955609 1j) A 677846 322154 42999 90284 32 29 6337191 44*0 955548 i\J .V i , ; | 678171 Mo 3218291 143025 90271 31 30 633984 ll. 955488 1U. 10 678496 . -i 54 2 321504: 43051 90259 30 31 9.634249 44 1 9.955428 10 1 9-678821 54 1 10. 321179!! 43077 90246 29 32 33 634514 634778 44! 44 955368 955307 10 i 679146 679471 J4, 1 54.1 54 1 320854 : 43 104 90233 320529 4b 130 90221 28 27 34 635042 44 n 955247 10 i 679795 54 1 320205; 43156 ;>0208 26 35 63530o 44 , U 43 9 955186 10 1 680120 04 . j. 319880 i 431S2 90196 25 36 635570 43 <4 955126 680444 54 o 3195561(43209 90183 24 37 635834 40 . y 43 9 955065 10 1 680768 O4 . U Mo 3 1 9232 l| 43235 90171 23 38 636097 40 . y 43 8 955005 10 1 681092 . V 3189081(43261 90158 22 39 40 636360 636623 43.8 954944 954883 io!i 681416 681740 53!g 53 9 318584 || 43287 318260 1143313 90146 90133 21 20 41 9.636886 40 7 9.954823 10 1 9.682063 cq q 10.317937 43340 90120 19 42 637148 4o . / 43.7 954762 10 1 682387 oo . y 53 9 317613 43366 90108 18 43 637411 43.7 954701 io!i 682710 53 8 317290 43392 90095 17 44 637673 43*7 95-1040 683033 53 8 316967 43418 90082 16 45 637935 43.6 954579 10* 1 683356 53 8 316644 43445 90070 15 46 638197 40 c 954518 10*2 683679 CO ft 316321 43471 90057 14 47 638458 4o . O 43 6 954457 10 2 684001 Oo . O xo 7 315999 43497 90045 13 1 48 638720 43*5 954396 10 2 684324 OO . / 5q 7 315676 43523 90032 12 49 638981 43 5 954335 10 2 684646 Oo . I xq 7 315354 43549 90019 11 50 639242 43 5 954274 10 2 684968 Oo . / cq 7 315032 43575 90007 10 51 9.639503 43 4 9.954213 10 2 9 . 685290 Oo , i cq ft 10.314710 43602 89994 9 52 639764 4O . 4 43 4 954152 10*2 685612 Oo . O xq f> 314388 43628 89981 8 53 640024 43 4 954090 10 2 685934 Oo . O cq f: 314066 43654 89968 7 54 640284 43 3 954029 10 2 686255 Oo . O CO iO 313745 43680 89956 6 55 640544 40 q 953968 10 2 686577 Oo . D 5q 5 313423 43706 89943 5 56 640804 4o . O 43.3 953906 10^2 686898 Oo . O 53 5 313102 43733 89930 4 57 641064 953845 10 2 687219 KO 5 312781 43759 89918 3 58 641324 963783 102 687640 oo . o KQ K 312460 43785 89905 2 59 641584 A q o 953722 1U . *> i (i q 687861 Oo . CO A 312139 43811 89892 1 60 641842 4o . 953660 117. O 688182 Oo . 4 311818 43837 89879 Cosine. Sine. Co tang. Tang. N. cos. N.siue. 64 Degrees. JLAIUHII. Log. Sines and Tangents. (26) Natural Sines. 47 Sine. D. 10" Cosine. D. 10" Tang. D. 10" Cotang. N. sine.jN. cos. o 9.641842 9.953660 9.688182 10.311818 : 43837 89879 ! GO 1 642101 43. 1 953599 10.3 688502 53.4 311498 ; 43863 89867 59 2 3 642360 642618 43.1 43.1 953537 953475 10.3 10.3 688823 689143 53 . 4 53.4 311 177 j 43889 310857 439 K) 89854 89841 58 57 4 642877 43.0 953413 10.3 689463 53.3 310537 43942 89828 56 6 643135 43.0 953352 10.3 689783 53 .3 KO O 310217 143968 89816 55 6 643393 43.0 953290 10.3 690103 OO. O 309897 43994 89803 54 7 643650 43.0 953228 10.3 690423 53.3 309577 44020 89790 53 8 643908 42.9 953166 10.3 690742 53.3 KO 9 309258 44046 89777 I 52 9 10 644165 644423 42!9 953104 953042 io!s 691082 691381 Oo . Z 53.2 CO O 308938 1 44072 308619 44098 89764 89752 51 50 11 9.644680 42.8 9.952980 10.3 9.691700 Oo. 2 CO 10.308300 44124 89739 49 12 644936 42.8 ACt Q 952918 10.4 1 A 1 692019 Oo . CO 307981 1 44151 89726 48 13 645193 42.0 952855 10.4 692338 Oo , CO 30 ?6b2i 44177 89713 47 14 645450 42.7 952793 10.4 692666 OO. 1 CO 1 307344: 44203 89700 46 15 645706 42.7 952731 10.4 692975 oo . CO 1 307025 44229 89687 45 16 17 645962 646218 42.7 42.6 40 j 952669 952603 10.4 10.4 693293 693612 OO . 1 53.0 KO /) 306707 44255 306388 "44281 89674 89C62 44 43 18 646474 4.D 952544 10.4 693930 Oo . U P-Q A 306070 44307 89649 42 19 646729 42.6 952481 10.4 694248 oo . U fro A 305752 44333 89636 41 20 646984 42.5 952419 10.4 694566 OO . U 305434 44359 89623 40 21 9.647240 42.5 9.952356 10.4 9.694883 52 .9 10-305117 44G85 89610 39 22 647494 42.5 952294 10.4 695201 52 .9 304799 44411 89597 38 23 647749 42.4 952231 10.4 10 4 695518 62.9 52 9 304482 44437 89584 37 24 648004 42.4 952168 695836 304164 44464 S9571 36 25 648258 42. 4 952103 10.5 6961531 303847 44490 89558 35 26 648512 42.4 952043 10.5 696470 ?o o 303530 44516 89545 34 27 648766 42 . 3 951980 10.6 696787 Br-J 303213 44542 89532 33 28 649020 % * 951917 10.5 1 i z, 807108 S i 302897 44568 S9519 32 29 649274 ; 951854 10. o 697420 ?~ 302580 44594 89506 31 30 649527 [7^ 951791 10.5 697736 \n i. 302264 44620 89493 30 31 9.649/81 vz,z 9 951 J28 10.5 9.698053 * t. 10-301947 44646 89480 29 32 650J34 42.2 951665 10.5 698369 t . V 301631 44672 89467 28 33 650287 42.2 951602 10.5 698685 52.7 301315 44698 89454 27 34 650539 ~, 951539 10.5 69901)1 u^.6 300999 44724 89441 26 35 650792 4z. i 951476 10.5 1 A K 699316 5:2.6 300684 44750 89428 25 36 651044 42. 1 951412 10. o 699632 ?H ^ 300368 44776 89416 24 37 651297 42. 951349 10.5 699947 o~.6 300053 44802 89402 23 38 651549 42.0 951286 10.6 i A r 700263 52.6 299737 44828 89389 22 39 651800 42.0 951222 10. U 700578 52 . 5 299422 44854 89376 21 40 652052 41 .9 951159 10.6 700893 52 . 5 c;O K 299107 44880 89363 20 41 9.652304 41 .9 9.951096 10.6 9.701208 0^.5 10-298792 44906 89350 19 42 652555 41 .9 41 Q 951032 10.6 701523 52 . 4 298477 44932 89337 18 43 652806 41.8 A\ Q 950968 10.6 701837 o~.4 298163 44958 89324 17 44 653057 41.8 950905 10.6 702152 52. 4 297848 44984 89311 16 45 653308 41.8 950841 10.6 702466 62 . 4 297534 45010 89298 15 46 653558 41.8 950778 10.6 702780 52.4 297220 45036189285 14 47 653808 41.7 950714 10.6 703095 52. 3 296905 45062 89272 13 48 654059 41.7 950650 10.6 703409 52. 3 296591 45088 89259 12 49 654309 41.7 950586 10.6 703723 52. 3 296277 45114 89245 11 50 654558 41 .6 950522 10.6 704036 52. 3 295964 45140 89232 10 61 9.654808 41.6 9.950458 10.7 9.704350 52 .2 -o o 10-295650 45166 89219 9 52 655058 41.6 950394 10.7 704663 oz .2 295337 45192 89206 8 53 54 655307 655556 41.6 41.5 950330 950366 10.7 10.7 704977 705290 52 . 2 52.2 KO Ct 295023 294710 4521889193 4524389180 7 6 55 655805 41.5 950202 10.7 705603 oz . 2 294397 45269 89167 6 56 656054 41.5 950138 10.7 705916 52. 1 294084 45295 89153 4 57 656302 41.4 950074 10.7 706228 52.1 293772 45321 89140 3 58 59 650551 656/99 41 .4 41.4 950010 949945 10.7 10.7 706641 706864 52. 1 52.1 293459 293146 45347 45373 89127 89114 2 1 60 657047 41.3 949881 10.7 707166 52.1 292834 45399 89101 Cosine. Sine. Cotang. Tang. N. cos. V 03 Degrees. 43 Log. Sines and Tangents. (27) Natural Sines. TABLE II. Sine. D. 10 Cosine. D. 10" Tang. 1). iU f (Jotang. ||N. siue. N. cos. 9.657047 41 *l 9.949881 107 9.707166 52 10.292834 45399 89101 60 1 657295 41.o A 1 Q 949816 1U . / 107 707478 xo n 292522 45425 89087 59 2 657542 41 . o A1 9 949752 IU . i 707790 un . U CO 292210 45451 89074 58 B 657790 41 . ^ 41 94 )688 1ft fi 7J8102 O^ . U KO 291898 45477 S9061 57 4 658037 A1 9 949623 1 A W 708414 C 1 Q 291586 i 45503 S9048 56 ! 6 6 658284 658531 41 . 41.2 A\ 1 949558 949494 4 U . O 10.8 1 A Q 708726 709037 01 . y 51.9 ci n 291274 290983 ] 45529 45554 89035 89021 55 54 7 658778 41 . 1 41 1 949429 IU. O 108 709349 01 . y EI n 290651 45580 89008 53 8 659025 41 . 1 A 1 1 949364 IU . o 108 709660 o i . y Mn 290340 45606 88995 52 9 659271 41 . 1 A 1 n 949300 IU . o 108 70^971 . y fil 8 290029 45632 88981 51 10 659517 41 . U A 1 n 949235 IU . Q i n ft 710282 Ol . o ei o 289718 45G58 88968 50 11 9.659/63 41 . U di n 9.949170 IU. o 1 8 9.710593 Ol . o K 1 Q 10.289407 45684 88955 49 12 660 J09 41 . U 949105 IU. o 710904 01 . o 289096 45710 88942 48 13 660255 40.9 4-0 Q 949040 10.8 108 711215 51 .8 K1 Q 288785 45736 88928 47 14 660501 948975 iu. o i o Q 711525 Ol . o ei 17 288475 45762 88915 46 15 600746 A 948910 IU. o 711836 1 . / 288164 45787 88902 45 16 660991 40.9 948845 10.8 1 A Q 712146 51 .7 287854 45813 88888 44 17 661236 40.8 Af\ 948780 10.8 i A n 712456 51.7 287544 45839 88875 43 18 661481 4U.o 948715 10. 9 712766 51.7 287234 45865 88862 42 19 661726 40.8 948650 10.9 713076 51 .6 286924 45891 88848 41 20 661970 40.7 u i ft 948584 10.9 713386 51 .6 286614 45917 88835 40 21 9.662214 4U.7 9.948519 10.9 9.713696 51 .6 10.286304 45942 88822 39 22 662459 A(\ "7 948454 10.9 1 A O 714005 51 .6 K1 ; 286996 45968 88808 38 23 662703 4U. / At\ 1 948388 iu.y 714314 51 .6 285686 45994 88795 37 24 662946 4U.D 4ft fl 948323 10. 9 I A q 714624 c 1 t; 285376 46020 88782 36 25 663190 4U . D AC\ (-* 948257 iu.y 714933 01.0 285067 46046 88768 35 26 27 663433 663677 4U.O 40.5 4 A K. 948192 948126 10. 9 10.9 715242 715551 61 .5 51.5 284758 28444.9 46072 46097 88755 88741 34 33 28 663920 4U.O 948060 10.9 715860 51 .4 284140 46123 88728 32 29 664163 40.5 Af\ K 947995 10.9 nf\ 716168 51 .4 283832 46149 88715 31 30 31 664406 9.664348 4U.5 40.4 947929 9.947863 .0 11.0 716477 9.716785 51 .4 51.4 283523 10.283215 46175 46201 88701 88688 30 29 32 664391 40.4 A(\ 4 947797 11 .0 nf\ 717093 ef-J 282907 46226 88674 28 33 665:33 4U . 4 947731 . u 7174011!^ 282599 46252 88661 27 34 665375 40.3 947665 11.0 717709! 9! 282291 46278 88647 26 35 665617 40.3 947600 11.0 no 718017 Ol .0 ri o 281983 46304 88634 25 36 665359 40 2 947633 . V no 718325 01 . o 281675 46330 88620 24 37 666100 947467 . u nf\ 718633 61 .0 281367 46355 88607 23 38 666342 A(\ 9 947401 . u 718940 51 .2 281060 46381 88593 22 39 666583 o 947335 11.0 U(\ 719248 51 .2 280752 46407 88680 21 40 666324 A n 1 947269 .U nf\ 719555 51.2 280445 46433 88566 20 41 9.667065 4U. 1 9.947203 .0 nf\ 9.719862 51 .2 10.280138 46458 88553 19 42 667305 40. 1 A A 1 947 \ 36 .u 720169 51 .2 279831 46484 88539 18 43 44 667546 667786 4U. 1 40.1 947070 947004 11.1 11.1 720476 720783 51.1 51.1 279524 279217 46510 46536 88526 88512 17 16 45 668027 40.0 4t\ f\ 946937 n .1 Hi 721089 51 .1 278911 46561 88499 15 46 47 668-267 668, i06 4U U 40.0 oq q 946871 946804 . i 11.1 721396 721702 51 . 1 f!-i 278604 278298 46587 46613 88485 88472 14 13 48 668/46 oy . y qq q 946738 . 722009 ";" 277991 466C9 88458 12 49 668J86 oy . y qq q 946671 . 722315 *} 277685 46664 88445 11 50 669, !25 oy . y qq q 946604 722621 g-g 277379 46690 88431 10 51 9.669464 oy . y qq o 9.946538 . , 9.722927 J}- 10.277073 46716 88417 9 52 <;<;:> 0:5 oy . o qq x 946471 . 723232 JJ J 276768 46742 88404 8 53 6691*42 oy . o 946404 723538 gj-g 276462 46767 88390 7 54 670181 qq 7 9.46337 11. 276156 46793 88377 6 55 670119 oy . / 946270 . 724149 ijjJj-9 275851 46819 88363 5 56 670;558 39.7 946203 11.2 724454 |"-jj 276546 46844 88349 4 57 670896 J9.7 946136 11.2 724769 !X X 275241 46870 88336 3 58 671134 39.7 946069 11.2 725066 ou.o 274936 46896 88322 2 59 671372 39.6 946002 11.2 725369 50.8 274631 46921 88308 1 60 671609 39.6 945935 11.2 725674 60.8 274326 46947 88295 dowine. Sine. Cotang. Tang. N. cos N.sine. C2 Degrees. TABLE II. Log. Sines and Tangents. (28) Natural Sines. 49 Sine. D. 10 Cosine. D. 10 Tang. D.10 Cotang. 1 | N. sine N. cos 9.671609 on A 9.945935 Ho 9.725674 Krt 8 10.274326 46947 88295 60 1 671847 OC7 . O on K 945868 . * 725979 OU.C 274021 46973 88281 59 2 672034 o9. O 94580!) 11.2 726284 50.fi 273716 46999 88267 58 3 672321 39.5 945733 11.2 726588 50.7 273412 47024 88254 57 4 672558 39. 5 945666 11.2 726892 50.7 273108 47050 88240 56 5 672795 39. 5 945598 11.2 727197 50.7 272803 47076 88226 55 6 673032 39. 4 945531 11.2 727501 50.7 272499 47101 88213 54 7 673268 39. 4 945464 11.2 727805 50-7 272195 47127 88199 53 8 673505 39. 4 945396 11.3 728109 50-6 271891 1 47153 88185 52 9 673741 39. 4 945328 11 .3 728412 50-6 271588 47178 88172 51 10 11 12 673977 9.674213 674448 39. 3 39.3 39.3 OQ O 945261 9.945193 945125 11.3 11.3 11.3 Uo 728716 9.72902C 729323 50-6 50.6 50.6 KA K 271284 10.270980 270677 47204 47229 47255 88158 88144 88130 50 49 48 13 674684 oy . - 945058 . 729626 50 -5 270374 47281 88117 47 14 674919 39. 2 OQ O 944990 11.3 no 729929 5Q.5 270071 47306 88103 46 15 675155 oy J OQ 944922 . no 730233 50-5 FiA K 269767 47332 88089 45 16 675390 oy . & OQ 1 944854 . o no 730535 O(J.O Kf\ ff 269465 47358 88075 44 17 675624 oy . i OQ -1 944786 o UQ 730838 DU.O 269162 47383 88062 43 18 675859 oy . i OQ 1 944718 .0 Uo 731141 50.4 Kn A 268859 47409 88048 42 19 676094 OJ. 1 OQ I 944650 . o 731444 O(J.4 268556 47434 88034 41 20 676328 oy . i OQ A 944582 11.3 731746 50.4 268254 47460 88020 40 21 9.676562 oy . o OQ A 9.944514 11.4 9.732048 50-4 10.267952 47486 88006 39 22 676796 jy . (j OQ A 944446 11.4 732351 50-4 267649 47511 87993 38 23 677030 oy . u OQ A 944377 11.4 732653 5Q. 3 267347 47537 87979 37 24 677264 oJ . U OQ Q 944309 11.4 UA 732955 50-3 267045 47562 87965 36 25 677498 oo . y OQ Q 944241 .* UA 733257 50-3 266743 47588 87951 35 26 677731 oo . y OQ Q 944172 .4 UA 733558 50-3 266442 47614 87937 34 27 677964 oo y OQ C 944104 .4 UA 733860 50-3 266140 4763987923 33 28 678197 oo . O OQ Q 944036 .1 HA 734162 50-2 265838 4766587909 32 29 678430 oo . o oQ G 943967 .4 734463 50-2 265537 4769087896 31 30 678663 oo . o OQ Q 943899 11.4 734764 50-2 265236 4771687882 30 31 9.678895 OO . O OQ 7 9.943830 11.4 9.735066 50-2 10.264934 4774187868 29 32 679128 oc t OQ 7 943761 11.4 735367 50.2 264633 4776787854 28 33 679360 OO. I OQ rf 943693 11.4 735668 50.2 264332 4779387840 27 34 35 679592 679824 oo. / 38.7 oQ {? 943624 943555 11.5 11.5 735969 736269 50.1 50.1 264031 263731 47818,87826 47844 87812 26 25 36 680056 OO .D OQ p 943486 11.5 736570 50.1 263430 47869 87798 24 37 680288 oo.o 38 6 943417 11.5 UK 736871 50.1 S;A i 263129 47895 87784 23 38 680519 38*5 943348 o UK 737171 Du. 1 ^A A 262829 47920 87770 22 39 680750 38 5 943279 UK 737471 OU. U ^A n 262529 47946 87756 21 40 680982 OQ * K 943210 . O He; 737771 ~>u. u 262229 47971 87743 20 41 9.681213 oo . O 00 K 9.943141 .0 9.738071 50.0 10.261929 47997 87729 19 42 681443 oo. O 00 A 943072 11.5 738371 50.0 * A A 261629 48022 87715 18 43 681674 oo. 4 00 ,. 943003 11.5 738671 oO.O 261329 48048 87701 17 44 681905 00.4 00 A 942934 11.5 738971 49.9 261029 48073 87687 16 45 682135 00.4 OQ A 942864 11.5 nc 739271 49.9 4n o 260729 48099 B7673 15 46 682365 oo . 4 OO 942795 . O U? 739570 4y .y 4n o 260430 4812487659 14 47 682595 oo , o 38 3 942726 .O Uc 739870 4y . y 4.Q Q 260130 48150 B7645 13 48 682825 OQ O 942656 . o Ua 740169 *iy . y AQ Q 259831 48175 B7631 12 49 683055 oo . o OQ 942587 . o Ua 740468 rty . y 4Q 8 259532 48201 37617 11 50 683284 OO . o OQ 9 942517 . o nf! 740767 4y . o 4.Q R 259233 48226 37603 10 51 .683514 oo , 4 OQ 9 9.942448 . O Ua 9.741066 4y . o 4.Q 8 0.258934 48252 37589 9 52 683743 oo . 4 OQ 9 942378 . o Uf; 741365 4y . o 4Q 8 258635 48277 37575 8 53 683972 oo , ^ OQ 942308 . D Uc 741664 iy .o 4Q 8 258336 48303 37561 7 54 684201 OO . w 00 1 942239 .O Uc 741962 4y , o 4Q 7 258038 48328 37546 6 55 684430 OO . 1 OQ 1 942169 .O nf* 742261 iy . / 257739 48354 37532 5 56 684658 OO. 1 OQ 1 942099 D no 742559 49.7 257441 48379 37518 4 57 684887 OO . 1 942029 .0 U/> 742858 49.7 257142 48405 37504 3 58 685115 38.0 941959 .0 743156 49. 7 256844 48430 37490 2 59 685343 38. 941889 11.6 743454 49. 7 256546 48456 37476 1 GO 685571 38.0 941819 11.7 743752 49.7 256248 48481 37462 Cosine. Sine. Cotaiig. Tang. N. cos. N.sine. / 61 Degrees. 50 Log. Sines and Tangents. (29) Natural Sines. TABLE II. Sine. D. 10" Cosine. |D. 10" Tang. D. 10" Cotang. N. sine. N. cos. 9.685571 9.941819 9.743752 0.256248 48481 87462 60 1 685799 38.0 941749 11 .7 744050 or 255950 48506 37448 59 2 686027 37.9 941679 11 .7 nrt 744348 or 255652 48532 37434 68 3 686254 QT f\ 941609 . 4 UT 744645 AQ f\ 255355 48557 37420 67 ! 4 686482 o i . y 0-7 Q 941539 , I U7 744943 4*J . O 255057 48583 87406 56 5 6 686709 686936 o / . y 37.8 0-7 Q 941469 941398 . / 11.7 U7 745240 745538 49*6 49 5 254760 254462 48608 48634 87391 87377 55 54 7 687163 / .0 QT Q. 941328 , / U7 745835 49 6 254165 48659 87363 53 8 687389 O I , C> QT Q 941258 . / nrj 746132 4Q 5 253868 48684 87349 52 9 687616 O/ . QT 1 941187 / Un 746429 *J . O 253571 48710 87335 51 10 687843 Ot . / 941117 . 4 nrj 746726 49 5 253274 48735 87321 50 11 .688069 017 17 9.941046 . / no 9.747023 49 4 10.252977 48761 87306 49 12 688295 O/ . / 940975 .O no 747319 49 4 252681 48786 87292 48 13 688521 *}?* 940905 .0 no 747616 4*7 .^I 49 4 252384 48811 87278 47 14 688747 O / . D QT fi 940834 . o UQ 747913 49 4 252087 48837 87264 46 15 688972 O/ .O OT tZ 940763 .0 Uo 748209 49 4 251791 48862 87250 45 16 689198 at .O 940693 .0 748505 49 *} 251495 48888 87235 44 17 689423 37.6 07 X 940622 11,8 UQ 748801 49 3 251199 48913 87221 43 18 689648 o i . O OT X 940551 . o UQ 749097 49 3 250903 48938 87207 42 19 20 689873 690098 o / , O 37.5 940480 940409 , o 11.8 no 749393 749689 49*3 250607 250311 48964 48989 87193 87178 41 40 21 22 3.690323 690548 37.5 37.4 Q<7 A 9.940338 940267 .O 11.8 Uo 9.749985 750281 49*3 AQ 10.250015 249719 4901487164 4904087150 39 38 23 690772 o/ .4 OT A 940196 .0 no 750576 **y . *. 4.9 Q 249424 49065 87136 37 24 690996 ol .4 940125 .0 750872 4s*? , ~> M\ o 249128 49090 87121 36 25 691220 37.4 07 o. 940054 11.9 nq 751167 49*2 248833 49116 87107 35 26 691444 o I . o OT O 939982 . j no 761462 4Q 9 248538 49141 87093 34 27 691668 o7 .0 939911 y UQ 751757 rit/ . M 4Q 9 248243 49166 87079 33 28 691892 37.3 OT O 939840 . y Un 752052 T:J . ^ 49 1 247948 49192 87064 32 29 692115 o7 .0 939768 .y nn 752347 4Q 1 247653 49217 87050 31 30 31 692339 9.692562 37 ..2 37.2 07 o 939697 9.939625 .y 11.9 Uq 752642 9.752937 T:*- , 1 49.1 49 1 247358 10.247063 49242 49268 87036 87021 30 29 32 692785 o 1 . 6 OT 1 939554 . y UQ 753231 49*1 246769 49293 87007 28 33 693008 Ol . 1 939482 . y no 753526 4Q 1 246474 49318 86993 27 ; 34 693231 37.1 OT 1 939410 .y Uq 753820 4*7 . 1 49 ft 246180 49344 86978 26 35 693453 Ol . i OT 1 939339 *y Uq 754115 **7 . U 49 ft 245886 49369 86964 25 36 693676 Ol , 1 939267 y 12 754409 T:*7 , U 49 246591 49394 86949 24 37 693898 37.0 OT A. 939195 754703 49 ft 245297 49419 86935 23 38 694120 o7 . U OT A 939123 |0*A 754997 **7 . U 4Q A 245003 49445 86921 22 39 694342 61 . U 939052 1 Q A 755291 T:*7 . U 49 ft 244709 49470 86906 21 40 694564 37.0 oft O 938980 U. U 1 Q f\ 755585 1*7 . U 48 9 244415 49495 86892 20 41 42 9.694786 695007 06. y 36.9 36 9 9.938908 938836 I .w . U 12.0 12 ft 9.755878 756172 48 . 9 48 9 10.244122 S43828 49521 49546 86878 86863 19 18 43 695229 938763 1* . U 1 2 ft 756465 4g g 243536 49571 86849 17 44 695450 oft ft 938691 !*$ . U 12 756759 48*9 243241 49596 86834 16 45 695671 OO.O Oft Q 938619 12 ft 757052 48*9 242948 49622 86820 15 46 695892 OO.O oft Q 938547 JU* . U 19 ft 757345 242655 49647 86805 14 47 690113 OO.O 938475 1> . U 12 757638 48*8 242362 49672 86791 13 48 696334 36.8 oft T 938402 19 1 757931 ^o .0 242069 49697 86777 12 49 696554 ot>. / oft 7 938330 i ^ . i 12 1 758224 48.8 241776 49723 86762 11 50 696775 oO . 1 938258 12* 1 758517 48 8 241483 49748 86748 10 51 9.696995 36. 7 9.938185 9.758810 40 . o 48 8 10.241190 49773 86733 9 52 697215 36. 7 938113 12 1 759102 4o . o 48 7 240898 49798 86719 8 53 697435 36.6 Oft ft 938040 12 1 769395 4o . I 48 7 240605 49824 86704 7 54 697654 OO. O 937967 1^ . L 759687 48 7 240313 49849J86690 6 55 01)7874 36.6 937895 19*1 759979 4o , / 48 7 240021 4987486675 6 56 698094 36.6 Oft K 937822 1-i. 1 19 1 760272 4o . * 48.7 239728 4989986661 4 67 698313 oO. o 937749 AZ , 1 -JO 1 760564 48*7 239436 4992486646 3 58 098532 36.5 oft K 937676 760856 4o . i 48 6 239144 4995086632 2 59 698751 OO.O Oft K 937604 10 1 761148 48 6 238852 49975 80617 1 60 698970 oO.O 937531 1 - . 1 761439 238561 5000U 80603 Cosine. Sine. Cotang. Tang. N. co* N.pim- 60 Degrees. TABLE 11. Log. Sines and Tangents. (30) Natural Sines. 51 Sine. D. 1U Oft A Cosine. D. 10 Tang. D. 10 Co tang. 10.238561 N. sint N. cos 60 9.698970 9.937531 19 1 9.761439 AQ f! 5000 86603 1 699189 OO T or* A 937458 1.0. J 1 O 761731 4o.O 238269 5002 86588 59 c 699407 OO. 4 qc A 937385 15*. 19 762023 48.6 AQ fc 237977 5005 86573 58 699626 OO . T Oft A 937312 IA , 1 9 762314 43. AQ ft 237686 50076 86559 57 ^ 699844 OO . T lfi 937238 1^ . 1 9 762603 4o .0 AQ 237394 5010 86544 56 700062 oo . * O 937165 iA . 762897 4o .O 237103 50126 86530 55 6 700280 oo. Of? 937092 12. 10 763188 48.5 AQ r^ 236812 5015 86515 54 7 700498 oo . < oc* 937019 !* 763479 4o.O 236521 50176 86501 53 8 700716 oo. 936946 12.5 763770 48. 1 236230 5020 86486 52 700933 oo. Oft 936872 12. 1 9 764061 48.5 AQ K. 235939 5022~ 86471 51 10 701151 oo . Oft 936799 \. , 19 764352 4o . O AQ / 235648 50252 86457 50 11 9.701368 OO . or* 9.936725 4w 1 O 9.764643 4o .< 10.235357 50277 86442 49 12 701585 oo. or 1 c 936652 \2i . 764933 48.^ 235067 50302 86427 48 13 701802 oo. oo 1 936578 12.3 765224 48.^ 234776 5032~ 86413 47 14 702019 OO. 1 or* 1 936505 12.3 765514 48.4 234486 50352 86398 46 15 702236 OO. I O^J 1 936431 12.3 765805 48.4 234195 5037" 86384 45 16 702452 Jo. i or* 1 936357 12.3 766095 48.4 233905 50403 86369 44 17 702669 OO. 1 936284 12.3 766385 48.4 233615 50428 86354 43 18 702885 36. qo A 936210 12.3 10 o 766675 48.3 AQ 233325 50453 86340 42 19 703101 OO v/ 936136 116. O 766965 4o.d 233035 50478 86325 41 20 703317 36.0 QJ ft 936062 12.3 10 o 767255 48.3 232745 50503 86310 40 21 9.703533 oo.U 9.935988 lie. a 9.767545 48.3 10.232455 50528 86295 39 22 23 703749 703964 35.9 35.9 QK Q 935914 935840 12.3 12.3 10 o 767834 768124 48.3 48.3 232166 231876 50553 50578 86281 86266 38 37 24 704179 oo.y q~ o 935766 1-6. o 1 r\ A 768413 48.2 231587 50603 86251 36 25 704395 oo.y QK Q 935692 ll.4 1O xi 768703 48.2 231297 50628 86237 35 26 704610 oo.y q- 935618 lii.4 10 1 768992 48.2 231008 50654 86222 34 27 704825 OO.O q~ Q 935543 lxi.4 in /i 769281 48.2 230719 50679 86207 33 28 705040 OO .0 q- Q 935469 12.4 1O yl 769570 48.2 230430 50704 86192 32 29 705254 oo.o q- Q 935395 lz.4 769860 48.2 230140 50729 86178 31 :;* 705469 OO. O- T 935320 12.4 770148 48. 1 229852 50754 86163 30 31 9.705683 00. / O" n 9.935246 12.4 9.770437 48.1 10-229563 50779 86148 29 32 705898 oo. i O- 1 1 935171 12.4 770726 48.1 229274 50804 86133 28 33 706112 01. I O~ 1 935097 12.4 771015 48.1 228985 50829 86119 27 34 706326 00. 1 93502-2 12.4 771303 48.1 228697 50854 86104 26 35 706539 35.6 934948 12.4 771592 48.1 228408 50879 86089 25 36 706753 35.6 934873 12.4 771880 48.1 228120 50904 86074 24 37 706967 35.6 934798 L2.4 772168 48.0 227832 50929 86059 23 38 707180 35.6 934723 L2.5 772457 48.0 227543 50954 86045 22 39 707393 35.5 Q" K 934649 12.5 772745 48.0 227255 50979 86030 21 40 707606 OO.O O" K 934574 12. 5 773033 48.0 226967 51004 86015 20 41 9.707819 OO.O OK e 9.934499 12.5 9.773321 48.0 10-226679 51029 86000 19 42 708032 OO.O OK A 934424 L2.5 773608 48.0 226392 51054 85985 18 43 708245 OO.4 O" /I 934349 L2.5 773896 47.9 226104 51079 85970 17 44 708458 oo.4 934274 12.5 774184 47.9 225816 51104 85956 16 45 703670 1 35.4 934199 12. 5 774471 47.9 225529 51129 5941 iO 46 708882 35.4 934123 12. 5 774759 47.9 225241 51154 5926 14 47 709094 35.3 OX Q 934048 12.5 1O K 775046 47.9 A^l O 224954 51179 5911 13 48 70930ci OO.O 933973 Lii.O 775333 47. y 224667 51204 5896 12 49 709518 35.3 933898 12.5 775621 47.9 224379 51229 5881 11 50 709730 35.3 O" O 933822 12.6 775908 47.8 4*y Q 224092 51254 5866 10 51 ) . 709941 OO.O 9.933747 L2.6 9.776195 7.0 0-223805 51279 5851 9 52 710153 35.2 933671 L2.6 776482 47.8 223518 51304 5836 8 53 710364 35.2 933596 L2.6 776769 47.8 223231 51329 5821 7 54 710575 35.2 933520 L2.6 777055 47.8 222945 51354 5806 6 55 71U786 35.2 933445 [2.6 777342 47.8 222658 51379 5792 5 56 710967 35.1 933369 i2.6 777628 47.8 222372 51404 5777 4 57 711208 35. 1 933293 [2.6 777915 47.7 222085 51429 5762 3 58 71M19 35. I 933217 ^2.6 778201 47.7 221799 51454 5747 2 59 711629 oo. 1 933141 .2.6 778487 47.7 221612 51479 5732 1 60 711839 35.0 933066 .2.6 778774 17.7 221226 "1504 5717 Cosine. Sine. Cotang;. Tang. N. cos. \.Kine. 59 Degrees. 52 Log. Sines and Tangents. (31) Natural Sines. TABLE II. Sine. |D. 10" Cosine. D. 10" Tang. D. 10" Cotang. N.sine.jN. cos. 9.711839 9.933036 9.778774 10.221226 61504185717 60 1 712050 35.0 qx A 932990 12.6 779060 47.7 47 7 220940 5162985702 59 2 712260 oO . U 932914 I ~ . / 779346 220654 5 1554 185687 58 3 712469 35.0 932838 12.7 779632 47.6 220368 51579185672 57 4 712679 34.9 932762 12.7 779918 47 .6 220082 61604185657 56 6 712889 34.9 932685 12.7 780203 47.6 219797 61628185642 55 6 713098 34.9 932609 12.7 780489 47.6 219511 51653 85627 54 713308 34.9 932533 12.7 780775 47.6 219225 51678 85612 53 8 713517 34.9 932457 12.7 781060 47 .6 218940 61703 85597 52 9 10 713726 713935 34.8 34.8 932380 932304 12.7 12.7 781346 781631 47.6 47.5 218654 218369 51728 51753 85582 85567 51 50 11 9.714144 34.8 9.932228 }% l 19. 781916 47.5 10.218084 51778 85551 49 12 13 14 714352 714561 714769 34.8 34.7 34.7 932151 932075 931998 li* . 7 12.7 12.8 782501 782486 782771 47.5 47.5 47.5 217799 217514 217229 51803 J85636 51828i8552l 51852J85506 48 47 46 15 714978 34.7 931921 12.8 783056 47.6 216944 51877185491 45 16 715186 34.7 .d 7 931845 12.8 19 S 783341 47.5 47 5 216659 5190285476 44 17 18 715394 715602 o4 . i 34.6 931768 931691 1-4 . O 12.8 783626 783910 47^4 216374 216090 61927185461 51952185446 43 42 19 715809 34.6 931614 12.8 784195 47.4 Art A 215805 51977J85431 41 20 716017 34.6 Q/1 fi 931537 12.8 784479 47 .4 4T A 215521 52002 85416 to 21 9.716224 o4.D 9.931460 12. 8 9.784764 4 / . 4 10.215236 52026 85401 39 22 716432 34.5 931383 12.8 785048 47.4 214952 52051 185385 38 23 716639 34.5 931306 12.8 785332 47.4 214668 62076 85370 37 24 716846 34.5 MR 931229 12.8 1O Q 785616 47.3 47 3 214384 52101 85355 36 25 26 717053 717259 . O 34.5 * A 931152 931075 1.4. y 12.9 1O O 785900 786184 47 . 3 47 1 214100 213816 52126 52151 85340 85325 35 34 27 717466 O^i . Q QA A 930998 i^ . y 786468 4/ . o 47 Q 213632 52175 85310 33 28 717673 O4 . *i 04 A 930921 12 . 9 786752 4 / . o 213248 52200185294 32 29 71787912? ? 930843 12.9 787036 47 . 3 212964 52225 85279 31 30 31 32 718085 9.718291 718497 34.3 34.3 0/1 Q 930766 9.930688 930611 12.9 12.9 12.9 787319 9.787603 787886 47.2 47.2 /i-7 o 212681 10.212397 212114 52250 52275 52299 85264 85249 85234 30 29 28 33 718703 ISTS 930533 12.9 788170 4 / . Z 47 O 211830 52324 85218 27 34 718909 O*. U 930456 1.2.9 788453 4 / . Z 211547 62349 85203 26 35 719114 34.3 930378 12.9 788736 47.2 211264 52374 85188 25 36 719320 34.2 930300 12.9 789019 47.2 210981 52399 85173 24 37 719525 34.2 Mo 930223 13.0 789302 47.2 Arj -t 210698 52423 85157 23 38 719730 .A 930145 13. 789585 4/ . 1 210415 62448 85142 22 39 719935 0/4 1 930067 13.0 789868 AIJ i 210132 52473 85127 21 40 720140 o4. i 929989 13.0 790151 47 . 1 /IT 1 209849 52498 85U2 20 41 9.720345 41 9.929911 13.0 9.790433 47.1 10.209567 52522 85096 19 42 720549 0/1 1 929833 13.0 790716 /IT 1 209284 52647 85081 18 43 720754 !J-i 929755 13.0 790999 47.1 /IT 1 209001 52572 85066 17 44 720958 iq? 929677 13.0 791281 47 . 1 208719 52597 85051 16 45 721 162 oT 929599 13.0 791563 47.1 208437 52621 85035 15 46 721366 ! ? 929521 13.0 791846 47.0 208154 52646 85020 14 47 721570, q7 ( } 929442 13.0 792128 47.0 207872 62671 85005 13 48 791774 JJ S 929364 13.0 792410 47.0 /IT A 207590 52696 84989 12 49 731978 :Q 929286 13 . 1 Q 792692 47.0 47 n 207308 52720 84974 11 50 722181 XT Q 929207 lo . 792974 4/ .U /IT A 207026 62745 84959 10 51 9.722385 XX a 9.929129 13. 9.793266 47.0 10.206744 52770 84943 9 52 722588 XX Q 929050 13. 793538 47.0 206462 52794 84928 8 53 722791 |-*{ 5 928972 13. 793819 46.9 206181 52819 84913 7 54 722994 XX 928893 13. 794101 46.9 205899 52844 84897 6 55 723197 *j a 928815 13. 794383 46.9 205617 52869 84882 5 56 723400 XX 2 928736 13. 794664 46.9 205336 52893 84866 4 57 723603 JJ 2 928657 13. 1 O 794945 46.9 205055 52918 84851 3 58 723805 "-L 928578 lo. 1 O 795227 46 . 9 204773 52943 84836 2 59 724007 ] XX ,1 928499 lo. 1 O 795508 46 . 9 1f Q 204492 5296 i 84820 1 60 724210, 928420 lo. 795789 46. o 204211 |i 62992 84805 "Cosine. I Sine. Cotang. Tang. lN.coB.JN.sine. T~ 58 Degrees. TABLE II. Log. Sines and Tangents. (32) Natural Sines. 53 Sine. D. 10" Cosine. |D. 10" Tang. (D. 10"j Cotaiig. N. sine. N . cos . 9.724210 9.928420 9.795789 AC Q 10.204211 52992 84805 60 1 724412 33.7 OO O 928342 13.2 100 796070 46.h Aft. ti 203930 53017 84789 59 2 724614 OO . / 928263 lo . Z 796351 4u .0 203649 53041 84774 58 3 724816 33.6 928183 13.2 796632 46.8 203368 53066 84759 57 4 725017 33.6 928104 13.2 796913 46.8 203087 53091 84743 56 5 725219 33.6 928025 13.2 797194 46.8 202806 53115 84728 55 6 725420 33.6 927946 13.2 797475 46.8 2u2525 63140J84712 54 7 8 725622 725823 33.5 33.5 927867 927787 13.2 13.2 797755 798036 46.8 46.8 202245 201964 53164J84697 5318984681 53 52 9 726024 33.5 927708 13.2 798316 46.7 201684 5321484666 51 10 726225 33.5 927629 13.2 798596 46.7 201404 5323884650 50 11 9.726426 33.5 9.927549 13.2 9.798877 46.7 4 n rj 10.201123 53263|84635 49 12 13 726626 726827 33.4 33.4 927470 927390 13.2 13.3 799157 799437 4o. / 46.7 .*-* n 200843 200563 53288 53312 84619 84604 48 47 14 727027 33.4 927310 13 . 3 799717 46 . 7 AC 17 200283 53337 84588 46 15 727228 33.4 927231 13.3 799997 4t>. 7 200003 53361 84573 45 16 727428 33.4 OO Q 927151 13.3 1 Q O 800277 46.6 AC 199723 53386 84557 44 17 727628 06 . o 927071 lo. O 1 Q o 800557 4b . D Ad fi 199443 53411 84542 43 18 727828 33 . 3 OO Q 926991 10. O 1 q 800836 4o . o 4f> fi 199164 53435 84526 42 19 728027 OO . O OQ 9. 92691 1 lo . o 1 q q 801116 4O . D Afi fi 198884 53460 84511 41 20 21 22 23 728227 9.728427 728626 728825 00 . 33.3 33.2 33.2 926831 9.926751 926671 926591 lo. d 13.3 13.3 13.3 801396 9.801675 801955 802234 4O . O 46.6 46.6 46.6 198604 10.198325 198045 197766 53484 84495 5350984480 5353484464 53558 84448 40 39 38 37 24 729024 33.2 926511 13.3 802513 46.5 197487 53583 84433 36 25 26 729223 729422 33.2 33.1 926431 926351 13.4 13.4 802792 803072 46.5 46.6 197208 196928 5360784417 5363284402 35 34 27 28 729621 729820 33.1 33.1 926270 926190 13.4 13.4 803351 803630 46.5 46.5 196649 196370 5365684386 5368184370 33 32 29 730018 33. 1 OQ n 926110 13.4 i q A 803908 46.5 AC r. 196092 53705J84355 31 :;o 31 730216 9.730415 OO. U 33.0 926029 9.925949 lo.4 13.4 804187 9.804466 40 . 46.5 195813 10.195534 53730J84339 53754184324 30 29 32 730613 33.0 oo n 925868 13.4 804745 46. 195255 53779^84308 28 33 730811 33.0 925788 13.4 805023 46. 194977 53804^4292 27 34 731009 33.0 925707 13.4 805302 46. 194698 63828 84277 26 35 731206 32.9 925626 13.4 805580 46. 194420 63853 84261 25 36 37 731404 731602 32.9 32.9 Of) Q 925545 925465 13.4 13.5 i q e 805859 806137 46. 46. /in 4 194141 193863 53877 [84245 53902184230 24 23 38 731799 O-i . i> 925384 lo , D 806415 40 . ^ 193585 53926 84214 22 39 731996 32.9 on W 925303 13.5 806693 46.3 AC O 193307 53951 84198 21 40 732193 32.0 or Q 925222 13 . 5 806971 46 .0 AC Q 193029 53975 84182 20 41 9.732390 d 2.o on Q 9.925141 13. 5 9.807249 46.o 10.192751 54000184167 19 42 732587 w.o O -x 925060 13.5 807527 46.3 192473 5402484151 18 43 44 732784 732980 32.0 32.8 or i1 924979 924897 13.5 13.5 807805 808083 46.3 46.3 192195 191917 5404984135 5407384120 17 16 45 733177 32.7 924816 13.5 808361 46.3 191639 54097 84104 15 46 733373 32.7 924735 13.5 1O C. 808638 46.3 AC a 191362 54122 84088 14 47 733569 32.7 924654 lo.b 808916 46.2 191084 54146 84072 13 48 733765 32.7 on O 924572 13.6 809193 46.2 190807 54171 84057 12 49 50 733961 734157 32. / 32.6 924491 924409 13.6 13.6 809471 809748 46.2 46.2 190529 190252 5419584041 5422084025 11 10 51 9.734353 32.6 9.924328 13.6 9.810025 46.2 10.189975 54244 ! 84009 9 52 734549 32.6 924246 13.6 810302 46.2 189698 54269 ! 83994 8 53 734744 32.6 924164 13.6 810580 46.2 189420 54293 83978 7 54 734939 32.5 924083 13.6 810857 46.2 189143 54317 83962 6 55 56 735135 735330 32.5 32.5 924001 923919 13.6 13.6 811134 811410 46.2 46.1 188866 188590 5434283946 54366:83930 5 4 57 735525 32. 5 923837 13.6 811687 46. 1 188313 54391 83915 3 58 735719 32. 5 923755 13.6 811984 46. 1 188036 54415 83899 2 59 735914 32.4 923673 13.7 812241 46. 1 187759 54440 83883 1 60 736109 32.4 923591 13.7 812517 46.1 187483 54464 83867 ""Cosine. Sine. Cotang. Tang. NTcos. N. si ne. ~i~~ 57 Degrees. 54 Log. Sines and Tangents. (33) Natural Sines. TABLE IT. Sine. IX 10 Cosine. I). 10 Tang. D. 10 Cotang. ,N. sine. N. cos 9.736109 OO A 9.923591 9.812517 10.187482 54464 83867 60 1 736303 62 . < 923509 13.1 812794 46.] 187206 64488 83851 59 2 736498 32.^ 923427 13.7 i q 7 813070 46.] 46 . ] 186930 64513 83835 58 c 736692 62 . 923345 lo . i 813347 186653 54537 83819 57 4 736886 32. i qo 923263 13.7 1 q 7 813623 46. C 186377 54561 83804 56 5 737080 o^ . t 923181 lo . i 1 q 7 813899 AT ( 186101 54586 83788 55 f> 737274 O-u . I 923098 lo. / 814175 4o .(. 185825 54610 83772 54 7 737467 32. J 923016 13.7 814452 46. ( 185548 54635 83756 53 8 737661 OO 922933 13.1 814728 46 . ( 185272 54659 83740 52 9 737855 62 . 922851 13.1 815004 46 . t 184996 54683 83724 51 10 738048 OO 922768 13.7 815279 46. 1 184721 54708 83708 50 11 9.738241 62 . oo 9.922686 13.!- 9.815555 46. ( 10.184445 54732 8369L 49 12 738434 62. OO 922603 13. 815831 45. i 184169 54756 83670 48 13 738627 62. 922520 13. 816107 45. i: 183893 54781 83660 47 14 738820 32. 1 922438 13. 816382 45.- 183618 54805 83645 46 15 739013 32.1 922355 13.8 1 q o 816658 45. t 45 ^ 183342 54829 83629 45 16 739206 qo 1 922272 lo.o 1 q o 816933 183067 64854 83613 44 17 739398 .> w . 1 922189 lo.o 817209 45. o 182791 54878 83597 43 18 739590 32. 1 oo f\ 922106 13. 817484 45. 182516 54902183581 42 19 739783 62. U 922023 13.8 817759 A- C 182241 54927 83565 41 20 21 739975 9.740167 32. 32.0 OO A 921940 9.921857 13.8 13.8 818035 9.818310 45!8 A z. Q 181965 10.181690 54951 54975 83549 83533 40 39 22 740359 d2.U OO A 921774 13.9 818585 45.0 A e Q 181415 54999 83517 38 23 740550 J2.0 921691 13.9 818860 45.8 181140 55024 83501 37 24 740742 31.9 921607 13.9 819135 45.8 180865 55048 83485 36 25 740934 31.9 01 o 921524 13.9 819410 45.8 A C Q 180590 5507283469 35 26 741125 oi .y 921441 13.9 819684 40.0 A K. Q 180316 5509783453 34 27 741316 31 .9 921357 13.9 819959 45.O A * Q 180041 5512183437 33 28 741508 31.9 921274 13.9 820234 4o .0 AS. Q 179766 55145 83421 32 29 741699 31.8 921190 13.9 820508 4o.o 179492 5516983405 31 30 741889 31.8 921107 13.9 820783 45.7 179217 5519483389 30 31 32 33 34 9.742080 742271 742462 742652 31.8 31.8 31.8 31.7 9.921023 920939 920856 920772 13.9 13.9 14.0 14.0 9.821057 821332 821606 821880 45.7 45.7 45.7 45.7 10.178943 178668 178394 178120 55218 83373 55-242 83356 55266183340 55291 ! 83324 29 28 27 26 35 36 742842 743033 31 .7 31.7 Q1 *7 920688 920604 14.0 14.0 4r\ 822154 822429 45.7 45.7 177846 177571 55315 ! 83308 55339 83292 25 24 37 743223 ol . / o-i ry 920520 .U 4f\ 822703 45.7 177297 55363 83276 23 38 743413 ol . / Q1 fi 920436 .u 822977 45. 7 A " K. 177023 55388 83260 22 39 40 743602 743792 ol .O 31.6 Q1 P 920352 920268 [4.0 14.0 823250 823524 4o . o 45.6 176750 176476 5541283244 55436 J83228 21 20 41 0.743982 ol .0 9.920184 [4.0 9.823798 45.6 10.176202 55460 83212 19 42 744171 31.6 01 r 920099 [4.0 824072 45.6 175928 5548483195 18 43 744361 ol .D Ol K 920015 .4.0 824345 45.6 176655 5550983179 17 44 744550 ol . 5 O1 K 919931 4.0 824619 45.6 175381 55533 83163 16 45 744739 ol .5 01 K 919846 4.1 824893 45.6 175107 6555783147 15 46 744928 ol , 01 K 919762 4. 825166 45.6 174834 6558183131 14 47 745117 Ol . ql K 919677- 4. 825439 45 . 6 174561 5560583115 13 48 745306 ol . O 919593 . 825713 45 .6 174287 55630 83098 12 49 745494 31.4 919508 4. 825986 45.5 174014 55654183082 11 50 745683 1.4 919424 4. 826259 45.5 173741 C5678|83066 10 51 9.745871 .4 9.919339 4. .826532 45.5 0.173468 55702 83050 9 52 746059 919254 4. 826805 1:5.5 173195 55726 83034 8 53 54 746248 746436 .3 919169 919085 4.1 4.1 827078 827351 15 . 6 5K 172922 172649 5575083017 5577583001 7 6 55 56 57 746624 746812 746999 .3 .3 3 919000 918915 918830 4. 1 4.1 4.2 827624 827897 828170 . o 45.5 45.4 fL A 172376 172103 171830 6579982985 5582382969 5584782953 5 4 3 58 747187 2 918745 40 828442 tO , 4 171558 6587182936 2 59 747374 o o 918659 2 828715 rO.4 171285 5589582920 1 60 747562 o .2 918574 4.2 828987 :5.4 171013 65919|82904 Cosine. Sine. Cotang. Tang. \. cos. JN. sine. 56 Degrees. TABLE II. Log. Sines and Tangents. (34) Natural Sines. 55 Sine. D. 10 Cosine. D. 10 Tang. D. 10 Cotang. | N.sine N. cos 9.747562 qi 9 9.918574 Ho 9.828987 AK. A 10.171013 55919 82904 60 1 747749 OJL 01 o 918489 . -w 1 4 O 829260 4O . 4 Af A 170740 55943 82887 59 2 747936 ol . qi o 918404 14 . Z 829532 4O . 4 A?\ A 170468 j 55968 82871 58 3 748123 918318 * 829805 4O . 4 170195 155992 82855 57 4 748310 31 .1 918233 14.2 830077 45.4 169923 56016 82839 56 5 748497 31 .1 918147 14.2 1 1 O 830349 45.4 A ~ O 169651 56040 82822 55 6 748683 31 . 1 918062 14. Z Ho 830621 4o.d A ~ O 169379 56064 8280* 54 7 748870 31 . 1 917976 . A 830893 4o .0 169107 56088 82790 53 8 749056 31.1 t ( \ 917891 14.3 14 q 831165 45.3 168835 56112 82773 52 9 749243 ol . U 917805 14 . o 831437 4o . <J 168563 56136 82757 51 10 749426 31 .0 ol A 917719 14.3 Uq 831709 45.3 AZ. o 168291 56160 82741 50 11 12 9.749615 749801 ol . U 31.0 9.917634 917548 . o 14.3 Ho 9.831981 832253 4O . d 45.3 10.168019 167747 56184 56208 82724 182708 49 48 13 749987 31 .0 on Q 917462 .0 Uq 832525 45.3 167475 56232 82692 47 14 750172 du.y 917376 . O Uq 832796 XX O 167204 56256 82675 46 15 750358 30.9 917290 . d 833068 4O . d 166932 56280 82659 45 16 750543 30.9 917204 14. 3 833339 45.2 166661 56305 82643 44 17 750729 30.9 917118 14.3 833611 45 .2 166389 56329 82626 43 18 750914 30.9 917032 14.4 833882 45 .2 166118 56353 82610 42 19 751099 30.8 916946 14.4 834154 45 .2 165846 56377 82593 41 20 21 22 751284 9.751469 751654 30.8 30.8 30.8 916859 9.916773 916687 14.4 14.4 14.4 834425 9.834696 834967 45 .2 45.2 45.2 165575 10.165304 165033 56401 56425 56449 82577 82561 82544 40 39 38 23 751839 30.8 qA Q 916600 14.4 144 835238 45.2 164762 56473 82528 37 24 25 752023 752208 ol> .0 30.7 916514 916427 14.4 14.4 HA 835509 835780 45.2 164491 164220 56497 56521 82511 82495 36 35 26 752392 30 . 7 916341 , 4 836051 4o.l 163949 56545 82478 34 27 752576 30.7 916254 14.4 836322 45.1 A K 163678 56569 82462 33 28 752760 30.7 916167 14.4 836593 4o. 163407 56593 82446 32 29 752944 30.7 916081 14.5 836864 45 . 163136 56617 82429 31 30 753128 30.6 915994 14.5 1 A & 837134 45 . 162866 56641 82413 30 31 9.753312 30.6 9.915907 14. 9.837405 4o. 10.162595 5(5065 82396 29 32 33 753495 753679 30.6 30.6 915820 915733 14.5 14.5 837675 837946 45. 45. 162325 162054 56689 56713 82380 82363 28 27 34 753862 *" * 915646 14.5 1/1 K 838216 45.1 AK -t 161784 56736 82347 26 35 754046 IK S 915559 14. O He 838487 4o . 1 AK f\ 161513 56760 82330 25 36 754229 JX K 915472 . o He 838757 40 . 161243 56784 82314 24 37 754412 30-5 915385 .O Ue 839027 45 .0 160973 56808 82297 23 38 754595 ^ 915297 . O He 839297 45 . 160703 56852 82281 22 39 754778^ ? 915210 .O 839568 45 .0 160432 56856 82264 21 40 754960 ,X 915123 14. 5 14 fi 839838 45.0 4K n 160162 56880 82248 20 41 9. 755143! ,[{: 9.915035 14. o 14 fi 9.840108 . 10.159892 56904 82231 19 42 755326 ,X l 914948 14. D 840378 AK f\ 159622 56928 82214 18 43 765508 Si ; 914860 14.6 Uf! 840647 40 .u A " n 159353 56952 82198 17 44 755690 !,X 7 914773 .0 840917 4o . U A A n 159083 56976 82181 16 45 755872 JX J 914685 14.6 Uc 841187 44. y A A r\ 158813 57000 82165 15 46 47 756054 JX-J 756236 S J 914598 914510 .0 14.6 H& 841457 841726 44. y 44.9 A A n 158543 158274 57024 57047 82148 82132 14 13 48 756418 fX 3 914422 .0 Uc 841996 44. y 44 9 158004 57071 82115 12 49 766600 IX 1 914334 .0 Uc 842266 A A Q 157734 57095 82098 11 50 756782 iq" 914246 . 1/1 *7 842535 44 . y A A O 157465 57119 82082 10 51 9.756963 \ Z 3.914158 14. / t A *1 9.842805 44 . y A A Q 10.157195 57143 82065 9 52 757144 Ifg ij 914070 14. / 843074 44 . y 156926 57167 82048 8 53 913982 14.7 843343 44.9 A A n 156657 57191 82032 7 54 757507 1*2 2 913894 14.7 Ury 843612 44.9 A A O 156388 57215 82015 6 55 767688 IS!, 913806 . 7 843882 44. y A A 156118 57238 81999 5 56 767869 irX } 913718 14. 7 844151 44. o A A W 155849 57262 S1982 4 57 768050 fX, 913630 14. 7 844420 44. a A A a 155580 57286 81965 3 58 758230 |X } 913541 14.7 Hry 844689 44. a 44 fi 155311 57310 81949 2 59 758411 JX t 913453 . / 844958 44 , o A A Q 155042 57334 81932 1 60 758591 dU>1 913365 14.7 845227 44. o 154773 57358 81915 ~ Cosine, i S5neT~ Cotang. Tang. N. cos. \ sine. 55 Degrees. 56 Log. Sines and Tangents. (35) Natural Sines. TABLE II. Sine. D. 1U Cosine. D. lu" Tang. D. 10" Cotang. j N. sine. N. cos. o 9.758591 9.913365 .845227 0.154773 57358 31915 60 1 758772 30.1 913276 [4.7 845496 44.8 A 4 U 154504 57381 31899 59 2 758952 30.0 913187 [4.7 845764 44.0 A A G 154236 57405 31882 58 3 759132 30.0 913099 14.8 846033 44.0 A A Q 153967 57429 31865 57 4 759312 30 913010 14. 8 UQ 846302 44.0 A A Q 153698 57453 31848 56 5 759492 30. v 912922 .0 Uo 846570 44.0 A 4 1 153430 57477 31832 55 6 759672 30.0 912833 .0 Uo 846839 44. / 153161 57501 31815 54 7 759852 29 . 9 912744 .0 847107 44.7 152893 57524 31798 53 8 760031 29.9 912655 14.8 -1 A tt 847376 44.7 152624 57548 81782 52 9 10 760211 760390 29.9 29.9 912566 912477 14.0 14.8 Uo 847644 847913 44.7 44.7 Ail 152356 152087 57572 57596 81765 81748 51 50 11 9.760569 29 . 9 9.912388 .0 Uo 9.848181 41. / 10.151819 57619 81731 49 12 760748 29. o 912299 .0 848449 44.7 151551 57643 81714 48 13 7(50927 29.8 orv Q 912210 14.9 848717 44.7 151283 57667 81698 47 14 761106 29.0 c%n Q 912121 14.9 i A n 848986 44.7 151014 57691 81681 46 15 761285 29.0 912031 14.9 U(\ 849254 44.7 150746 57715 81664 45 16 761464 29.8 911942 .y 849522 44.7 150478 57738 81647 44 17 761642 29.8 911853 14.9 849790 44.7 A i 1 150210 57762 81631 43 18 19 761821 761999 29.7 29.7 911763 9U674 14.9 14.9 U(\ 850058 850325 44.6 44.6 149942 149675 57786 57810 81614 81597 42 41 20 762177 29.7 911584 .9 850593 44.6 A A Z 149407 57833 81580 40 21 9.762356 29.7 9.9H495 14.9 Ur\ 9.850861 44.D 10.149139 57857 81563 39 22 23 24 762534 762712 762889 29.7 29.6 29.6 911405 911315 911226 ,9 14.9 15.0 1 PC A 851129 851396 851664 44.6 44.6 44.6 A A (Z 148871 148604 148336 57881 81546 6790481530 5792881513 38 37 36 25 763067 ay. 6 911136 lo. u 1 K A 851931 44. 148069 57952 81496 35 26 763245 .49. D 911046 lo. u 1 K A 852199 44.6 A A (^ 147801 57976 81479 34 27 763422 29.6 910956 lo. U 1 K A 852466 44. D A A fi 147534 57999 81462 33 28 763600 29.6 910866 lo . 852733 44. o 147267 58023 81445 32 29 763777 29.5 910776 15.0 i K A 853001 44.5 A A K 146999 58047 81428 31 30 31 763954 9.764131 29^5 910686 9.910596 10 . U 15.0 853268 9.853535 44 . 44.5 146732 10-146465 5807081412 5809481395 30 29 32 764308 29.5 910506 15.0 853802 44.5 146198 5811881378 28 33 764485 29.5 910415 15.0 854069 44.5 A A K 145931 58141 81361 27 1 34 764662 29.4 910325 15.0 854336 44. 145664 58165 81344 26 35 764838 29.4 910235 15.1 854603 44.5 145397 58189 81327 25 36 765015 29.4 910144 15.1 854870 44. 5 145130 58212 81310 24 37 765191 29.4 910054 15.1 855137 44.5 144863 58236 81293 23 38 765367 29.4 909963 15.1 855404 44. 5 144596 58260 81276 22 39 765544 29.4 909873 16.1 855671 44.5 144329 58283 81259 21 40 765720 29.3 909782 15.1 855938 44.4 144062 58307 81242 20 41 9.765896 29.3 9.909691 15.1 9.856204 44.4 10-143796 58330 81225 19 42 766072 29.3 909601 15.1 856471 44 .4 143529 58354 81208 18 43 766247 29.3 909510 15. 1 856737 44.4 143263 58378 81191 17 44 766423 29.3 909419 15.1 857004 44.4 142996 58401 81174 16 45 766598 29.3 909328 15.1 857270 44.4 142730 58425 81157 15 46 766774 29.2 909237 15.2 857537 44.4 142463 58449 81140 14 47 766949 29.2 909146 15.2 857803 44.4 142197 58472 81123 13 48 767124 29.2 909055 15.2 858069 44.4 141931 58496 81106 12 49 767300 29.2 908964 15.2 858336 44.4 141664 58519 81089 11 50 767475 29.2 908873 15. 858602 44.4 141398 58543 81072 10 61 9.767649 29.1 9.908781 15. 9.858868 44.; 10-141132 58567 81055 9 52 767824 29.1 908690 15. 859134 44,; 140866 58590 81038 8 53 767999 29. 1 908599 15.- 859400 44J 140600 58614 81021 7 54 768173 29. 1 908507 15. 859666 44.J 140334 58637 81004 6 55 768348 29.1 908416 15. 859932 44.; 140068 58681 80987 5 56 768522 29. C 908324 15.; 860198 44.; 139802 58684 80970 4 67 768697 29. C 908233 15. c 860464 44.; 139536 58708 80953 3 58 768871 29.0 908141 15.^ 861)730 44.; 139270 58731 80936 2 59 769045 29.0 908049 15. J 860995 44.; 139005 58755 80919 1 60 769219 29.0 907958 16. 861261 44.; 138739 58779 80902 Cosine. Sine. Cotang. Tang. N. cos. N.sine 54 Degrees. TABLE II. Log. Sines and Tangents. (36) Natural Sines. 57 Sine. D. 10" Cosine. D. 10" Tang. D. 10" Cotang. N. sint;. N. cos. 1 o 9.769219 769393 769566 29.0 28.9 9.907958 907866 907774 15.3 15.3 9.861261 861527 861792 44.3 44.3 10.138739J 58779 138473 J58802 138208 58826 80902 80885 80867 60 59 58 3 4 769740 769913 28.9 28.9 907682 907590 15.3 15.3 86203S 862321. 44 2 4-i. 2 1 37942 i ! 58849:80850 137677 5887380833 57 56 j 5 770087 28.9 98 Q 907498 15.3 1 e q 862589 44.2 Ail O 137411 58896 80816 55 6 770260 ^o . y 90 Q 907406 10 . o 1 e. q 862854 i . Z A A O 137146 58920 80799 54 7 770433 ^O , O 98 8 907314 10 . | K A 863119 44 . MO 136881 58943 80782 53 8 770606 -o . o 98 8 907222 10. 4 1 X A. 863385 . ~ A A O 136615 58967 80765 52 9 770779 ^o . o 98 8 907129 lo.4 1 K A 863650 44 . ^ A A O 136350 58990 80748 51 10 770952 ZfJ . O 98 8 907037 iO . 4 IX A 863915 44 . Z A A O 136085 5901480730 50 11 9.771125 Z& . o OQ Q 9.906945 ID. 4 1 K. A 9.864180 44 . Z A A O 10.135820 59037 80713 49 12 771298 -iO . O QQ 7 906852 10. 4 i p- A 864445 44 . Z A A O 135555 59061 80696 48 13 14 771470 771643 -*-o . / 28.7 OQ n 906760 906667 1O.4 15.4 864710 864975 44 . Z 44.2 135290 135025 59084 80679 59108180662 47 46 15 771815 ~!o. 7 OQ n 906575 15.4 865240 44. 134760 59131J80644 45 16 771987 iio.7 OQ o 906482 15.4 1C A 865505 44. 134495 6915480627 44 17 772159 ^o.7 OU rf 906389 15.4 IK K. 865770 44. 134230 5917880610 43 18 772331 2o.7 906296 15. o 1 K K 866035 44. 133965 59201 80593 42 19 772503 28. 6 OQ r* 906204 15. o 1 p- e 866300 44. A A 133700 59225 80576 41 20 772675 Zfj . (j oQ r* 906111 10 . o 1 {- K 866564 44. 133436 59248 80558 40 21 9.772847 ZV> . O OQ f> 9.906018 15. o IK Z. 9.866829 44 . A A 10.133171 59272 80541 39 22 773018 Zo . D 905925 IO" 867094 44 . 132906 59295 80524 38 23 24 773190 773361 28.6 28.6 98 Pi 905832 905739 15.5 15.5 1^5 867358 867623 44. 44. 132642 132377 5931880507 5934280489 37 36 25 773533 *o . o OQ * 905645 10 . o IK K. 867887 44. 132113 59365 80472 35 26 773704 *o . O OQ e 905552 lo . o 1 * p^ 868152 44.1 131848 59389 80455 34 27 28 29 30 773875 774046 774217 774388 ijo.5 28.5 28.5 28.5 905459 905366 905272 905179 lo.o 15.5 15.6 15.6 868416 868680 868945 869209 44.0 44.0 44.0 44.0 131584 131320 131055 130791 5941280438 59436:80422 5945980403 59482180386 33 32 31 30 31 9.774558 28.4 OQ A 9.905085 15.6 i - fi 9.869473 44.0 A \ n 10.130527 5950680368 29 32 774729 ^O . 4 OQ A 904992 10 . u 1 K. R 869737 44 . U 130263 59529|80351 28 33 774899 o , 4 OQ A 904898 lo.o 1 K f\ 870001 44.0 129999 5955280334 27 j 34 775070 Zo .4 OQ A 904804 15. D 1 K K 870265 44.0 129735 5957680316 26 35 775240 Ho .4 98 A 904711 15 . o IK f; 870529 44.0 129471 5959980299 25 36 775410 Zo . 4 OQ O 904617 iO . O 1 - c 870793 44 . A A A 129207 5962280282 24 37 775580 ^o , o OQ O 904523 IO . D If P 871057 44. U 128943 59646 80264 23 38 775750 2o . d 904429 o.o 1^ *y 871321 44.0 128679 59669 80247 22 39 775920 28.3 OQ Q 904335 0. / 1 - o 871585 44.0 128415 59693 80230 21 40 776090 ~o. o OQ 904241 lo. / 1 n 7 871849 44.0 A<1 Q 128151 59716 80212 20 41 9.776259 Zo . o OQ q 9.904147 J O . < 1 n 7 9.872112 *o .y /!*} Q 10.127888 59739 80195 19 42 776429 Zo . O 9,8 9 904053 IO . I I ^ 7 872376 4o.y 127624 69763 80178 18 43 44 776598 776768 -^o . Z 28.2 OQ O 903959 903864 10.1 15.7 1 PI 7 8 i 2640 872903 43 .9 43.9 ,1Q O 127360 127097 5978680160 6980980143 17 16 45 776937 Zo . Z 903770 10.4 1r ^ 873167 4o .y 126833 59832 80125 15 46 777106 28.2 28 9, 903676 0. / 1^7 873430 43.9 4q q 126570 59856 80108 14 47 777275 z& . z 903581 IO , I 873694 4o . y 126306 59879 80091 13 48 49 777444 777613 28. 1 28.1 98 1 903487 903392 15.7 15.7 1 K C 873957 874220 43.9 43.9 A Q 126043 125780 ! 59902180073 ! 59926 80056 12 11 50 777781 ^o . 1 28 1 903298 1O.O 15 8 874484 4<_> . y 40 Q 125516 59949180038 10 51 9.777950 9.903202 1" Q 9.874747 4o . J 10.125253 69972 80021 9 52 53 54 55 56 57 778119 778287 778455 778624 778792 778960 28. 1 28.1 28.0 28.0 28.0 28.0 OQ A 903108 903014 902919 902824 902729 902634 O.o 15.8 15.8 15.8 15.8 15.8 1 ^ Q 875010 875273 875536 875800 876063 876326 43.9 43.9 43.8 43.8 43.8 43.8 A O Q 124990 124727 124464 124200 123937 123674 59995 | 60019 j 60042 ! 60065 ! 60089 160112 80003 79986 79968 79951 79934 79916 8 7 6 5 4 3 58 779128 Zo. 902539 lo . o 876589 4d.O 123411 60135 79899 2 59 779295 28 . 902444 15 .9 876851 43.8 123149 60158 79881 1 60 779463 27 . 9 902349 15.9 877114 43.8 122886 60182 79864 Cosine. S7ne7~ Co tang. Tang. N. cos N.sine. 53 Degrees. 58 Log. Sines and Tangents. (37) Natural Sines. TABLE II. i Sine. D. 10 Cosine. |D. 10 Tang. D. 10 Cotang. ||N .sine N. cos 9.779463 27 ^ 9.902349 i K n 9.877114 10.122886 60182 79864 60 1 779631 902253 15. IK n 877377 43.8 122623 60205 79846 59 2 779798 27 . 97 Q 902158 15. 1 K Q 877640 43.8 122360 60228 79829 58 3 779966 61 ,y 902063 lo.y 1 K n 877903 4d.c 122097 60251 79811 57 4 780133 97 Q 901967 15.9 i K Q 878165 43.8 121835 60274 79793 56 5 780300 M / . y 97 8 901872 10. y 1 K C 878428 ,40 Q 121572 60298 79776 55 6 780467 * / . o 97 S 901776 10. y -i K r 878691 4d . O 121309 60321 79758 54 7 780634 1 , O OT O 901681 lo.b 878953 43.8 121047 60344 79741 53 8 780801 21 . o 97 R 901585 15.9 1 ^ Q 879216 43.7 120784 160367 79723 52 9 780968 41 .0 901490 lo. y IK Q 879478 A3 120522|i60390 79706 51 10 781134 c*n * o 901394 lo . a 879741 4d. 120259 60414179688 50 11 9.781301 27.o 9.901298 16. C 9.880003 43. 10.119997 60437 79671 49 12 781468 27.7 901202 16. C 880265 43. 119735 60460 79658 48 13 781634 27.7 901106 16.0 880528 43. 119472 60483 79635 47 14 781800 27.7 901010 16.0 880790 43. 119210 60506 79618 46 15 781966 27.7 900914 16.0 881052 43. 118948 60529 79600 45 16 782132 27.7 900818 16.0 1 n n 881314 43. 118686 60553 79583 44 17 782298 27.7 900722 lo. U f r * f\ 881576 43. 118424 60576 79565 43 18 782464 27.6 r\r* r; 90&626 lo. U 1 r* f\ 881839 43 . 118161 60599 79547 42 19 782630 27. O 900529 lo. U i r* " A 882101 43 . 117899 60622 79530 41 20 782796 27.6 900433 lo. U 882363 43. 117637 60645 79512 40 21 9.782961 27.6 9.900337 16.1 9.882625 43.6 10.117375 60668 79494 39 22 783127 27.6 900242 16. 1 882887 43.6 117113 60691 79477 38 23 783292 27.6 900144 16.1 883148 43.6 116852 60714 79459 37 24 783458 27.5 900047 16. 1 883410 43.6 116590 60738 79441 36 25 783623 27.5 899951 16.1 883672 43.6 1 16328 60761 79424 35 26 783788 27.5 899854 16.1 883934 43.6 116066 60784 79406 34 27 783953 27.5 899757 16.1 884196 43.6 115804 6080? 79388 33 28 784118 27.5 899660 16.1 884457 43.6 115543 60830 79371 32 20 784282 27.5 899564 16. 1 884719 43.6 115281 60853 79353 31 39 784447 27.4 899467 16.1 884980 43.6 115020 60876 79835 30 31 9.784612 27.4 9.899370 16.2 9.885242 43.6 10.H4758 60899 79318 29 32 784776 27.4 899273 16.2 885503 43.6 114497 60922 79300 28 33 784941 27.4 899176 16.2 885765 43.6 114235 60945 79282 27 34 785105 27.4 899078 16.2 886026 43.6 113974 60968 79264 26 35 785269 27.4 898981 16.2 886288 43.6 113712 60991 79247 25 36 785433 27.3 898884 16.2 886549 43.6 113451 61015 79229 24 37 785597 27.3 898787 16.2 886810 43.5 113190 61038 79211 23 38 785761 27.3 898689 16.2 887072 43. 5 112928 61061 79193 22 39 785925 27.3 898592 16.2 887333 43.5 112667 61084 79176 21 40 41 42 786089 . 786252 786416 27.3 27.3 27.2 898494 9.898397 898299 16.2 16.3 16.3 887594 9.887855 888116 43.5 43.5 43.5 A O C 112406 10.112145 111884 61107 79158 6113079140 6115379122 20 19 18 43 786579 27.2 898202 16.3 888377 4o . 5 111623 61176 79105 17 44 786742 27.2 898104 16.3 888639 43 . 5 111361 61199 79087 16 45 786906 27.2 O Y O 898006 16.3 1 d O 888900 43 .5 A Q K iinoo 61222 79069 15 46 787069 21 ,2 897908 Lo . d 889160 4d.O A Q K 11 0840 i 61245 79051 14 47 787232 27.2 897810 L6. 3 889421 4o . O 110579 61268 79033 13 48 787395 27. 1 897712 [6. 3 889682 43 . 5 110318 61291 79016 12 49 787557 27.1 897614 [6.3 889943 43.5 110057 61314 78998 11 50 787720 27. 1 897516 I6.3 890204 To 109796 6133? 78980 10 51 .787883 27. 1 3.897418 L6.3 .890465 4d. 4 0.109535 61360 78962 9 52 53 788045 788208 27.1 27.1 897320 897222 16.4 L6.4 890725 890986 43.4 43.4 JO A 1U9275 61383 109014: 61406| 78944 78926 8 7 54 55 788370 788532 27. 1 27.0 897123 897025 ^6.4 16.4 891247 891507 T:<-> . 4 43.4 108753 ! 61429 108493 161451 78908 78891 6 5 66 788694 27.0 896926 16.4 6 A 891768 43.4 108232 j 61474 78873 4 57 788856 27.0 896828 . 4 892028 43.4 107972 i 61497 "8855 3 68 789018 27.0 896729 6.4 892289 4d.4 107711 61520 f8837 2 59 789180 27.0 896631 6.4 892549 43.4 107451 61543 C8819 1 60 789342 27.0 896532 6.4 892810 43 . 4 107190 61566 "8801 Cosine. Sine. Cotang. Tang. IX cos. M.sine. 52 Degrees. TABLE II. Log. Sines and Tangents. (38) Natural Sines. 59. Sine. D. 10" Cosine. D. 10" Tang. D. 10" Cotang. j N. sine.jN. cos. 9.789342 9.896532 9.892810 10.107190 61566 78801 60 1 789504 26.9 896433 16.4 893070 43.4 106930 61589 78783 59 2 789665 26.9 896335 16.5 893331 43.4 106669 61612 78765 58 3 789827 26.9 896236 16.5 893591 43.4 10b409 61635 78747 57 4 789988 26.9 896137 16.5 893851 4 : . 4 106149 61658 78729 56 6 790149 26.9 896038 16.5 894111 43.4 105889 61681 78711 55 6 790310 26.9 895939 16. 5 894371 43.4 105629 61704 78694 7 790471 26.8 895840 16.6 1 ft K 894632 43.4 105368 61726 78676 53 8 790632 26.8 895741 Ib.o 1 ft K 894892 43.3 105108 61749 78668 52 9 790793 26.8 895641 Ib. o 895152 43.3 104848 61772 78640 51 10 790954 26.8 895542 16. 5 895412 43.3 104588 61795 78622 50 11 9.791115 Oft Q 9.895443 16.5 9.895672 43.3 10.104328 61818 78604 49 12 791275 2b .0 oft ry 895343 16.6 1 ft ft 895932 43 .3 104068 61841 78586 48 13 791436 2b .7 895244 Ib.b 1 ft ft 896192 43.3 103808 61864 78568 47 14 791596 26 . 7 895145 Ib.b 1 ft ft 896452 43.3 103548 61887 78550 46 15 791757 26 .7 895045 Ib.b 896712 43 .3 103288 61909 78532 45 16 791917 *?" 894945 16.6 896971 43.3 103029 61932 78514 44 17 792077 26 . 7 894846 16.6 897231 43 .3 102769 61955 78496 43 18 792237 26 .7 Oft ft 894746 16.6 1 ? ft 897491 43 .3 102509 61978 78478 42 19 792397 26. b 894646 16. b 897751 43-3 102249 62001 78460 41 20 792557 26.6 894546 16.6 898010 43.3 101990 62024 78442 40 21 9.792716 26.6 Oft ? 9.894446 16.6 9.898270 43-3 10.101730 62046 78424 39 22 792876 2b .b 894346 16.7 898530 43.3 101470 62069 78405 38 23 793035 | 2 894246 16.7 Ifi 7 898789 43.3 40 o 101211 62092 78387 37 24 793195 * " 894146 ID . 899049 100951 62115 78369 36 25 793354 26.5 26 5 894046 16.7 16 7 899308 43.2 100692 62138 78351 36 26 793514 893946 1 ft i 899568 xo 100432 62160 78333 34 27 793673 26 . 6 Oft K. 893846 16. / 1 ft *7 899827 43 .2 100173 62183 78315 33 28 793832 *b .0 893745 16. i 1 ft "7 900086 43 .2 099914 62206 78297 32 29 793991 26.5 Oft E 893645 16. t 1ft 7 900346 40 9 099654 62229 78279 31 30 794150 -*O . O Oft /I 893544 ID . / 1 ft 1 900605 Ho . Z 099395 62251 78261 30 31 |9. 794308 xib .4 Oft A 9.893444 Ib. t 1ft Q 9.900864 43 .2 43 ^ 10.099136 62274 78243 29 32 794467 oft 4 893343 ID . o 1 ft ft 901124 098876 62297 78225 28 33 794626 ZD -H 893243 ib.o 1 ft Q 901383 40 9 098617 62320 78206 27 34 794784 Oft /I 893142 ID . o 1ft 901642 9 098358 62342 78188 26 35 794942 I*"" 893041 ID . o 1ft 901901 4Q Q 098099 62365 78170 25 36 795101 1 ; 892940 ID . o 1ft 8 902160 HO . Z 40 9 097840 62388 78152 24 37 796359 ~rj 892839 ID . o 1 ( \ H 902419 Ho . Z 40 9 097581 62411 78134 23 38 796417 IS a 892739 ID . o 1ft C 902679 Ho . Z 40 9 097321 62433 78116 22 39 795575 J a 892638 ID . o 1 ft ft 902938 Ho . Z 40 9 097062 62456 78098 21 40 795733 *J? a 892536 ID . o 1 ft Q 903197 Ho . Z 096803 62479 78079 20 41 9. 795891 if?^ 9.892435 ID . o 1 ft Q 9.903455 40 i 10.096545 62502 78061 19 42 796049 f? f 892334 ID . y 903714 HO . 1 40 -i 096286 62524 78043 18 43 796206 , * 892233 Ib . 9 1 ft Q 903973 HO . I 40 i 096027 62547 78025 17 44 796864 S ! 892132 ID . y 1 ft O 904232 HO . 1 40 i 095768 62570 78007 16 45 796521 IT* 892030 ID . y 904491 HO . 1 A Q . 095509 62592 77988 15 46 47 48 796679 g J 796836 2 J 796993 fj o 891929 891827 891726 16.9 16.9 16.9 904750 905008 905267 Hd.l 43.1 43.1 095250 094992 094733 62615 62638 62660 77970 77952 77934 14 13 12 49 797150 ** 891624 16.9 905526 43.1 094474 62683 77916 11 50 797307 ** 891523 16.9 905784 43. 1 094216 62706 77897 10 51 9.797464 * : 9.891421 17.0 9.906043 43.1 10.093957 62728 77879 9 52 797621 * : 891319 17.0 906302 43.1 093698 62751 77861 8 53 797777 9^ 891217 17.0 nfi 906560 43.1 A Q 093440 62774 77843 7 54 797934 oft 891115 U 906819 HO . 093181 62796 77824 6 ; 55 798091 * : 891013 17.0 907077 43 . 092923 62819 77806 G 56 798247 * : 890911 17.0 907336 43 . 092664 62842 77788 4 57 798403 ~ * 890809 17.0 907594 43 . 092406 62864 77769 3 58 798560 fj!*U 890707 17.0 nn 907852 43 . 40 092148 62S87 77751 2 59 798716 JJ-J 890605 . U 908111 HO . 091889 62909 77733 1 60 798872 26 890.303 17.0 908369 43.0 091631 62932 77715 Cosine. SiuR. Cotang. Tang. : M . cos". N.sine- r 51 Degrees. 60 Log. Sines and Tangents. (39) Natural Sines. TABLE II. _ Sine. D. 10 Cosine. D. 10 Tang. L>. 10 Cotang. N. sine N. cos o 9.798772 9.890503 9.903369 10.091631 62932 77715 60 799028 26. C 890400 17. t 1 7 1 908628 43. ( 091372 62955 77696 59 < 799184 9R f 890298 1 / . 1 ni 908886 40 f 091114 62977 77678 58 i 799339 ~? g 890195 . 1 ni 909144 4t3 . 1. 40 ( 090856 63000 77660 57 i 799495 ORfl 890093 . L ni 909402 4o . I. 40 f 090598 63022 77641 56 6 799651 o- q 889990 . 1 ni 909360 4o . (_ 40 i 090340 63045 77623 55 6 799806 Ofi fl 889888 . 1 1 7 1 909918 4o . L 090082 63068 77605 54 799962 2X Q 889785 1 / . 1 17 1 910177 40 / 089823 63090 77586 53 8 800117 o.y 889682 1 . A 910435 4o . I. 039565 63113 77568 52 9 800272 25.9 OK Q 889579 ni 910693 40 / 089307 63135 77550 51 10 800427 ZO . O c*~ O 889477 . J n-t 910951 4o . I. 089049 63158 77531 50 11 9.800582 2o.o 9.889374 . I 9.911209 43 . C A O A 10.088791 93180 77513 49 800737 25.8 OK O 889271 17. 911467 4o . I 40 A 038533 63203 77494 48 13 800892 zo.o 889168 , 911724 4o . I 088276 63225 77476 47 14 801047 25.8 889064 17.x 911982 43. C 038018 63248 77458 46 15 801201 25.8 888961 17. 912240 43. C /iO A 037760 63271 77439 45 16 801356 25.8 888858 . 912498 4d.O A O A 087502 63293 77421 44 17 801511 25.7 888755 17. 912756 43.1 087244 63316 77402 43 18 801665 25.7 888651 17. 913014 43. C *O A 036986 63338 77384 42 19 801819 25.7 888548 17. 913271 42.9 086729 63361 77366 41 20 801973 25.7 888444 17. 913529 42.9 086471 63383 77347 40 21 9.802128 25.7 9.888341 17.3 no 9.913787 42 . <j 42 9 10-086213 63406 77329 39 22 802282 2o 7 838237 . o 914044 035956 63428 77310 38 23 802436 25.6 888134 17.3 17 3 914302 42.9 42 ^) 085698 63451 77292 37 24 802589 2o.b 888030 nr> 914560 085440 63473 77273 36 25 802743 25.6 887926 .O no. 914817 42 . 9 42 9 035183 63496 77255 35 26 802897 O" ft 887822 . O no 915075 42 9 034925 63518 77236 34 27 803050 ZO- D 887718 .0 1 7 q 915332 4Q A 084668 63540 77218 33 28 803204 25.6 887614 J. . o n 915590 4z . y 084410 63563 77199 32 29 30 803357 803511 25.6 25.5 887510 887406 .3 17.3 H A 915847 916104 42.9 42.9 084153 083896 63585 63603 77181 77162 31 30 31 9.803664 25.5 9.887302 .4 9.916362 42. 9 10-083638 63630 77144 29 32 803817 25.5 887198 17.4 n A 916619 42.9 083381 63653 77125 28 33 803970 25.5 887093 .4 nA 916877 42.9 083123 63675 77107 27 34 804123 25-5 886989 .4 917134 42.9 082866 63698 77088 26 35 804276 25-5 886885 17.4 917391 42. 9 082609 63720 77070 25 36 804428 25-4 886780 17.4 n4 917648 42.9 /to n 082352 63742 77051 24 37 804581 25.4 886676 . 4 nA 917905 4^.y 082095 63765 77033 23 38 804734 25-4 886571 . 4 918163 42.9 081837 63787 77014 22 39 804886 25.4 886466 17.4 ni 918420 42.8 40 c 081580 63810 76996 21 40 805039 25-4 OK A 886362 . 4 nt 918677 z . o 081323 63832 76977 20 41 42 9.805191 805343 ZO -4 25.4 9-886257 886152 . o 17.5 9.918934 919191 42. 8 10-081066 i 63854 0808091163877 76959 76940 19 18 43 805495 25-3 886047 17.5 919448 o 080552 63899 76921 17 44 805647 25.3 885942 17 . 5 919705 42.8 080295 63922 76903 16 45 805799 25.3 885837 17 . 5 919962 42.8 080038 63944 76884 15 46 47 48 805951 806103 806254 25.3 25.3 25.3 885732 885627 885522 17.5 17.5 17.5 920219 920476 920733 42.8 42.8 42.8 079781 079524 079267 63966 76866 63989(76847 6401176828 14 13 12 49 806406 25.3 885416 17.5 nK 920990 42.8 079010 | 64033 76810 11 50 806557 zo . z 885311 . o 921247 4z. o 078753 64056 76791 10 51 .806709 25.2 9.835205 17.6 .921503 42.8 0-078497 64078 76772 9 52 53 806860 807011 25.2 25.2 885100 884994 17. 6 17.6 921760 922017 42. 8 078240 077983 j 64100 64123 76754 76735 8 7 54 807163 25.2 884889 17.6 9222/4 42.8 077726 64145 76717 6 55 807314 25.2 884783 17.6 922530 42.8 077470 64167 76698 5 56 57 58 807465 807615 807766 25.2 25.1 25.1 884677 884572 884466 17.6 17.6 17.6 92278 7 923044 92330J 42 8 42.8 42.8 077213l!64190 076956 64212 076700 64234 76679 76661 76642 4 3 2 59 807917 25. 1 17.6 923557 42.8 076443 ; 64256 76623 1 60 808067 25. 1 884254 L7.6 923813 42.7 076187 jj 64279 "6604 Cosine. Sine. Cotang. Tang. N. cos. S.sinc. 50 Degrees. TABLE II. Log. Sines and Tangents. (40) Natural Sines. 61 "7 Sine. D. 10" Cosine. |D. 10" Tang. D. 10" Cotang. N .sine. N. cos. 9.808067 9.884254 177 9.923813 10.076187 64279 76604 60 1 808218 of i 884148 1 / . / 924070 42.7 075930 64301 76586 59 2 808368 S)K 1 884042 17.7 177 924327 42. 7 AO 7 075673 64323 76567 58 3 808519 >- > , 1 883936 1 . 924583 4^5 . / 075417 64346 76548 57 4 808669 25 , 883829 17.7 924840 42. 7 075160 64368 76530 56 5 808819 o- n 883723 17.7 925098 42.7 074904 64390 76511 55 6 7 808969 809119 25^0 O" A 883617 883510 17.7 17.7 925352 925609 42. 7 42.7 074648 074391 64412 64435 76492 76473 54 53 1 8 809269 zo , 883404 17.7 nm 925865 42.7 074135 64457 76455 52 9 809419 O Q 883297 . 1 no 926122 42. / 073878 64479 76436 51 10 809569 24.9 883191 .8 926378 42.7 073622 64501 76417 50 11 9.809718 24. 9 OA Q 9.883084 17.8 no 9.926634 42.7 AO fjf 10.073366 64524 76398 49 12 809868 ^4, y 882977 .0 926890 4^ , I 073110 64546 76380 48 13 810017 24,9 24 9 882871 17.8 no 927147 42.7 072853 64568" 76361 47 14 810167 O A O 882764 .0 927403 42.7 072597 64590 76342 46 15 810316 24, y 018 882657 17.8 927659 42. 7 072341 64612 76323 45 16 810465 24.O OA 8 882550 17.8 i n Q 927915 42. 7 072085 64635 76304 44 17 810614 24. o 882443 1 r.C! no 928171 4O 7 071829 64657 76286 43 18 810763 9A8 882336 .0 928427 4O7 071573 64679 76267 42 19 810912 24. o 882229 i .y 928683 Ao fj 071317 64701 76248 41 20 811061 OA 8 882121 17.9 1 * < 1 928940 4;$ . / 4O *7 071060 64723 76229 40 21 9.811210 OS 9.882014 17.9 9.929196 Z.I 10.070804 64746 76210 39 22 811358 c\ ft 881907 17.9 929452 AO 070548 64768 76192 38 23 24 811507 811655 24 24.7 O \ 7 881799 881692 17.9 17.9 929708 929964 42! 7 070292 070036 64790 64812 76173 76154 37 36 25 811804 24. i O 4 fj 881584 17.9 930220 42. 6 069780 64834 76135 35 26 27 811952 812100 24. i 24.7 O A **1 881477 881369 17.9 17.9 930475 930731 42.6 42.6 069525 069269 64856 64878 76116 76097 34 33 28 812248 24. i 881261 17.9 930987 42.6 069013 64901 76078 32 29 812396 24.7 881153 18.0 931243 42.6 068757 64923 76059 31 30 812544 24.6 881046 18.0 1 Q A 931499 42.6 068501 64945 76041 30 31 9.812692 OAfi 9.880938 lo . U 1 Q A 9.931755 42 . 6 10.068245 64967 76022 29 32 33 34 812840 812988 813135 24> 24.6 OH ft 880830 880722 880613 lo . 18.0 18.0 1 Q A 932010 932266 . 932522 42!e 42.6 067990 067734 067478 64989 65011 65033 76003 75984 75965 28 27 26 35 813283 24. 880505 lo . 932778 42 . 6 067222 65055 75946 25 36 813430 C\A K. 8o<3397 18.0 933033 42.6 066967 65077 75927 24 37 813578 24. 5 880289 18.0 933289 42. o 066711 65100 75908 23 38 813725 OA K 880180 18. 1 1 Q 1 933545 42. o 066455 65122 75889 22 39 813872 24. O OH C 880072 lo . 1 933800 42.O 066200 65144 75H70 21 40 814019 24.5 879963 18. 934056 42.6 065944 65166 75851 20 41 9.814166 OA K. 9.879855 18. 1 Q 9.934311 42.6 10.065689 65188 75832 19 42 814313 24, O 879746 lo . 934567 4 % w.D 065433 65210 75813 18 43 814460 24.5 879637 18. 934823 42.6 065177 65232 75794 17 44 814607 24,4 OH H 879529 18. 935078 42.6 064922 65254 75775 16 45 814753 24. 4 O .4 A 879420 18. 935333 42.6 034667 65276 75556 15 46 814900 24.4 OH H 879311 18. 935589 42.6 4O ft 064411 65298 75738 14 47 815046 24,4 879202 18. 935844 42 . o AC) ; 064156 65320 75719 13 48 815193 A A 879093 18.2 936100 42 . 063900 65342 75700 12 49 815339 24.4 878984 18.2 936355 42.6 063645 65364 75680 11 50 815485 24.4 878875 18.2 936610 42.6 063390 1 165386 75661 10 51 9.815631 24.3 OA Q 9.878766 18.2 10 O 9.936866 42.6 40 x 10.063134 65408 75642 9 52 815778 24. o 878656 lo . 2 937121 iJ.o 062879 65430 75623 8 53 54 815924 816069 24.3 24.3 878547 878438 18.2 18.2 937376 937632 42.5 42.5 062624 062368 65452 65474 75604 75585 7 6 55 56 816215 816361 24.3 24.3 878328 878219 18.2 18.2 937887 938142 42.6 42.5 062113 061858 65496 ! 65518 75566 75547 5 4 57 816507 24. 3 OH. O 878109 18.3 938398 42 . 5 061602 !i 65540 75528 3 58 816652 24. 2 877999 18.3 938653 42.5 061347^65562 75509 2 59 816798 24. 2 877890 18.3 938908 42.5 061092 05584 75490 1 60 816943 24.2 877780 18.3 939163 42.5 060837 i:65uU6 7547 I Cosine. Sine. Cotang. Tang. ! N. cos. X.siue. ~ 49 Degrees. 62 Log. Sines and Tangents. (41) Natural Sines. TABLE LT. Sine. D. 10 Cosine. D. 10 Tang. D. 10 Cotang. i N. sine X. cos 9.816943 9.877780 100 9.939163 10.0(50837 65606 75471 60 1 817088 Of 877670 lo . ^ 10 Q 939418 4 n 060582 6562fc 75452 59 2 817233 *4* - 877560 lo . L 939673 ~t* . O 10 K 000327 65650 75433 58 3 817379 24.2 877450 18. c 939928 -tZ . O 060072 1 65672 75414 57 4 817524 24.2 877340 18. c 940183 42.5 >1O & 059817 ! 65694 75395 56 6 6 817668 817813 24.1 24.1 877230 877120 18. J 18.4 940438 940694 4J. O 42.6 40 K 059562 059306 ! 657 16 65738 75375 75356 55 54 7 817958 24.1 877010 18.4 940349 Z .0 40 rx 059061 65759 75337 53 8 818103 24. 1 876S99 18.4 941204 Z . o to ^ 058796 ! 05781 75318 52 9 818247 24. 1 876789 18.4 941458 iZ . O 058542 : 65803 75299 51 10 818392 24.1 876678 18.4 941714 42 . c 058286 65825 75280 50 11 9.818536 24. 1 9.876568 18.4 9.941968 /ID K 10.058032 65847 75261 49 12 818681 24. C 876457 18.4 942223 4vJ .O 49 ^ 057777 ! 65869 75241 48 13 818825 24.1 876347 18.4 942478 057522 65891 75222 47 14 818969 24. C 876236 18.4 942733 42 .5 Act f 057267 65913 75203 46 15 819113 24. C 876125 18.5 942988 vZ . o 057012 65935 75184 45 16 819257 24. C 876014 18.5 943243 42 . 5 056757 65956 75165 44 17 819401 24.0 875904 18.5 943498 42.5 056502 65978 75146 43 18 819545 24.0 875793 18.5 943752 42.5 056248 66000 75126 42 19 819689 23.9 875682 18.5 944007 42 .5 055993 66022 76107 41 20 21 819832 9.819976 23.9 23.9 875571 9.875469 18.6 18.6 944262 9.944517 42.5 42.5 055738 10.055483 66044 ! 66066 75088 75069 40 39 22 23 820120 820263 23.9 23.9 875348 875237 18.5 18.5 944771 945026 42 .5 42.4 055229 054974 66088 166109 75050 75030 38 37 24 820403 23.9 875126 18.5 10 a 945281 42.4 054719 66131 75011 36 25 820550 23.9 875014 18. o 945535 44 054465 166153 74992 35 26 820693 23.8 874903 18.6 945790 44 054210 1 66175 74973 34 27 820836 23.8 874791 18.6 946045 Act A 053955 | 66197 74953 33 28 8-20979 23.8 874680 18.6 946299 4ii.4 053701 66218 74934 32 29 821122 23.8 874568 18.6 946554 42.4 053446 ! 66240 74915 31 30 821265 23.8 874456 18.6 946808 42 .4 053192 166262 74896 30 31 32 9.821407 821550 23.8 23.8 J. 874344 874232 18.6 18.6 9.947063 947318 42.4 42.4 Act A 10.052937 052682 66284 66306 74876 74857 29 28 33 821693 23.8 OQ 7 874121 18.7 1 R 7 947572 412. 4 42 4 052428 66327 74838 27 34 821835 Jo. / 874009 J.O . 1 947826 052174 166349 74818 26 35 821977 23.7 873896 18.7 948081 42. 4 051919 66371 74799 25 36 822120 23.7 873784 18.7 948336 42.4 051664 OO.S93 74780 24 37 822363 23.7 873672 18.7 948590 42.4 051410 66414 74760 23 38 39 822404 82-2546 23.7 23.7 873560 873448 18.7 18.7 948844 949099 42.4 42.4 05 1 1 56 j 166436 050901 166458 74741 74722 22 21 40 822688 23.7 OQ 1* 873335 18.7 10 ry 949353 42.4 42 4 050647| 166480 74703 20 41 9.822830 Jo . D 9.873223 lo .7 9.949607 1 0. 050393 i 1 66501 -4683 19 42 822972 23.6 873110 18.7 949862 42.4 050138 ! 66523 -4663 18 43 823114 23.6 872998 L8.8 950116 42.4 049884 166545 -4644 17 44 823255 23.6 872885 18.8 950370 42 . 4 049630 66566 *4625 16 45 823397 23.6 872772 L8.8 950625 42. 4 049375 6658.S "4600 15 46 823539 23.6 872659 18.8 950879 42. 4 049121 : 66610 -4586 14 47 823680 23.6 872547 L8.8 951133 4J.4 048867 66632 4567 13 48 823821 23.5 872434 [8.8 951388 42.4 0486121:66663 4548 12 49 823963 23.5 872321 18.8 951642 42.4 048358 66675 45-2-2 11 50 824104 23.5 872208 18.8 951896 42.4 048104 66697 74509 10 51 52 9.824245 824386 23.6 23.5 9.872095 871981 18.8 18.9 J. 952 150 952405 42.4 42.4 0.047850 66718 047595 66740 74489 74470 9 8 53 824527 23.5 871868 [8.9 952659 42.4 047341 66762 74451 7 54 824668 23.5 871755 [8.9 95-2913 12.4 047087 66783 74431 6 55 824808 23.4 871641 [8.9 953167 t2 . 4 046833 66805 74412 5 56 824949 23.4 871528 [8.9 8 9 953421 L2. 3 [2 3 046579| 60827 74392 4 57 826090 ~o . 4 871414 953675 046325 66848 74373 3 58 59 825230 825371 23.4 23.4 871301 871187 .8.9 18.9 8f\ 953929 954183 12. 3 42.3 [O Q 046071 i 66870 045817 66891 74353 ?4334 2 1 60 825511 23 .4 871073 . y 954437 t Z . o 045563 | 66913 74314 Oosine. Sine. Cotang. Tang. i N. cos. V.sine. 48 Degrees. TABLE 11. Log. Siiies and Tangents. (42) Natural Sines. 63 Sine. D. 10 Cosine. I). 10"| Tung. D. 10" Cotang. N. sine. N. cos. o 9.825511 9.871073 9.954437 10.045563 66913 74314 60 1 825651 23.4 870960 19.0 954691 42.3 045309 66935 74295 59 2 825791 23.3 870846 19.0 954915 42.3 /1O } 045055 66956 74276 58 3 825931 23.3 870732 19.0 955200 4J .0 044800 66978 74256 57 4 826071 23.3 870618 19.0 955454 42.3 044646 166999 74237 56 6 826211 23.3 870504 19.0 955707 42.3 044293 167021 74217 55 6 7 826351 826491 23.3 23.3 870390 870276 19.0 19.0 955961 956215 42.3 42.3 044039 043785 67043 74198 67064174178 54 53 8 826631 23.3 870161 19.0 95(5469 42.3 043531 67086 74159 62 9 826770 23.3 oo o 870047 19.0 956723 42.3 ACt O 043277 67107 74139 51 10 826910 2o .2 8li9933 19. 956977 1_ .0 043023 67129 74120 50 11 9.827049 23.2 9.869818 19. 9.957231 42.3 10.042769 67151 74100 49 19 827189 23.2 869704 19. 957485 42.3 042515 67172 74080 48 13 827328 23.2 869589 19. 957739 42.3 042261 67194 74061 47 14 827467 23 . 2 869474 19. 957993 42.3 042007 67215 74041 4(5 15 827606 23.2 869360 19. 958246 42.3 041754 67237 74022 45 1G 827745 23.2 869245 19. 958500 42.3 041500 67268 74002 44 17 827884 23.2 869130 19. 958754 42.3 041246 i 67280 73983 43 18 828023 23.1 869015 19. 959008 42-3 040992 67301 73963 42 19 828162 23. 1 868900 19.2 959262 42.3 040738 87323 73944 41 20 828301 23.1 868785 19.2 959516 42.3 040484 67344 73924 40 21 9.828439 23.1 9.868670 19.2 9.959769 42.3 10.040231 67366 73904 39 22 828578 23.1 868555 19.2 960023 42.3 039977 67387 73885 38 23 24 828716 828855 23 . 1 23.1 868440 86H324 19.2 19.2 960277 960531 42.3 42.3 039723 039469 67409 67430 73865 73846 37 36 25 828993 23.0 868209 19.2 960784 42.3 039216 67452 73826 35 i 26 829131 23.0 868093 19.2 961038 42.3 038962 67473 73806 34 27 829269 23.0 867978 19.2 961291 42.3 038709 (57495 73787 33 28 829407 23.0 867862 19.3 961545 42.3 038455 67516 73767 32 29 829545 23.0 867747 19.3 961799 42.3 038201 67538 73747 31 30 829683 23.0 867631 19.3 962052 42.3 037948 67559 73728 30 31 32 9.829821 829959 23.0 22.9 9.867515 867399 19.3 19.3 9.962306 962560 42.3 42.3 10.037694 037410 67580 1 67602 73708 73688 29 28 33 830097 22.9 OO O 867283 19.3 962813 42.3 037187 1| 67623 73(5(59 27 34 830234 ZA -9 867167 19.3 963067 42 .3 036933 j 67645 73(549 26 35 3G 830372 830509 22.9 22.9 90 q 867051 866935 19.3 19.3 in A 963320 963574 42.3 42.3 AC) O 036(580 03(542(5 67666 67688 TIJG- iJ 73(510 25 24 37 38 830646 830784 <& d 22.9 OO Q 866819 866703 LJ , 4 19.4 in A 963827 964081 *t^t . O 42.3 4O Q 036173 035919 67709(73590 6773073570 23 22 39 830921 . y 866586 Iv .4 964335 .w . O 035665 6775273551 21 40 831058 22.8 866470 19.4 964588 42.3 035412 6777373531 20 41 9.831195 22.8 9.866353 19.4 9.964842 42.2 10.035158 67795)73511 19 42 831332 22.8 no Q 866237 19.4 965095 42.2 034905 6781673491 18 43 831469 22 .0 866120 19.4 965349 42 .2 034651 67837173472 17 44 831606 lg-2 866004 19.4 965602 42.2 034398 67859 73452 16 45 831742i^-g 865887 19.5 965855 42-2 034145 678HO 73432 15 46 47 48 831879 832015 832152 1 Z^i -O 22.8 22.7 865770 865653 865536 19. B 19.5 19.5 966109 966362 966616 42.2 42.2 42.2 033891 033638 033384 67901 67923 67944 73413 73393 73373 14 13 12 49 832288 865419 19.5 966869 42.2 033131 67965 73353 11 50 832425 865302 19.5 967123 42.2 032877 67987 73333 10 51 9 883661 g. 9.865185 19.5 9.967376 42.2 10.032624 68008 73314 9 52 883697 Sri 865068 19.5 967629 42 .2 032371 68029 73294 8 53 832833 "* 864950 19.5 9(57883 42.2 032117 68051 73274 7 54 882969 jfj fi 864833 19.5 968136 42.2 031864 68072 73254 6 55 833105 g J 864716 19.6 968389 42.2 031611 68093 73234 5 56 833241 ^ 864598 19.6 968(543 42.2 031357 68115 73215 4 57 833377 g J 864181 19.6 968896 42.2 031104 (5813(5 73195 3 58 833512 g-J 864363 19.6 969149 42.2 030851 <;81. : i*< 7:3175 <2 59 833648 :~~7 864245 19.6 969403 42.2 030597 (581 7!) 73155 T 60 833783 r^ b 864127 19.6 969656 42.2 030344 68200 /3136 Cosine. Bine. Cotang. Tang. H N. cos N.sine 47 Degrees. Log. Sines and Tangents. (43) Natural Sines. TABLE II. Sine. D. 10 Cosine. D. 10 Tang. |D. 10" C.itsmg. jj.N.sine. N. cos o 9.833783 oo r 9.864127 9.969656 40 o 10.030344 1 68200 73135 60 ] 833919 -w w . t 09 r 864010 1 (\ p 969909 z . 2 40 o 030091 68221 73 IK 59 IT 834054 834189 22! 5 OO K 863892 863774 1LJ . u 19.7 970162 970416 2 . 2 42.2 029838 029584 68242 ! 68264 73096 73076 58 57 i 834325 22, 5 99 t 863656 19.7 1O T 970669 42.2 /io o 029331 68285 73056 56 5 834460 ~ w . OO K. 863538 iy . 7 970922 4^. 2 029078 68306 73036 55 6 834595 <**> . 99 - 863419 in T 971175 AVr 028825 68327 73016 54 r t 834730 <6-6. D OO K 863301 iy . 7 in T 971429 40 o 028571 68349 72991 53 8 834865 * &.O 863183 iy . 7 971682 2.2 028318 68370 7297( 52 9 834999 22 4 863064 1 q r 971935 42 2 028065 68391 72957 51 10 835134 862946 iy . t i q o 972188 027812 68412 72937 50 11 9.835269 99 A 9.862827 iy . o in Q 9.972441 40 O 10.027559 68434 72917 49 12 835403 . T OO A 862709 iy .0 in Q 972694 TC-* , 2 AC) f> 027306 68455 72897 48 13 835538 Z* . 4 862590 iy . c 972948 4^ , 2 027052 68476 72877 47 14 835672 22.4 OO A 862471 19. 1 Q t 973201 42,2 /1O O 026799 68497 72857 46 15 835807 2& . 4 09 A 862353 iy . c 19 S 973454 4*5 , 2 42 2 026546 68518 72837 45 16 835941 . 4 862234 973707 026293 68539 72817 44 17 836075 22.4 OO Q 862115 10 u 973960 42.2 /1O O 026040 68561 72797 43 18 836209 *+ o OO O 861996 iy .0 1 Q Q 974213 4^ . 2 025787 68582 72777 42 19 20 836343 836477 22. o 22.3 OO Q 861877 861758 iy . o 19.8 1 U O 974466 974719 42!2 025534 025281 68603 68624 72757 72737 41 40 21 9.836611 22 . o OO Q 9.861638 iy . y in n 9.974973 49 9 10.025027 68645 72717 39 22 836745 22 . o 831519 iy . y 975226 Q2.2 024774 68666 72697 38 23 836878 22.3 861400 19 .9 975479 42.2 024521 68688 72677 37 24 837012 oo o 861280 1 Q n 975732 Af) O 024268 68709 72657 36 25 837146 22.2 oo o 861161 iy . y in n 975985 42 . 2 024015 68730|72637 35 26 837279 22 . 2 99 9 861041 iy . y i q q 976238 42 2 023762 68751 72617 34 27 837412 - . M 99 9 860922 iy . y 1 q q 976491 /IO O 023509 68772 72597 33 28 837546 22 . 99 9 860802 iy . y IQ Q 976744 4-^ . 2 023256 68793 72577 32 29 30 31 837679 837812 9.837945 22. A 22.2 22.2 860682 860562 9.860442 iy . y 20.0 20.0 976997 977250 9.977503 42^2 42.2 023003 022/50 10 022497 68814 68835 68857 72557 72537 72517 31 30 29 32 838078 22.2 OO 1 860322 20 . on n 977756 42.2 022244 68878 72497 28 33 838211 22 . 1 860202 *U . U 978009 42.2 021991 68899 72477 27 34 35 838344 838477 22. 1 22.1 OO 1 860082 859962 20.0 20.0 20 978262 978515 42.2 42.2 42 2 021738 021485 6892072457 6894172437 26 25 36 838610 w W . 1 OO 1 859842 978768 021232 68962 72417 24 37 838742 2& 1 859721 20.0 979021 42.2 0-20979 68983 72397 23 38 39 838875 839007 22. 1 22.1 OO 1 859601 859480 20! 1 >M 1 979274 979527 42.2 020726 020473 69004 69025 72377 72357 22 21 40 839140 22 . 1 859360 <(J . 1 on i 979780 4^.2 020220 69046;72337 20 41 .839272 22.0 99 n 859239 20 . 1 on i 9.980033 42.2 42 ^ 0.019967 69067(72317 19 42 839404 - - . U 859119 *U. 1 980286 019714 6908872297 18 43 839536 22.0 858998 20. 1 On 1 980538 42.2 019462 69109 72277 17 44 839668 22. 858877 2\J . 1 930791 42. 2 019209 6913072257 16 45 839800 22.0 858756 n n 981044 4~. 1 018956 69151 72236 15 46 839932 22.0 858635 20 , "2 on o 981297 42. 1 AC) I 018703 6917272216 14 47 48 49 840064 840196 840328 22. 21.9 21.9 858514 858393 858272 20J2 20.2 981550 981803 982056 42 . i 42.1 42.1 018450 0181971 017944 69193 72196 69214172176 69235 ! 72 156 13 12 11 50 840459 21.9 858151 20. 2 982309 42. 1 017691 6925672136 10 51 .840591 21.9 .858029 20. 2 .982562 1:2. 1 0.0174381 6927772116 9 52 840722 21.9 857903 20.2 982814 1 ^ f 01 7186 ; 6929872095 8 53 840854 21.9 857786 2J.2 983067 016933:16931972075 7 54 840985 21 .9 857665 20.2 933320 42. 1 016U80 69340172055 6 55 56 841116 841247 21 .9 21.8 857543 85-422 20. 3 20.3 983573 983826 42. 1 42.1 016427;j 69361 ! 72035 016174 69382^72015 5 4 57 841378 21 .8 85730J 20.3 934079 42. 1 015921 69403:71995 3 58 59 841509 841640 21 .8 21.8 85 71 78 | 20.3 20.3 984331 984584 42. 1 42.1 015669 694-2471974 015416 b9445 71954 2 1 841771 21.8 85J934 20.3 984837 42. 1 015163 69466 j 7 1934 ( ..sine. Sine. Cotang. Tang. : N. cos. N.siu<;. 46 Degrees. TABLE II. Log. Sines and Tangents. (44) Natural Sines. 65 Sine. D. 10" Cosine. D. 10" Tang. D.,10" Cotang. N. sine. N. cos. 9.841771 9.856934 9.984837 10.015163 69466 71934 60 1 841902 21.8 856812 20.3 985090 42. 014910 69487 71914 59 2 842033 21.8 856690 20.3 985343 42. 014657 69508 71894 58 3 842163 21.8 856568 20.4 985596 42. 014404 69529 71873 57 4 842294 21.7 856446 20.4 985848 42. 014152 69549 71853 56 5 842424 21.7 856323 20.4 986101 42. 013899 69570 71833 55 6 842555 21.7 856201 20.4 986354 42. 013646 69591 71813 54 7 842G85 21.7 856078 20.4 986607 42.1 013393 69612 71792 53 8 842815 21.7 855956 20.4 986860 42.1 013140 69633 71772 52 9 842946 21.7 855833 20.4 987112 42. 1 012888 69654 71752 51 10 843076 21.7 855711 20.4 987365 42.1 012635 69675 71732 50 11 9.843206 21.7 9.855588 20.5 9.987618 42. 1 10.012382 69696 71711 49 12 843336 21.6 O1 * 855465 20.5 90 Ft 987871 42.1 49 1 012129 69717 71691 48 13 843466 21. b 855342 **\J . O 90 fi 988123 Q-6 . 1 49 1 011877 69737 71671 47 14 843595 21 .6 O1 ? 855219 ^U. O 90 K 988376 4^. 1 49 1 011624 69758 71650 46 15 843725 yi.b 855096 A\J . 9O f\ 988629 4^. 1 49 011371 69779 71630 45 16 843855 21.6 854973 ^\J . on K 988882 4^. 49 011118 69800 71610 44 17 843984 21.6 854850 4\J . 989134 * . ACt 010866 69821 71590 43 18 844114 21.6 854727 20.5 989387 42 . 010613 69842 71569 42 19 844243 21.5 854603 20.6 989640 42. 010360 69862 71549 41 20 844372 21.5 854480 20.6 989893 42. 010107 69883 71529 40 21 9.844502 21.5 9.854356 20.6 9.990145 42. 10.009855 69904 71508 39 22 844631 21.5 854233 20.6 990398 42.1 009602 69925 71488 38 23 844760 21.5 854109 20.6 990651 42.1 009349 69946 71468 37 24 844889 21.5 853986 20.6 990903 42.1 009097 69966 71447 36 25 845018 21.5 853862 20.6 991156 42.1 008844 69987 71427 35 26 845147 21.5 853738 20.6 991409 42.1 008591 70008 71407 34 27 845276 21.5 853614 20.6 991662 42.1 ACt 1 008338 70029 71386 33 28 845405 21.4 853490 20. 7 991914 42.1 in . 008086 70049 71366 32 29 845533 21.4 853366 20.7 992167 42.1 ACt 1 007833 70070 71345 31 30 845662 21.4 853242 20.7 992420 42.1 OQ7580 70091 71325 30 31 9.845790 21.4 9.853118 20.7 9.992672 42. 10-007328 70112 71305 29 3-2 845919 21.4 852994 20.7 992925 42. 007075 70132 71284 28 33 846047 21.4 852869 20.7 993178 42. 006822 70153 71264 27 34 846175 21.4 852745 20.7 993430 42. 006570 70174 71243 26 35 846304 21.4 852620 20.7 993683 42. 006317 70195 71223 25 36 846432 21.4 852496 20.7 993936 42. 006064 70215 71203 24 37 846560 21.3 852371 20.8 994189 42. 005811 70236 71182 23 38 846688 21.3 852247 20.8 994441 42. 005559 70257 71162 22 39 846816 21.3 852122 20.8 994694 42. 005306 70277 71141 21 40 846944 21.3 851997 20.8 994947 42. 005053 70298 71121 20 41 9.847071 21.3 9.851872 20.8 9.995199 42. 10-004801 70319 71100 19 42 847199 21.3 851747 20.8 995452 42. 004548 70339 71080 18 43 847327 21.3 851622 20.8 995705 42. 1 004295 70360 71059 17 44 847454 21.3 851497 20.8 995957 42.1 004043 70381 71039 16 45 847582 21.2 851372 20.9 996210 42.1 003790 70401 71019 15 46 847709 21.2 851246 20.9 996463 42.1 003537 70422 70998 14 47 847836 21.2 851121 20.9 996715 42. ACt 003285 70443 70978 13 48 847964 21.2 850996 20.9 996968 4.4. ACt 003032 70463 70957 12 49 848091 21.2 850870 20.9 997221 <Z . ACt 002779 70484 70937 11 50 848218 21.2 850745 20.9 997473 Q" . ACt 002527 70505 70916 10 -51 9.848345 21.2 9.850619 20.9 9.997726 41* . ACt 10.002274 70525 70896 9 52 848472 21.2 850493 20.9 997979 4. ACt 002021 70546 70875 8 53 848599 21.1 850368 21.0 998231 4z. 001769 70567 70855 7 54 848726 21.1 850242 21.0 998484 42. 001516 70587 70834 6 55 848852 21.1 850116 21.0 998737 42. 001263 70608 70813 5 56 848979 21.1 849990 21.0 998989 42. 001011 70628 70793 4 57 849106 21.1 849864 21 .0 999242 42. ACt 000758 70649 70772 3 58 849232 21. 1 849738 21.0 999495 *. ACt 000505 70670 70752 2 59 849359 21. 1 849611 21 .0 999748 QX> . ACt 000253 70690 70731 1 60 849485 21.1 849485 21.0 10.000000 4-i. 000000 70711 70711 Cosine. Sine" Co tang. Tang. N. cos. N.pinc. 45 Degrees. 66 LOGARITHMS TABLE III. LOGARITHMS OF NUMBERS. FROM 1 TO 200, INCLUDING TWELVE DECIMAL PLACES. N. Log. N. Log. N. Log. 1 2 3 4 5 000000 000000 301029 995664 477121 254720 602059 991328 698970 004336 41 42 43 44 45 612783 856720 623249 290398 633468 455580 643452 676486 653212 513775 81 82 83 84 85 908485 018879 913813 852384 919078 092376 924279 286062 929418 925714 6 7 8 9 10 778151 250384 845098 040014 903089 986992 954242 509439 Same as to 1. 46 47 48 49 50 662757 831682 672097 857926 681241 237376 690196 080028 Same as to 5. 86 87 88 89 90 934498 451244 939519 252619 944482 672150 949390 006645 Same as to 9. 11 12 13 14 15 041392 685158 079181 246048 113943 352307 146128 035678 176091 259056 51 52 53 54 55 707570 176098 716003 343635 724275 869601 732393 759823 740362 689494 91 92 93 94 95 959041 392321 963787 827346 968482 948554 973127 853600 977723 605889 16 17 18 19 20 204119 982656 230448 921378 255272 505103 278753 600953 Same as to 2. 56 57 58 59 60 748188 027005 755874 855672 763427 993563 770852 011642 Same as to 6. 96 97 98 99 100 982271 233040 986771 734266 991226 075692 995635 194598 Same as to 10, " 21 22 23 24 25 322219 2947 342422 680822 361727 836018 380211 241712 397940 008672 61 62 63 64 65 785329 835011 792391 699498 799340 549453 806179 973984 812913 356643 101 102 103 104 105 004321 373783 008600 171762 012837 224705 017033 339299 021189 299070 26 27 28 29 30 414973 347971 431363 764159 447158 031342 462397 997899 Stnno as to 3. 66 67 68 69 70 819543 935542 826074 802701 832508 912706 838849 090737 Same as to 7. 103 107 108 109 110 025305 865265 029383 777685 033423 755487 037426 497941 Same as to 11. 31 32 33 34 35 491361 693834 505149 978320 518513 939878 531478 917042 544068 044350 71 72 73 74 75 851258 348719 857332 496431 863322 860120 869231 719731 875061 263392 111 112 113 114 115 045322 978787 049218 022670 053078 443483 056904 851336 060397 840354 36 37 38 39 40 556302 500767 568-201 724067 579783 596617 591064 607026 Same as to 4. 76 77 78 79 80 880813 592281 886490 725172 892094 602690 897627 091290 Same as to 8. 116 117 118 119 120 064457 989227 068185 861746 071882 007306 075546 961393 Same as to 12. OF NUMBERS. 67 N. Log. N. 148 149 150 151 : 152 Log. N. Log 121 122 123 124 125 082785 370316 086359 830675 089905 111439 093421 685162 096910 013008 170261 715395 173186 268412 176091 259056 178976 947293 181843 587945 175 176 177 178 179 243038 048686 245512 667814 247973 266362 250420 002309 252853 030980 126 127 128 129 130 100370 545118 103803 720956 10/209 969G48 110589 710299 Same as to 13. 153 154 155 i 156 I 157 184691 430818 187520 720836 190331 698170 193124 588354 195899 652409 180 181 182 183 184 255272 505103 257678 574869 260071 387985 262451 089730 264817 823010 131 132 133 134 135 117271 295656 120573 931206 123851 640967 127104 798365 130333 768495 158 159 160 161 162 198657 086954 201397 124320 204119 982656 206825 876032 209515 014543 185 186 187 188 189 267171 728403 269512 944218 271841 606536 274157 849264 276461 804173 136 137 138 139 140 133538 908370 136720 567156 139879 086401 143014 800254 146128 035678 163 164 165 166 167 212187 604404 214843 848048 217483 944214 220108 088040 222716 471148 190 191 192 193 194 278753 600953 281033 367248 283301 228704 285557 309008 287801 729930 141 142 143 144 145 149219 112655 152288 344383 155336 037465 158362 492095 161368 002235 168 169 170 171 172 225309 281726 227886 704614 230448 921378 232996 110392 235528 446908 195 196 197 198 199 290034 611362 292256 071366 294466 226162 296665 190262 298853 076410 146 147 164352 855784 167317 334748 173 174 238046 103129 240549 248283 LOGARITHMS OF THE PRIME NUMBERS FROM 200 TO 1543, INCLUDING TWELVE DECIMAL PLACES. N. Log. N. Log. N. Log. 201 203 207 209 211 303196 057420 307496 037913 315970 345457 320146 286111 324282 455298 277 281 283 293 307 442479 769064 448706 319905 451786 435524 466867 620354 487138 375477 379 383 389 397 401 678639 209968 583198 773968 589949 601326 598790 506763 603144 372G20 223 227 229 233 239 348304 863048 356025 857193 359835 482340 367355 921026 378397 900948 311 313 317 331 337 492760 389027 495544 337546 501059 262218 519827 993776 527629 900871 409 419 421 431 433 611723 308007 622214 022966 624282 095836 634477 270161 636487 896353 241 251 257 263 269 3820 T 7 042575 399673 721481 409933 123331 419955 748490 429762 280002 347 349 353 359 367 540329 474791 542825 426959 647774 705388 555094 448578 1 564666 064252 439 443 449 457 461 642424 520242 646403 726223 652246 341003 659916 200070 663700 926390 271 432969 290874 373 571708 831809 463 666580 991018 68 LOGARITHMS N. Lug. N. Log. N. TfrT 1181 1187 1193 1201 Log. 467 479 487 491 499 6,)9olo 6805t,6 680335 513414 687528 961215 691081 492123 698100 545623 821 823 827 829 839 914343 157119 915399 835212 917505 509553 918554 530550 923761 960829 ObS5c6 895072 072249 807613 074450 718955 076640 443670 OJ9543 007385 503 609 521 623 541 701567 985056 706717 782337 716837 723300 718501 688867 733197 285107 853 857 859 863 877 930949 031168 932980 821923 933993 163831 936010 795715 942099 593356 1213 1217 1223 1229 1231 083860 800845 085290 678210 087426 458017 089551 882866 090258 052912 647 657 663 669 671 737987 326333 745855 195174 750508 394851 755112 266393 756636 108246 881 883 887 907 911 944975 908412 945960 703578 947923 619832 957607 287060 959518 376973 1237 1249 1259 1277 1279 092369 699609 096562 438356 100025 729204 108190 896808 106870 642460 577 687 593 699 601 761176 813156 768638 101248 773054 693364 777426 822389 778874 472002 919 929 937 941 947 963315 611386 988015 713994 971739 590888 973589 623427 976349 979003 1283 1289 1291 1297 1301 108226 656362 110252 917337 110926 242517 112939 986066 114277 296540 607 613 617 619 631 783138 691075 787460 474518 790285 164033 791690 649020 800029 369244 953 967 971 977 983 979092 900638 985426 474083 987219 229908 989894 563719 992553 517832 1303 1307 1319 1321 1327 114944 415712 116275 587564 120244 795568 120902 817604 122870 922849 641 643 647 653 659 806858 029519 808210 972924 810904 280669 814913 181275 818885 414594 991 997 1009 1013 1019 996073 654485 998695 158312 003891 166237 005609 445360 008174 184006 1361 1367 1373 1381 1399 133858 125188 135768 514554 137670 537223 140193 678544 145817 714122 661 673 677 683 691 810201 459486 828015 064224 830588 668685 8344-20 703682 839478 047374 1021 1031 1033 1039 1049 0090-25 742087 013258 665284 014100 321520 016615 547557 020775 488194 1409 1423 1427 1429 1433 148910 994096 153204 896557 154424 012366 155032 228774 156246 402184 701 709 719 727 733 845718 017967 850646 235183 856728 890383 861534 410859 865103 974742 1051 1061 1063 1069 1087 021602 716028 025715 383901 026533 264523 028977 705209 036229 644086 1439 1447 1451 1453 1459 158060 793919 160468 531109 161667 412427 162265 614286 164055 291883 739 743 751 757 761 888644 488395 870988 813761 855639 937004 879095 879500 881384 656771 1091 1093 1097 1103 1109 037824 750588 038620 161950 040206 627575 042595 512440 044931 546 119 1471 1481 1483 1487 1489 167612 672629 170555 058512 171141 151014 172310 968489 172894 731332 769 773 787 797 809 885926 339801 888179 493918 895974 732359 901458 32139G 907948 521612 1117 1123 1129 1151 1153 018053 173116 050379 756261 052693 941925 061075 323630 061829 307295 1493 1499 1511 1523 1531 174059 807708 175801 632866 179264 464329 182699 903324 184975 190807 811 909020 854211 1163 065579 714728 1543 188365 926053 OF NUMBERS. 69 AUXILIARY LOGARITHMS, N. lg. W. Log. i.ouy 003891106237 - 1 1.0009 000390689248 ^ .008 003460532110 1 . 0008 000347296684 .007 003029470554 1 . 0007 000303899784 .006 002598080685 1.0006 001)260498547 .005 002166061766 A 1 . 0005 000217092970 >B .004 001733712775 1 . 0004 000173683057 .003 001300933020 1 . 0003 000130268804 .002 000867721529 1 . 0002 000086850211 1.001 000434077479 1 . 0001 000043427277 N. Log. : N. Log. . 00009 . 00008 . 00007 . 00008 . 00005 . 00004 .00003 . 00002 . 00001 000039083266 000034740691 000030398072 0000-26055410 000021712704 000017371430 0000130-28638 000008085802 0001104342923 1.000009 . 000008 .000007 .000006 . 000005 . 000004 1 . 000003 1.000002 1.000001 OOOOJ3908628 000003474338 000003040047 000002605756 000002171464 000001737173 000001302880 OU0000868587 000000434294 - N. Log. 1.0000001 1.00000001 1.000000001 1 . 0000000001 000000043429 (n) 000000001343 (o) 000000000434 (p) 000000000043 (q ) ?7i=0.4342944819 log. 1.637784298. By the preceding tables and the auxiliaries A, B, and C, we can find the logarithm of any number, true to at least ten decimal places. But some may prefer to use the following direct formula, which may be found in any of the standard works on algebra: Log. (z-fl)==log.z+0.8685889638/ -_L > ) The result will be true to twelve decimal places, if z be over 2000. The log. of composite numbers can be determined by the combination of logarithms, already in the table, and the prime numbers from the formula. Thus, the number 3083 is a prime number, find its loga rithm. We first find the log. of the number 3082. By factoring, discover that this is the product of 46 into 67. 70 NUMBERS. Log. 46, 1.6627578316 Log. 67, 1.8260748027 Lo. 3082 3.4888326343 Log. 3083=3. 6165 NUMBERS AND THEIR LOGARITHMS, OFTEN USED IN COMPUTATIONS. Circumference of a circle to dia. 1 } Log. Surface of a sphere to diameter IV =3.14159265 0.4971499 Area of a circle to radius 1 ) Area of a circle to diameter 1 = .7853982 1.8950899 Capacity of a sphere to diameter 1 = .6235988 1.7189986 Capacity of a sphere to radius 1 =4.1887902 0.6220886 Arc of any circle equal to the radius = 5729578 1.7581226 Arc equal to radius expressed in sec. = 206264"8 5.3144251 Length of a degree, (radius unity) =.01 745329 2.2418773 12 hours expressed in seconds, = 43200 4.6354837 Complement of the same, =0.00002315 5.3645163 360 degrees expressed in seconds, = 1296000 6.1126050 A gallon of distilled water, when the temperature is 62 Fahrenheit, and Barometer 30 inches, is 277. ^VV cubic inches. ,/277.274= 16.651 542 nearly. 277 27 =18.78925284 V 231 =15.198684. .775398 _ J28~2~ = 16. 792855. 282 ._= 18.948708. .785398 The French Metre -3.2808992, English feet linear mea sure, =39.3707904 inches, the length of a pendulum vi brating seconds. UNIVERSITY OF CALIFORNIA LIBRARY BERKELEY Return to desk from which borrowed. This book is DUE on the last date stamped below. MAR 2 1950 APR 5 H 50 LD 21-100m-9, 48(B399sl6)476 Vp L/ 79^1007 QR 5SI