,H, making a continued line DH : then by (th. 1) the angles ABD-\-ABU= 2B (2 R indicates two But by (hy.) ABD-{-AB0=2R right angles.) By subtraction ABU— AB 0=0 That is, the angle OBJS is zero ; and DBO is a continued line ; or BO falls on BK Q. JE. D, THEOREM 3. J^ two straight lines intersect each other, the opposite vertical angles are equal. If AB and CD intersect each other at SI, we are to demonstrate that the angle AEG equals its opposite angle DEB, and AED = CEB. As ABB is a right line, ^A is ex- actly in the opposite direction from UB ; and for the same reason S!0 is opposite in direction from UD ; therefore, the difference in direction between EA and jE'C is equal to the difference in direction between UB and UD; or by (def. 7), the angle AEC^DEB. In the same manner we can show that the angle AED= OEB. Q. E. D. Otherwise : Let AEO=z, AED=y, and DEB=x; then we are to show that x=z. As AB is a right line, and DE falls upon it, we have, by (th. 1), x-\-y=2li Also, . . . .z-\-y=2B By subtraction. By transposition. .x—z=0 . x=z Q.E.D, BOOK I. 17 THEOREM 4. If a atraigJU line falls across two parallel straight lines y the sum of the two interior angles on the same side of the crossing line is eqital to two right angles. Let AB and CD be two paral- lel lines, and EF running across them ; then we are to demonstrate that the angle BGH+GHD=2R. Because Q-B and HD are parallel, they are equally inclined to the line EFy or have the same difference of direction from that line : Therefore J FGB= J GffD. To each of these equals add the J BGH. Then FGB-^BGH= GHD-\-BGH. But by (th. 1 ) the first member of this equation is equal to two right angles : that is, the two interior angles GHD and BGB are together equal to two right angles. §. E, JD, THE OREM 5. Jf a straight line falls across two parallel straight lines, the interior alternate angles are equal; and also the opposite exterior angles. On the supposition that AB and CD are parallel^ (see last figure), and EF falls across them, we are to demonstrate 1st. That the J ^6^^=the alternate J GffD. 2d. That A GF=^EHD ; or FGB^ CHE. By the definition of parallel lines we have FGB^GHD But FGB=AGff (ih. 3) Hence AGH=GHD (ax. 1) Q. E. D. 2d. The J FGB=GHD. But GHD=CHE (th. 3); there- fore, FGB=^ CHE. In the same manner we prove that A GF is equal to EHD. Q. E. D, THEOREM 6. If a straight line falls across two parallel siraiglU lines, the exterior angles are equal to the interior opposite angles on the same side of the crossing line. 2 IS GEOMETRY. If AB and CD are parallel, (see last figure), and UF crosses them, then we are to prove that the exterior j FGB= QHD And . . . . AGF=CHa For ... . AGH=FGB (th. 3) Also. . . . AGff=GJII) {th. 5) Hence FGB=GIID (ax. I) In the same manner we prove that A GF= CHG. Q. E. D. THEOREM 7. If a straight line falls across two other straight lines, and makes the mm of the two interior angles on the same side equal to two right angles, the two straight lines must be parallel. Let ^i^ be the line falling across the lines AB and CD, making the two angles BGE-{-GBD=^totwo right angles ; then we are to demon- strate that AB and CD must be parallel. As FF is a right line, and BA meets it, the two angles (th. 1 ) FGB+BGII=2li By (hy.) . GHD^BGH=^R By subtraction, FGB—GHD—^, That is, there is no differ- ence in the direction of GB and HD from the same line EF; but when there is no difference in the direction of lines (def 13) the lines are parallel ; therefore, AB and CD are parallel. Q. E. D. THEOREM 8. Parallel lines can never meet, however far they may be produced. If the lines AB and CD (see last figure) should meet at any distance on either side of EF, they would there form an angle ; and if they formed an angle they would not run in the same direc- tion ; and not running in the same direction, they would not be parallel ; but by (hy.) they are parallel ; therefore they cannot meet §. E. D, BOOK I 19 THEOREM 9 If two straigU lines are parallel to a third, they are parallel to each other. If AJB is parallel to UF, and CD also parallel to JEFy then we are to show that AB is parallel to CD. Because AB and FFare parallel, they make equal angles with the line HG (def. 13, 2) ; and because CD and FF are parallel, those two lines make equal angles with the line II&. Hence AB and CD, making equal angles with another line that falls across them, they are therefore parallel (def. 7). Q. F. D. THEOREM 10 If two angles have their sides parallel, the two angles will he equal. Let the two angles be A and DBF) AC parallel to DB, and ^^ parallel to BF. On that hypothesis we are to prove that the angle A=DBF. Produce DB, if necessary, to meet ^^in O, Then . ^ DBF=J DOff Also . . JiA^ADQH Therefore DBF=A (th. 6) (th. 6) (ax. 1) Q, E. D. Scholium, When ^ZT extends in the opposite direction, it is still parallel to BF ; but the angle then is the supplemental angle to DBF) that is, equal to FBQ, JO GEOMETRY. THEOREM 11. If any side of a triangle be produced, the exterior angle is equal to the sum of the two interior opposite angles ; and the sum of the three angles is equal to two right angles. Let ABCh& any triangle. Pro- duce AB to D. Then we are to show that the angle CBD= _i A -j-the angle C ; also, that the Bu- gles A-{- C-\- CBA=2li. From B conceive BU drawn parallel U) AC ; Then EBD= J. A (th. 6) By(th. 6) CBE=j^ (alternate angles). • By addition j CBD==A-\-C Q. E. D. To each of these equals add the angle CBA, and we have CBD-{- CBA=A-{- C7+ CBA But . . CBD-\-CBA=2E (th. 1) Therefore A-{-C-{- CBA=2B (ax. 1 ) That is, the three angles of the triangle are, together, equal to two right angles ; and this triangle represents any triangle ; there- fore, the sum of the three angles of any triangle is equal to two right angles. Q. E. B. Cor. 1 . As the exterior angle of any triangle is equal to the sum of the two interior and opposite angles, therefore it is greater than either one of them. Cor. 2. If two angles in one triangle be equal to two angles in another triangle, the third angles will also be equal, (ax. 3), and the two triangles equiangular. Cor. 3. If one angle in one triangle be equal to one angle in another, the sums of the remaining angles will also be equal (ax. 3). Cor. 4. If one angle of a triangle be right, the sum of the other two will also be equal to a right angle, and each of them singly will be acute, or less than a right angle. Cor. 5. The two least angles of every triangle are acute, or each less than a right angle. BOOK I. 21 THEOREM 12. In any quadrangle the sum of all the four inward angles is equal -to four right angles. Let ABCD be a quadrangle ; then the sum of the four inward angles A-\-B-\- C-\-D is equal to four right angles. Let the diagonal AQ he drawn, dividing the quadrangle into two triangles, ABO, ADC; then, because the sum of the three angles of each of these tri- angles is equal to two right angles (th. 11), it follows that the sum of all the angles of both triangles which make up the four angles of the quadrangle, must be equal to four right angles (ax. 2). §. E. D. Cor. 1 . Hence if three of the angles be right angles, the fourth will also be a right angle. Cor. 2. And if the sum of two of the four angles be equal to two right angles, the sum of the remaining two will also be equal to two right angles. SCHOLIUM. ^V-7^ In any figure hounded by right lines and angles, the sum of all the interior angles is equal to tioice as many right angles as the figure has sides, less four right angles. Let ABCDE be any figure; then the sura of all its inward angles, A-\- B-\-C-\-D-\-E, is equal to twice as many right angles, wanting four, as the ficnire has sides. o For, from any point P, within it, draw lines PA, PB, PC, C equal, then will the angle ABC=ADC, and KG be a converse side. Place the two A's so that the given angles will come together at A, and lie on the oppo- site sides of the line A C. Then because AB=AD, ABD is an isos- celes A, and the line -4 (7 which bisects the angle A is perpendicular to BD and bisects BD (th. 1 5, cor. 1 ). Now BCarndDC must termuiate in the same point C, because BC= DC (th. 15, scholium), therefore, AC is common to the two A's ABC, ADC; and the A's are identical. Q. £J. D. Scholium. There are, in fact, two cases in this theorem, because BC=BE, and DC=^DE, giving two pair of A's. X X Bkf [ -^D BOOK I. 35 THEOREM 18. Th^ difference of any two sides of a triangle is less than the third side. Let ABC be the A, and let ^C7 be greater than AB; then we are to prove that A G — AB is less than BO. As a straight line is the shortest distance be- tween two points, Therefore, . AB+B Cy AC. From these unequals subtract the equals AB=AB, and we have J5(7> AC—AB. (ax. 5). Q. E. D. THEOREM 19. When two triangles have all three of the sides in one triangle equal to all three in the other, each to each, the two triangles will be identical and have equal angles opposite equal sides. In two triangles, as ABC and ABDy on the supposition that the side AB of the one =^.6 of the other, AC==AD, and BC=^BDy we are to demonstrate that the angle ACB=ihe angle ADB, BAC= BAD, and ABC=ABD. Conceive the two triangles to be joined together by their longest equal sides, and draw the line CD. Then, in the triangle A CD, because the side AC is equal to AD by (hy.), the angle A CD is equal to the angle ADC (th. 15). In like manner, in the triangle BCD, the angle BCD is equal to the angle BDC, because the side BC is equal to BD. Hence, then, the angle A CD being equal to the angle ADC, and the angle BCD to the angle BDC, by equal additions the sum of the two angles A CD, B CD, is equal to the sum of the two AD C, BDC (^x. 2) ; that is, the whole angle ACB is equal to the whole angle BDA . 26 GEOMETRY. Since then the two sides, A C, CB, are equal to the two sides AD, DBy each to each, by (hy.), and their contained angles -4 C5, ADB, also equal, the two triangles ABQ, ABB, are identical (th. 13), and have their other angles equal, the angle BAC to the angle BAB, and the angle ABC to the angle ABB. Q. E. D, THEOREM A. J^ there he two triangles which have the two sides of the one equal to the two sides of the other, each to each, and the included angles uneqiial, the third sides will he unequal, and the greater side unll helong to the triangle which has the greater included angle. Let ABC be one A, and A CD the other A- Let AB and AC oi the one A be equal to AD and AC oi the other A- But the angle BA C greater than the angle DA C ; then we are to prove that the biase BC is greater than the base CD. Conceive the two As joined together so that the shorter sides will be common to them. As AB—AD, ABD is an isosceles A, from the vertex A draw a line bisecting the angle BAD. This line must meet BC, and will not meet CD, because the J BAC is greater than the _J DAC, and be perpendicular to BD (th. 15). From U, where the perpendicular meets BC, draw ED. ^ Now .... BE=ED (th. 15, scholium). • Add to each EC, then BC=ED-{-EC * But DE-i-EC is greater than DC; Therefore . . BCy DC. Q. E, D. • THEOREM 20. A perpendicidar is the shortest line that can he drawn from any point to a straight line ; and if other lines be drawn from the same point to the same straight line, the greater vMl he at a greater distance from ike perpendicular ; and lines at equal distances from the perpendicular^ en opposite sides, ar6 equcd. BOOK I. 27 Let A be any point witliout the line DE; and let AJB be the perpen- dicular ; A Cy AD, and AJS oblique lines: then, if JBO is less than JSJ), and BC=BE, we are to show, 1st. That AB is less than AC. 2d. AC less than AT). 3d. AC=AE. In the triangle ABO, as AB is perpendicular by (hy.), the angle ABC is a right angle ; then, as it requires the other two angles of the triangle (th. 11) to make another right angle, the angle ACB, is less than a right angle ; and as the greater side is always oppo- site the greater angle, AB is less than A C; and as AO is any line differing from AB, therefore AB is the least of any line drawn from A, 2d. As the two angles ACB and ACB (th. 1 ) make two right angles, and A CB less than a right angle, therefore A CD is greater than a right angle ; consequently, the _] D is less than a right angle ; and, therefore, in the A A CD, AD is greater than A C, or AC is less than AD. 3d. In the As ABC and ABU, AB is common, and CB=BB, and the angles at B, right angles ; therefore, by (th. 15) ^ C=A£J. Q. U. D, THEOREM 21. The opposite sides, and the opposite angles of any parallelogram, are equal to each other. Let ABDC be a parallelogram. Then we are to show that AB=CD, AC=BD, the an- gle A=D, and the angle ACD=ABD. Draw a diagonal, as CB ; then, because AB and CD are parallel, the alternate an- gles ^5(7 and BCD (th. 5) are equal. For the same reason, as A C and BD are parallel, the angles A CB and CBD are equal. Now, in the two triangles ABCandBCD, the side CB is common, and The J ACB=J CBD . . (1) or>H )BCD=JABO . (2) 28 GEOMETRY. Therefore, the third angle A— the third angle D (th. 11), and by (th. 13) the two As are equal in all respects ; that is, the sides opposite the equal angles are equal ; or, AjB= CD, and A C—BD. By adding equations (1) and (2), (ax. 2), we have the angle ^Ci> = the angle ABD ; therefore, the opposite sides, &c. §. E. D. Cor. 1 . As the sum of all the angles of the quadrilateral is equal to four right angles, and the angle A is always = to the opposite angle D; if, therefore, Ais o. right angle, D is also a right angle, and all the angles are right angles. Cor. 2. As the angle ABD, added to the angle A, gives the same sum as the angles of the £\ ACB ; therefore, the two ad- jacent angles of a parallelogram make two right angles ; and this corresponds with the 2d point of theorem 12. THEOREM 22. If the opposite sides of a quadrilateral are equal, they are also parallel, and the figure is a parallelogram. Let ABDC represent any quadrilateral, and on the supposition that AC^=BD, and AB=- CD, we are to prove that AC is parallel to BD, and AB parallel to CD. Draw the diagonal CB ; then we have two triangles ABC, and CDB, which have the common side CB; and AC of the one=jBZ) of the other, and AB of the one= CD of the other ; therefore by (th. 19) the two As are equal, and the angles equal, to which the equal sides are opposite ; that is, the angled CB =the angle CBD, and these are alternate angles ; and, therefore, by (th. 5), AC is parallel to BD; and because the angle ABC= BCD, AB is parallel to CD, and the figure is a parallelogram Q. E. D. Cor. In this, and also in (th. 21), we proved that the two As which make up the parallelogram are equal ; and the same would be true if we drew the diagonal from A to D; and in general we may say, thai the diagonal of any parallelogram bisects the parallelogram. BOOK I. 29 THEOREM 2 3. The lines which join the corresponding extremities of two equal and parallel straight lines, are themselves equal and parallel ; and the figure thus formed is a parallelogram,. On the supposition that AB is equal and parallel to CD (see last figure), we are to show that AC will he equal and parallel to BD ; and that will make the figure a parallelogram. Join CB; then because AB and CD are parallel, and CB joins them, the alternate angles ABC and BCD are equal, and the side AB=CD, and CB common to the two As ABC and CDB ; therefore by (th. 13) the two triangles are equal; that is, AC= BD, the angle A=D, and A CB= CBD; hence, AC\% also parallel to BD; and the figure is a parallelogram. Q. E. D, THEOREM 24. Parallelograms on the sam£ base, and between the same parallels, are equal in surface^ Let ^^^Cand ABFD be two par- allelograms on the same base AB, and between the same parallel lines AB and CD; then we are to show that these two parallelograms are equal. Kow CE and FD are equal, because they are each equal to AB (th. 21 ); and if from the whole line (7i>we take, in succession, (7^ and .Pi), there will remain (ax. 3) ED=CF; but FB=CA. and AF=BD (th. 21) ; hence we have two As, CAF and FBD, which have tlie three sides of the one equal to the three corresponding sides of the other, each to each; and therefore by (th. 19) the two As CAF and FBD are equal. If from the whole figure we take away the A CAF, the parallelogram ABDF remains ; and if from the whole figure the other triangle FBD be taken away, the parallel- ogram ABFC will remain ; that is, from the same quantity, if equals are taken (ax. 3), equals will be left ; or the parallelogram ABDF=ABFC. Q. E, D. M. F m ad GEOMETRY. THEOREM 25. Triangles on the same base, and bettoeen the same parallels^ are equal {in respect to area or surface). Let the two As ABE and ABF have the same base AB, and between the same parallels AB and CD ; then we are to show that they are equal in surface. From B draw a dotted line, BD^ parallel to AF ; and from A draw a dotted line A (7, parallel to BE ; and produce EF both ways, if necessary, to C and D; then the parallelogram ABFD=t\\Q parallelogram ABCE (th. 24). But the A ABE is half the parallelogram ABCE, and the A ABF is half the parallelogram ABDF; but halves of equals are equal (ax. 7) ; therefore the A ^^^=the A ABF, Q. E. D. THEOREM 26. Parallelograms on equal bases, and between the sanw parallels, are equal in area. Let ABCJ), and EFGff, be two par- allelograms on equal bases, AB and EF, and between the same parallels ; then we are to show that they are equal in area. As AB=EF=JIO ; but lines which join equal and parallel lines, are themselves equal and parallel (th. 23) ; therefore, if AB and ^6^ be joined, the figure AB Gil is a parallelogram = to ABCD (th. 24) ; and if we turn the whole figure over, the two parallel- ograms HEFQ and RGB A, will stand on the same base, HG, and between the same parallels ; therefore, HGEF=^HGBA ; and consequently (ax. 1) ABCD^EFGH. Q. E. D. Cor. Triangles on equal bases, and between the same parallels, are equal ; for, join BD and EG, the A ABD is half of the par- allelogram A G; and the A EFG is half of the equal parallelogram FH; therefore, the A ^^i>=the A EFG (ax. 7). BOOK I. 31 THEOREM 27. If a tnangle and a parallelogram he upon the same or equal bases, and between the same parallels, the triangle wUl be half the parallelogram. Let ABC be a A, and ABLE a parallel- ogram, on the same base AB, and between the same parallels ; then we are to show thai the A ABC is half of ABDE. Draw the diagonal UB to the parallelo- gram ; then, because the two As ABC and ABE are on the same base, and between the same parallels, they are equal (th. 25) ; but the A ^^^is half the parallelogram ABDE (cor. to the 22) ; therefore the A ABC is half of the same parallelogram (ax. 7). Q. E, i>. THEOREM 28. The c&mplementary parallelograms of any parallelogram which are abotU its diagonal, are equal to each other. Let AC he Si parallelogram, and BD its diagonal ; take any point, as E, in the diagonal, and from it draw lines parallel to its sides ; thus forming four parallelograms. We are now to show thai the comple- mentary parallelograms AE and EC, are equal. By corollary to theorem 22 we learn that the A ADB=A DBG. Also by the same (cor.) a=6, and c=d; therefore by addition . . . a-\-c=^b-\-d. Now from the whole AADB take the sum of the two As {a-\-c), and from the whole A DBC ioke the equal sum {b-^-d), and the remainders AE and EC are equal (ax. 3). Q. E. D. THEOREM 29. The sides of a parallelogram vfill inclose the greatest space when the angles are right angles. 32 GEOMETRY. Lei ABDC be a right angled parallelogram, and ABha an ob- lique angled parallelogram of equal sides to the other ; then we are to show that the right angled 'parallelogram ABDC is greaier than the oth^r, ABba. We take Aa=A C, Then Aa is less than AE, because the per- pendicular A (7, or its equal Aa^ is less than any oblique line AE (th. 20) ; therefore the line ab is between the two parallels AB and CF. The parallelogram ABDC=ABFE ; because they are on the same base AB, and between the same parallels (th. 24) ; but the parallelogram ABba is but part of the parallelogram ABFE ; therefore, ABFE, or its equal ABDC, is greater than ABba ; but the parallelogram ABba has the same length of sides, respectively, as the parallelogram ABDC ; therefore the side, &c. Q. E. D. Cor. It is evident, then, that the area of the parallelogram ABba will become less and less as its angles become more and more oblique ; and greater and greater as its angles become nearer and nearer to right angles. Scholium. All parallelograms (indeed all figures) are referred to square units for their measurement, and the unit may be taken at pleasure ; it may be an inch, a foot, a yard, a rod, a mile, &c., according as convenience and propriety may dictate. For example, the parallelogram ABD C is measured by the number of linear units in CD, multiplied into the number of linear units in A C ; the product will be the square units in ABDC ; for conceive CD com- posed of any number of equal parts — say five — and each part some unit of linear measure, and AC composed of three such units, and from each point of division on CD draw ^OHBHII^S fines parallel U) AC ; and from each point of IHHHHHI division on AC draw fines parallel to CD or ■■HHHHI AB ; then it is as obvious as an axiom that the IsSHSBI parallelogram will contain 5X3=15 square B^^^^^KfiB units ; and in general the areas of right angled parallelograms are found by multiplying the base by the altitude. Right angled parallelograms are called rectangles (def. 26), and the altitude of any parallelogram, whether right angled or not, is the perpendicular distance between its opposite sides. BOOK I. 33 THEOREM 30. The area of any plane triangle is measured by the product of its base into half its altitude ; w half the base into the altitude. Let ABQ represent any triangle, AB its base, and AD at right angles to AB its alti- tude ; then we are to show that the area of ABO is equal to the product of AB into one half of AD ; or the half of AB into AD. On AB construct the rectangle ABED; and the area of this rectangle is measured by AB into AD (scholium to th. 29) ; but the area of the A ABQ is one half this rectangle (th. 27) ; therefore, &c. Q. E. D, THEOREM 31. The area of a trapezoid is measured by the half sum of its parallel sides, multiplied into the perpendicular distance between them. Let ^^i) (7 represent any trapezoid, and draw the diagonal BO, which di- vides it into two triangles, ABG and BCD: CD is the base of one tri- JOigle, and AB may be considered as the base of the other ; and EF is the common altitude of the two triangles. Now by the last theorem the area of the triangle CDB is=^ CDXEF; and the area of the A ABC=^ABXEF; therefore, by addition, the area of the two As, or of the trapezoid, is equal to U^B+ CD) X EF, Q. E. D, THEOREM 32. If there be two lines, one of which is divided into any number of parts, the rectangle contained by the two lines is equal to the several rectangles contained by the undivided line, and the several parts of (he divided line. 34 GEOMETRY. Let AB be one line, and AD the other ; and suppose AB divided into any number of parts at the points U, F, G, &c. ; then the whole rectangle of the two lines is AIT, which is measured by AB into AD; and the rectangle AL is measured by AJS into AD; and the rectangle ^^is measured by JEF into EL, which is equal to EF into AD; and so of all the other partial rectangles ; and the truth of the proposition is as obvious as that a whole is equal to the sum of all its parts ; and requires no other demon- stration than an explanation of exactly what is meant by the words of the text. THEOREM 33. . If a straight line he divided into any two parts , the sqimre of the whole line is egtccU to the sum of the squares of the two parts, and twice the rectangle contained by the parts. Let AB be any line divided into any two parts at the point ; then we are to show that the square on AB is equal to the sum of the squares on AC and CB, and twice the rec- tangle of AC into CB. On AB describe the square (or con-, ceive it described) AD. Through the point C conceive CM drawn parallel to BD; and take BH=BC; and through H draw J522V parallel to ABy and CH is the square on CBy by direct construction. As AB=BD, and CB=Bff, therefore, by subtraction, AB — CB=BD—BH; or AC=ffD. But NK==ACy being opposite sides of a parallelogram ; and for the same reason KM^=HD; therefore (ax. 1), NK=KM ; and the figure NM is a square on iVX equal to a square on A 0. But the whole square on AB is com- posed of the two squares CH, NM, and the two complements or rec* tangles ^^and KD; and each of these is ^(7 in length, and BO in width ; and each has for its measure A C into CB; therefore the whole square on AB is equal to A C''-\-B C'-^lA CX CB. Q. K D, This may be proved algebraically, thus : irnmnimnniniiiiiuutMimmffii BOOK I. 35 "Let w represent any whole right line divided into any two parts a and b; then we shall have the equation w=a -{-b By squaring w^z=a^-i-b'^-]'2ab, Q. E. D. Scholium. If a =5^ then w'=4a^ which shows that the square of any whole line is four times the square of half of it. THEOREM 31. The square on the difference of two lines is equal to the sum of the squares of the two lines, diminished by twice the rectangles contained by the lines. Let AB represent the greater line, -6 (7 a lesser line, and A C their difference. We are now to show that the square on AG is equal to the sum of the squares on AB and B C, diminished by tvnce the rectangle contained by AB into BC. On AB conceive the square AF to be de- scribed ; and on CB conceive the square BL described ; and on -4(7 describe the square ACGM; and pro- duce MG to K. As aC=A C, and CL= CB; therefore, by addition, ( G0-{- GL), or GL, is equal [AC-{-CB), or AB. Therefore the rectangle GE is AB in length, and CB in width ; and is measured by AB into BC. Also AH'=AB, and AM— AC; therefore by subtraction MH =^CB; and as MK=AB, the rectangle UK is AB in length, and CB in width, and it is measured by AB into CB; and the two rectangles 6^^^ and JIir,&re together equal to 2ABXBC. Now the squares on AB and BC make the whole figure AHFELC ; and from this whole figure, or these two squares, take away the two rectangles HK and GE, and the square on ^ (7 only will remain ; that is, AC^=AB^-^BC^--2ABXB0. Q. E. D. This may be proved algebraically, thus: 36 GEOMETRY. Let a represent one line, h another and lesser line, and d their difference ; then we must have this equation : d=a — b By squaring . . d^=a^-\-lP-^2ah, THEOREM 35. The difference of the squares of any two lines is equal to the rec* tangle contained hy the sum and difference of the lines. Let AB be one line, and A C the other, and on them describe the squares AD, AM; then the difference of the squares on AB and on AC is the two rectangles UF and FC, We are now to show that the measure of these rectangles may be expressed hy (AB-j-AC) into (AB — AC). The rectangle EF has ED, or its equal AB, for its length ; the other has MC, or its equal A C, for its length ; therefore, the two together (if we conceive them put between the same parallel lines) will have {AB-^-AC) for the length; and the common width is CB, which is equal to (AB — A C); there- fore, AB'--AC^={AB^AC)X{AB—AG). Q. E. D, This is proved algebraically thus : Put a to represent one line, and b another ; Then a-\-b is their sum, and a — b their difference ; and . . (a'\-b)X(a—i)=a^—b\ Q. E. D. THEOREM 36. The square described on the hypotenuse of any right angled triangle is equal to the sum of the squares on the other two sides. Let ABC represent any right angled tri- angle, the right angle at B. We are to show that the square on AG is equal to the sum of two squares ; one on AB, the other on BC. Conceive the three squares, AD, AT, and BM, described on the three sides. Through the point B, draw BXE perpendicular to A (7, and produce it to meet the line 6*7 in K. Produce AF to meet GI in H. If ML be BOOK I. ^ produced, it will meet the point K, and /^ZJTwill be a right angled parallelogram ; for its opposite sides are parallel, and all its angles right angles. The angle BA ^ is a right angle, and the angle XAH is also a right angle ; and from these equals if we subtract the common angle BAH, the remaining angle, BA (7, must be equal to the re- maining angle GAH. The angle ^ is a right angle, equal to the angle ABC; and AB=AQ; therefore, the two As ABC and AGJI are equal, and AH=^AC, But AG=AF; therefore AH =AF. Now the two parallelograms, AJE and ^iT are equal, be- cause they are upion equal bases, and between the same parallels, FH and JEJ^(th. 26). But the square AI, and the parallelogram -4 JST are equal, because they are on the same base, AB, and between the same parallels, AB and OIC; therefore the square AI, and the parallelogram AF, being both equal to the same parallelogram AF, are equal to each other (ax. 1 ). In the same manner we may prove the square BM equal to the rectangle iV"i>/ therefore, by addition, the two squares Aland BM, are equal to the two parallelograms AF and 2^D, or to the square AD. Q. F. D. Scholium. The two sides AB and BC may vary, while AC remains constant. ^5 may be equal to BC ; then the point iV would be in the middle of A C. When AB is very near the length of AC, and BCyerj small, then the point ^V falls very near to C. Now, as the parallelograms AF and JVD (while AC remains unchanged) depend for their relative magnitudes on the position of the point iV, on the line A C, the area AF must be to the area JVB as the line ^^Y to ^C ; that is, the square on AB, must be to the square on BC, as the line AN to the lin£ NC. ANOTHER DEMONSTRATION OF THEOREM 36. Let ABC be a right angled triangle, right angled oXA. Call AB, a, AC, b, and B C, h : then we are to show that a^-(-5^=A^. Produce AB to D, making BD=AG; and produce A C to F, making CF=AB : then AD^AF; and each of these lines is (a +i), and the whole square ^^is the square of (a+b), and by (th. 33) is a^+b^+2ab. 88 GEOMETRY. From B draw BG 2ki right angles to CB ; and from C draw CF at right angles, the same line CB ; then BG and CF must be parallel, and join FG. We must now prove that the four triangles* in the square AH are all equal, and that CGBF is the square on CB. As the two angles CBA and CBD make two right angles^ (th. 1), and CBG is a right angle by construction, therefore the two angles CBA and GBD make one right angle. But CBA and A CB (cor. 4, th. 11) are also equal to a right angle ; and from these equals take the angle CBA, and the angle GBD = the angle A CB, But the angle A= the angle D; both right angles, and BJ) was made equal to AG; therefore, the two triangles^ ABC and GBJ), having a side and two angles equal, are in all respects equal, and CB=BG. In the same manner we prove BG=GF; and there- fore CG is a square on CB, and the four triangles are each equal to ABC, and each triangle has for its measure ^ab. The measure of two of these is ab, and the four is 2ab. Now . Also . By subtraction By transposition AD^=a^-^b^-\-2ab AI)^=k^-\-2ab ■=a^^b'^h^ h^ =a^+b\ Q.F,D. Cor. From this equation we may have A*— a2=5^•or, (k+a) {h-a)^I^. THEOREM 37. In any obtuse angled triangle the square of the side opposite the obtuse angle is greater than the sum of squares on the other two sides, by twice the rectangle of the base, and the distance of the per" pendicular from the obtuse angle. Let ABC be any obtuse angled A, obtuse angled at B, Represent the side opposite B by b ; opposite -4 by a / and opposite G by c (and let this be a general form of notation) : also represent the perpendicular by p, and J)B by X. Now we are to show that b'^=a^-\' c^-\-2ax. By(th. 36) . . . jt>'+(a+ar)2=52 Also . . . p'-i' «'=*c» BOOK I. 89 By expanding equation (1), and subtracting (2), we have By transposition b^==a^+c^+2ax, Q. E, D. S(,nolium. This equation is true, whatever be the value of x, and X may be of any value less than CD. When x is very small, B is very near />, and the line c is very near in position and value to jt?. When a:=0, c becomes j?, and the angle ABG becomes a right angle, and the equation becomes IP=a^-{-c^, corresponding to (th. 36). THEOREM 38. In avy triangle, the square of a side opposite an acute angle is less than the square of the base, and the other side, hy twice the rectangle of the base, and the distance of the perpendicular from the actiie angle. Let ABC, eith- er figure, represent any triangle ; G the acute angle, CB the base, and AD the perpen- dicular, which falls either without or on the base. Then we are to prove that AB* ^CB'+AC^—^CBX CD. As in (th. 37), put AB=c, AC=h, CB=a, BD=x, AD=p; and when the perpendicular falls without the base, as in the first figure, CD=^a-]-x ; when it falls on the base, CD=a — x. Considering the first figure, and by the aid of (th. 36), we have the following equations : p'+(a+x)'=b' (1) p'+x''=c' (2) By expanding (I), and subtracting (2), we have a^-i-2aj:=b^—c^ By adding a^ to both members, and transposing c', we have c'-{-(2a^-\-2ax)=b^-\-a^ By transposing the vinculum, and resolving it into factors, ^^c have c':=a'+b'--2a(a+x). Q. K D. 40 GEOMETRY. Considering the other figure, we have (1) (2) By subtraction a^ — 2ax =b^ — c^ By adding a^ to both members, and transposing c^ we have . c^=b'^'\-a'—2a(a—x). Q. E. D. THEOREM 39. If in any triangle a line be drawn from any angle to the middle of the opposite side, tmce the square of this lin£, together with twice the square of half the side bisected, will be equal to the sum of the squares of the other two sides* Let ABC be a triangle, its base bisected in M. Tlien we are to prove that 2AM2+2CM2=AC2+AB^ Draw AD perpendicular to the base, and call it p. Put AC=:by AjB=c, CB=2a ; then CM=:a, and MB=a. Make MD=x ; then CD =za-{-x, and DB=a — x. Put AM=m, Now by (th. 36) we have the two following equations : ^24-(a— a;)2=c2 (1) ' p^-\-(a^xy=b^ (2) By addition . 2p2+2ar2-f-2a2=62+c^ ^viip''-^x'=m Therefore 2m''-\-2a''=b''^-c\ Q. E. D. THEOREM 40. The two diagonals of any parallelogram bisect each other; and the sum of their squares is equal to the sum of the squares of all the four sides of the parallelogram. Let ABCD be any parallelogram, and draw its diagonals A C and BD, We are now to show, 1st. Thai AE =EC, DE=EB. 2d. That AC^+BD^ =AB2+BC=»+DG='+AD^ BOOK I. 41 1. The two triangles ABB and BBC are equal, because AB xnDC, the angle ABE — the alternate angle EDO, and the vertical angles at E are equal ; therefor*^, AE, the side opposite the angle ABE, is equal to EC, the side opposite the equal angle EDC : also EB, the remaining side of the one A is equal to ED, the remaining side of the other triangle. 2. As ADC'\?> a triangle whor-e base AC \% bisected in E, we have, by (th. 39), , ckc, &c. ; and whatever relation or ratio A is of E, the same ratio B will be of D; and whatever relation j5 is of A, the same relation D will be of E. This shows that the means may be changed, or made to change places. Or, . . . A: E=B : i), which is the former pro- portion with the middle terms or means changed. The Jirst and third of four magnitudes are called the antecedents ; the second and fourth, the consequents. A simple relation or ratio exists between any two magnitudes of the same kind ; but a proportion, in the full sense of tlie term, must consist of four quantities. When the two middle quantities are equal, as, A:B=B:0 then the three quantities. A, B, and C, are said to be continued proportionals ; and B is said to be the mean proportional between A and C; and C is said to be the third proportional to A and B. In the proportion A : B= C : J), the last D is said to be the fourth proportional to AylB, and 0. By the same rule of expression, A may be called the first pro- portional, B the second, and C the third ; for either one can be found when the other three are given, as we shall subsequently explain.. When quantities have the same constant ratio from one to the other, they are said to be in continued proportion, Thus: the numbers 1, 2, 4, 8, 16, . Some numeral relation, or ratio, must exist between A and B. Let that ratio be represented by r; that is, B must equal rA. But, by the definition of proportion, the same relation must exist between Cand D as between A and B; or D=rC. Then by substitution we have A:rA==C:rC. The product of the extremes is rCA, and that of the means is ArC; obviously the same. Q. E. D, 46 GEOMETRY. THEOREM 3. If three magnitudes he continued proportionals y the product of the extremes is equal to the squxire of the mean. Let Ay By and C represent the three magnitudes : Then . .A: B=B : C, by the definition of proportion. But by theorem 2 (book 2), the product of the extremes is equal to the product of the means ; that is, -4X (7=ji5^. Q. E. D. THEOREM 4. Equimultiples of any two magnitudes have the same ratio as the magnitudes themselves ; and the magnitudes and their equimultiples wxiy therefore form a proportion. Let A and B represent the magnitudes, and mA and mB their equimultiples. Then . . . A'.B=mA:mB 7? •>» 7? H For the ratio of -4 to ^ is _, and of mA to mB is — =— , the A mA A same ratio ; therefore, », and ^": ^": : c": J>* A By the hypothesis, — = y-. Raising both members of this equation to the nth power, and BOOK II. 49 Changing this to the proportion A^ : B*:: C*: JD* B D' A G Resuming again the equation -D="n» ^^'^ taking the nth root -4* C" of each member, we have — j— ==— j-. Converting this equa- B D tion into its equivalent proportion, we have I i L L » n n n A\ B ::(7 :i> Now by the first part of this theorem, we have mm 5 2 n n n n A \B \\C \ D w* representing any power whatever, and n representing any root. THEOREM 11. If four magnitudes he proportional, also four others ^ their com- pound, or product of term hy term, vMlform a proportion. Admitting that . A'. B:\ O :D And . . . X: V:: M : N We are to show that AX: BY: ; MG : ND A G From the first proportion, ^-=7^ From the second, ~v^^l^ Multiply these equations, member by member, and AX^MG BY ND Or . . AX:BY'.:MG:ND The same would be true in any number of proportions. THEOREM 12. Taking the same hypothesis as in (th.ll), we propose to show, that a proportion may be formed by dividing one proportion by the other, term by term. By hypothesis, . A : B : : G : D And . . . X:Y:iM:2f 4 50 \ GEOMETRY. Multiply extremes and means, AD'=BC (1) And J!fX=Mr (2) Divide (1) by (2), and . d.x^=Jx J Convert these four terms, which make two equal products, into a proportion, and we shall have By comparing this with the given proportions, we find it com- posed of the quotients of the several terms of the first proportion, divided by the corresponding term of the second. THEOREM 13. Iffov,r magnitudes he proportional ^ we may multiply the first couplet or the second couplety the antecedents or the consequents, or divide them by the same factor, and the results will he proportional in every case. Suppose . . . . A : B : : C: J) Multiply extremes and means, and AI>=JBO (1) Multiply this equation by J/, and MAD=MBC Now, in this last equation, MA may be considered as a single term or factor, or MD may be so considered. So, in the second member, we may take 3fB as one factor, or MC. Hence, we may eonvert this equation into a proportion in four different ways. Thus, as . • JuA I juB '. C .D Or as A :B :: MC MD Or as MAiB : '.MC .J) Or as A :MB: C '.MD If we resume the original equation (1), and divide it by any number, M, in place of multiplying it, we can have, by the same •ourse of reasoning. A B . M' M' : C:D A:B : C D 'MM i-': 4- .4. -4 BOOK II. 51 THEOREM 14. If three magnitiules are in continued proportion, the first m to the third, as the sqitare of the first is to the square of the second. Let w4, J5, and (7, represent three proportionals. Then we are to show that A : 0=A^ : B* By (th. 3) AC^B" Multiply this equation by the numeral value of A, then we hav§ A^C^AB" This equation gives the following proportion : A : (7=^' : B\ §. E. D, THEOREM 15. If any one of the four mar/mtudes which fomi a proportion^ be effaced or unknown, it can he rentored by means of the other three. Let A : B=C : D represent a proportion, and suppose D un- known ; then represent it by ^r That is . . A\ B=C : x The ratio between A and B is the same as between C and x. Represent the ratio between A and B hy r; and as r is always a numeral, whatever quantities are represented by A and B, X therefore, p^=r; or x-=rC ; which shows that x or D must be of the same name as C When A and B are not commensurable, the ratio is expressed . B ^ CB , by ~j and rc=— -; or, m numbers, the product of the second and third terms divided by the first, will give the fourth, which is the rtde of three in arithmetic. In short, as An Tin A ^^ j^ ^^ n ^^ 4 n ^^ AD^BC, ^=-^-, -5=-^. ^^~B~* *^^-^=^- 52 GEOME TRY THEOREM 16. Parallelograms, and also triangles, having the same or equal atti- tudes, are to one another as their bases. Let a represent the number of units, and part of a unit in BC, and h the num- ber of units and part of a unit in BD. Also let^ represent the units and parts of a imit in the perpendicular, AB. Now by (scholium to th. 29 book 1), the parallelogram ABCE=pa, and the parallelogram ABDF=ph ; and as magnitudes must be proportional to themselves, ABCE : ABDF=pa : pb But . . a : b=pa : pb (th. 4 book 2) Therefore (th. 6 book 2), we have ABCE : ABDF=a : b. Q. E, I). Car 1. As triangles on the same base and altitude as parallel- ograms are halves of parallelograms ; and as the halves of quan- tities are in the same proportion as their wholes ; therefore The . . ABPC : A BQD=a : b. Cot. 2. When parallelograms and triangles have the same or equal basis, they will be to each other as their altitudes ; for the proportion ABCE \ ABDF=pa\pb, as above, is always true; and when a becomes equal to b and p, and p different, Then . . ABCE: ABDF=Fa:pa Or . , ABCE:ABDF-^P :p, that is, as their perpendicular altitudes. THEOREM 17. Lines drawn parallel to the base of a triangle, ctU the sides of the triangle proportionally. Let ABC be any triangle, and draw 2)-£^ parallel to the base BG; then we are to show thai AD : BB=AE : EC. Join DC and BE. The triangle DEB = the A DEC, because they are on the same base, DE, and be- tween the same parallels, DE and -B(7 (th. 25 book 1). BOOK II. iS Represent the triangle ADE by T, DEB by x, DEC by y; then x=y. Now, as the triangles T'and x may be considered as having AD and DB for bases, and the perpendicular distance of the point E from AB for altitudes, therefore, by (th. 16, book 2). AD:DB==T:x By reasoning in the same manner in reference to the triangles T and y, they having their common vertex in jO, we have the proportion AE : EC=T\y. AE\EC=T AD:DB=:T Therefore But -A But x=y Therefore, (th. 6, book 2) AE : EC^AD : DB Or AD: DB=AE : EG, Q. E. D, Cur. Considering AEB as one triangle, and AED another, having their common vertex mE; and in the same manner, ADC as one, and ADE another, whose vertex is i>, then we may have AB : AD=AG : AE For, by taking the proportion AD : DB=AE : EC And by composition, (th. 8 book 2), we have AB : AD=AC :AE. THEOREM 18. Similar trianyles have their sides, about the equal angles^ pro}Jorti" i^ a * This proposition has had a symbolical proof, in theorem 2 book 2, but wo deem it important to give this geometrical demonstration. 56 GEOMETRY. Place A and B at right angles with each other, and draw the hypotenuse. Also place C and D at right angles to each other, and draw its hypotenuse. Then bring the two triangles together, so that C «hall be at right angles with jB, as represented in the figure. Now, these two As have each a right j , and the sides about the equal angles, proportional ; that is, A : B=C : D; therefore, (th. 20, b. 2), the two As are equiangular, and the acute angles which meet at the extremities of B and C, are=to a right angle, and the lines B and C make another right angle, by construcUon ; therefore, the extremities of Ay B, C, and D, are in one right line (th. 2 b. 1), and that line is the diagonal of the parallelogram cb. Hence, the complementary parallelograms about this parallel- ogram are equal (th. 28, b. I) ; but one of these is B long, and and C wide, and the other D long, and A wide ; therefore, BXC=AXI). Q.E.J). Cor. When B=0 then A'D=B^, and B is the mean propor- tional between A and D. THEOREM 22. Similar triangles are to one another as the squares of tlieir like sides. Let ABC, and DBF, be two similar or equian- gular triangles. Then we are to prove that ABG'.DEF=AB^'.DE^ By the similarity of the triangles, we have, AB But, . . AB Hence, , . AB^ . DE^=^AB'LC .DE^MF But, by (th. 30, b. 1), AB'LC is double the area of the A ABC, DE'MF'is double of the A DFF. Therefore, ^ ABC : A DFF : : AB'LC : DE^MF (Th. 6, b. 2). " '* = AB^\DE\ Q. E, D. DE=^LG \MF DEz=:AB: BE BOOK II. 57 \4J^ THEOREM 23. The perimeters of similar figures are to one another as their like sides ; and their areas are to one another as the squares of their like sides. Let ABODE, and ahcde, be two similar figures ; then we are to show that EA is to ea as the sum of all the sides EA-1- AB, <&c.y is to ea+ab, c&c, and that the area of one is to that of the other, as EA^ to ea', (yr AB' to ab'. As the figures are exactly similar by bypotbesis, "whatever rela- tion AB is to EAy the same relation ah will be to ea ; and if we take AB=mEA^ 'cD^ ""ea r ^^®^ ^® °^^* ^^^ I>E=qEA^ Now, by (th. 7, b. 2), AE : ea=:EA-{-mEA, d;c. That is, EA : ea=P : jp, P and p representing the perimeters of the figures. As the two figures are exactly similar, whatever part the triangle EAB is of one whole, the same part the triangle eah is of ^ho other whole ; therefore, EAB : eab=EABCDE : eabcde. But by (th. 22, b. 2) EAB : eab=AB^ : ah" Therefore, by (th. 6, b. 2), EABCDE : eahcde^AB^ : ah\ Q, E. D, ' ah=m(ea) be = n(€a) cd=pUa) Je= glea) €a-\-meay chord, C the center of the circle, and CD perpendicular to AB ; then we are to prove that AD=BD, and AU=JSB. As C is the center of the circle, AC=CB, and CD is common to the two As A CD and BCD, and the angles at D being right angles, therefore the two As ADC and BDC are identical, and AD=DB, which proves the first part of the theorem. Now as AD=DB, and DU common to the two spaces, ADE and DEB, and the angles at D, right angles, if we conceive the sector CBE turned over and placed on GAE, CE retHining its posi- tion, the point B will fall on the point A, because AD=DB ; then the arc BE will fall on the arc AE; otherwise, there would be points in one or the other arc unequally distant from the center, which is impossible ; therefore, the arc AE = the arc EB. Q. E. D, THE OREM 2. Equal angles, at the center are subtended by equal chords, (See figure to last theorem). Let the angle AGE=zECB, then the two isosceles triangles, A CE, and ECB, are equal in all respects, and AE=EB. Q. E. D, THEOREM 3. In the same circle, or in equal circles, equal chords are equally distant from the center. Let AB and EF be equal chords, and C the center of the circle. From C, draw CO and CH perpendicular to the respective chords. These perpen- diculars will bisect the chords (th. 1, b. 3), and we shall have AG=EH, W$ ar§ now to sh^w that C0=: CH. BOOK III. 63 In the two As, A CG and ^Cff, we have ^C= CA, A G=E£r, and the angle ^= the angle O, both being right angles ; tliere- fore, the two triangles ACQ, and ECU, are identical, and CG=CH. q.E.D. We may demonstrate this theorem analyticalli/, and more generally ^ 9^ follows : Let ^^ represent the half of any chord, and put it equal to C. r*ut £[C==F, and CE=Ii; R representing the radius of the nircle. Then, by (th. 36, b. 1), we have C^-{-P^=zR^ (1) Also let A G represent the half of any other chord, and put it tqual to c; and put its distance from the center equal to p; then, c2+i?2=^2 (2) By equating the first members of (1) and (2), we have this general equation : C''-{-P''=c'-\-p^ (3) Now, if (7=c, that is, the chords equal, then P'^=p^^ or P=p, the perpendiculars will be equal ; and if P='p, then C==c; that is, chords equally distant from the center, are equal. Equation (3) is true, under all circumstances, and if we suppose C greater than c, then P will be less than p; that is, the greater the chord, the nearer it will be to the center. For if is greater than c, let d be their difference ; Then, . . C=c-\-d, and C^=c^-}-2cd+d^ And substitute this value of C^ in equation (3), and we have, c'+^cd+d^+P^^c^+p^ By canceling c^ we have, 2cd-\-d^-{- P'^—p^ That is P^ is less than p^y because it requires ^cd-^-d^ to make equality ; and if P^ is less than p^, P is less than p; thai is, the greater chord is at a less distance from the center. Cor. If the chord C runs through the center, then P, in equa- tion (3), equals 0, and (7'=c^4-i>^- ^^* P^==c^-\-p^, by equation (2), or C^=B?, or C=jR, or the semichord becomes the radius, a.s it manifestly should, in that case. THEOREM 4. If any line he drawn tangent to a circle, and from the point of con- tact a line he drawn to the center of the circhy the tangent and this radiits will form a right angle. A tangent line can meet the circle only at one point, for if the 64 GEOMETRY. line meets the circles in two points, and is still a tangent, it follows that the portion of the circumference between the two points, is a right line ; but no part of a circumference is a right line, but a continued curve line ; and whenever a right line meets a circle in two points, it must ctU the circle, and therefore cannot be a tangent. Now let ABO be a tangent line, touching the circle at the point B, and draw the radius, EB, and the line EC, and JSA. Now we are to show that EB is perpendicular to ABC. Because B is the only point in the line ABO which touches the circle, any other line, as UO, or EA, must be greater than UB; therefore, £JB is the shortest line that can be drawn from the point U to the line AO; therefore, UB is the perpendicular to AG (th. 20, b. 1). Q. K D. THEOREM 5. In the same circle, or in equal circles, equal chords subtend or stand on equal portions of the circumference. Conceive two equal circles, and two equal chords drawn within them. Then conceive one circle taken up and placed upon the other, in such a position that the two equal chords will fall on, and exactly coincide with each other; and then the circles must coincide, because they are equal ; and the two segments of the two circles on each side of the equal chords, must also coincide, or the circles could not coincide ; and magnitudes which coincide, or exactly fill the same space, are in all respects equal (ax. 9). Therefore Q. E. D, THEOREM 6. Thrmngh three given points, not in the same straight line, one cir- cumference can be made to pass, and but one. Join AB and BO. If a circle is made to pass through the two points A and B, the line AB will be a chord to such a circle ; and if a chord is bisected by a line at right angles, the bisecting line will pass through the center of the circle (th. 1, b. 3) ; therefore, if we bisect the line AB, BOOK III. 65 and draw DF at right angles from the point of bisection, any circle thai can pass through the points A and B, must have its center some- where in the line DF. And, by reasoning in the same way (after we draw £0 at right angles from the middle point of JBC), any circle that can pass through the points B and C, must have its center somewhere in the line FG. Now, if the two lines, DF, and FG, meet in a common point, that point will be a center, from whence a circle can be drawn to pass through the three points, A, By and (7, and DF and FG will always meet, unless they are parallel, and if they are parallel, it follows that AB and BC must be parallel (definition 1 3), or be in one and the same straight line ; but this can never be the case while the three given points, A^ JB, and (7, are not in the same straight line ; therefore the two lines will meet, and from the point H, at which they meet, a circle, and only one circle, can be drawn, passing through the three given points. Q. F. D, THEOREM 7. Jf tivo circles touch each other internally, or externally, the two centers and point of contact shall be in one right line. Let two circles touch each other internally, as represented at A, and through the point A, conceive AB to be a tangent, at the common point. Now, if a line, perpendic- ular to AB, be drawn from the point A, it must pass through the center of either circle (th. 4, b. 3); and as there can be but one perpendicular from the same point, (th. 20, b. 1), therefore. A, C, and D, the point of contact, and the two centers, must be in one and the same line. Q. F. D. Next, let the circles touch each other externally, and from the point of contact conceive the common tangent, AB, to be drawn. Then a line, AC, perpendicular to AB, will pass through the center of tlie external circle, (th. 4, b. 3), and a perpendicular, AD, from the same point, A, will pass through the center of the 66 GEOMETRY. other circle ; hence, JBA C and BAD are together equal to two right angles; therefore (7, A, D, is one continued line (th. 2, b. 1). Q.U.J), Cor. When two circles touch each other internally, the distance between their centers is equal to the diflference of their radii ; and when they touch each other externally, the distances of their centers are equal to the sum of their radii. THEOREM 8. An angle at the circumference of any circle is measured hy half the arc on which it stands. In this work it is taken as an axiom that any angle standing at the center of a circle is measured by the arc on which it stands ; and we now proceed to show that the angle at the circumference, is half the angle at the center. Let A CB be an angle at the center, and D an angle at the circumference, and at first suppose 2> in a line with A C. We are now to show that the angle ACB is double the angle D. Join DB, and the A DCB is an isosceles triangle ; for Ci>= CB; and as its exte- rior angle, A CB, is equal to the two inte- rior angles, i>, and CBD, (th. 11, b. 1), and these two angles equal to each other ; therefore, A CB is double the angle at D; but ACB is measured by the arc AB; therefore the angle D is measured by half the arc AB. Now let D be not in a line with A C, but at any point on the circumference (ex- cept on AB), and join DC, and produce it to^. Now by the first part of this theorem. The angle . ECB=^EDB Also, . . ECA=ite angles at C and D, aie together measured by half of the whole circumierence, or by 180 degrees, or by two right angles. Q. E. D. THEOREM 11. An angle formed hy a tangent and a chord, is measured hy one half of the intercepted arc. Let AB be a tangent, and AD a chord, and ^ the point of contact ; then we are to show that the angle BAD is measured by half the arc AED. From A, draw the radius AC ; and from the center, C, draw CE perpendicular to AD. The aagle BAD-\-DA (7=90° (th. 4, b. 3) Also, C+i)^(7=90° (cor. 4, th. 11, b. I) Therefore, by subtraction, BAD — (7=0 By transposition, the angle BAD=0. But the angle (7, at the center of the circle, is measured by the arc AE, the half of AED; therefore, the equal angle, BAD, is also measured by the arc AE, the half of AED. Q. E. D. THEOREM 12. An angle formed by a tangent and a chord, is equal to an an^le in the opposite segment of the circle. Let AB be a tangent, and AD a chord^ and from the point of contact. A, draw any angles, as A CD, and AED, in the seg- ments. Then we are to shoio that the angle BAD=ACD, and GAD=AED. By the last theorem, the angle BAD is measured by half the arc AED; and as the angle ACD (th. 8, b. 3) is measured by half of the same arc, therefore the angle BAD=ACD BOOK III. 69 Again, as AEDG is a quadrilateral, inscribed in a circle, the sum of the opposite angles, AGD-\-AED=^2 right angles, (th. 10, b. 3) Also, the angles BAD-\-DAG=^ right angles, (th. 1, b. 1) By subtraction (and observing that BAD has just been proved equal to A CD), vre have, AI!J)^J)AG=0 Or, • • • AED=DA G, by transposition. Q. E. D, THEOREM 13. Parallel chords^ or a tangent and a parallel chord, intercept equal a/rcs on the circumference. Let AB and GD be two parallel chords, and draw the diagonal, AD; and because AB and GD are parallel, the angle DAB = the angle ADG (th. 5, b. 1); but the angle DAB has for its measure, half of the arc BD; and the angle ADG has for its measure, half of the arc AG (th. 8, b. 3); and because the angles are equal, the arcs are equal ; that is, the arc BD— the arc ^(7. q.E.D, Next, let EF be a tangent, parallel to a chord, GD, and from the point of contact, G, draw GD, By reason of the parallels, the angle GDG = the angle DGF, But the angle CDG has for its measure, half of the arc GO (th. 9, b. 3); and the angle DGF has for its measure, half of the arc GD (th. 11, b. 3); therefore, these equal measures of equals must be equal ; that is, the arc GG = the arc GD, Q, E. D, THEOREM II. When two chords intersect each other within a circle, the angle thus formed is measured by half the sum of the two intercepted arcs. 70 GEOMETRY Let -45 and CD intersect each other within the circle forming the two angles, £, and E^y with their opposite vertical and equal angles. Then we are to show, that the angle E is measured by the half sum of the arcs AC-j-BD; and the angle E^ is measured by the half sum of the arcs AD-j-CB. First, draw AF parallel to CD; then, by reason of the parallels, the angle BAF=E. But the angle BAF is measured by half of the arc FDB; that is, half of the arc BD, plus half of the arc A C; because FD=A C (th. 1 3,b. 3). Now, as the sum of the angles, E-\-F^y make two right angles, that sum is measured by half the whole circumference. But the angle Ey alone, as we have just determined, is mea- sured by half the sum of the arcs BD-\-A C; therefore, the other angle, -fi'S is measured by half of the other parts of the circum- ference, AD-^ CB, Q. E. D. THEOREM 15 When two chords intersect, or meet each other without a circle, the angle fhtcs formed is measured hy half the difference of the intercejpted arcs. Draw AF parallel to CD; then, by reason of the parallels, the angle E, made by the intersection of the two chords, is equal to the angle BAF. Biit the angle BAF is measured by half the arc BF; that is, by half the diflference between the arcs BD and A C. Q. E. D. N. B. Prolonged chords, to meet without the circle, as ED, and EB, are called secants. They are geometrical, and not trigonometrical secants. BOOK in 71 THEOREM 16. The angle formed hy a secant and a tangent, is measured ly half the difference of the intercepted arcs. Let CB be a secant, and CD a tangent. We are now to show that the angle formed at C, is measured hy half of the difference of the arcs BD and DA. From A, draw AE parallel to CD; then the angle BAE=C. But the angle BAE is measured by half of the arc BE (th. 8, b. 3); that is, by half of the diflference be- tween the arcs BD and AD; for the arc AD=DE, and BD—DE=BE; therefore the equal angle, (7, is measured by half the arc BE, Q. E. D, THEOREM 17. When two chords intersect each other in a circle, the rectangle of the segments of the one, will be egtml to the rectangle of the segments of the other. Let AB and CD be two chords intersect- ing each other in E, Then we are to show that the rectangle A EX EB= CEX ED. Join AD and CB, forming the two tri- angles A ED and CEB, which are equi- angular, and therefore similar ; for the angles B and D are equal, because they are both measured by half the arc A C. Also the angles A and C are equal, because each is measured by half the same arc, DB; and the angle AED—CEB, because they are vertical angles; hence, the triangles, AED and CEB are equiangular. But equiangular triangles have their sides, about the equal angles, proportional (th. 18, b. 2); therefore, AE and ED, about the angle E, are pro- portional to CE and EB, about the same angle. That is, . . AE : ED : : CE : EB Or (th. 21, b. 2), AEXEB=^ EDXEC. Q. E. D. 72 GEOMETRY. Scholium. When one chord is a diameter, and the other at right angles to it, the rectangle of the segments of the diameter is eqvxd to the square of half the other chord; cyr half of the bisected chcyrd is a mean proportional between the segments of the diameter. Y n >f For ADXDB==FDXDE. But if AB passes through the center, C, at right an- gles to FE, then FD=DE (th. 1, b. 3), and in the place of FD, write its equal, DF, in the last equation, and we have, ADXDB=DF' Or, . ADxDEi'.DE'.DB Put, DE=x, CD=y, and CE=^R, the radius of the circle. Then ^i>=R— y, and DB=Ji-\-g. With this notation, ADXDB, Becomes, . . (JR — g)(Ii-\-g)=x^ Or, ... . Ii'—y^=a^ Or, . . . . . B^=x^+7/^ That is, the square of the hypotenuse of the right angled triangle, DOE, is equal to the sum of the squares of the other two sides. , THEOREM 18. If from any point withmii a circle, any number of secants be dravm, the rectangle formed hy any one secant and its external segment, will be equal to the rectangle of any other secant, and its external segment. Let AB, A C, AD, EX CD=ACX CB But DEX CD=AI>XDB, by (th. 17, b. 3), which, being sub- tituted, we have, CJD'+AD X DB=A OX CB. Q. E, J), THEOREM 21. The rectangle of the two diagonals of any quadrilateral inscribed in a circle, is equal to the sum of the two rectangles of the opposite sides. Let ABCD be a quadrilateral in a circle ; then ice are to show that ACX BD=:ABX DC-\-ADxBC, From C, let CE be drawn so that the angle DCE shall be equal to angle ACB; and as the angle BAC is equal to the angle CDE, both being in the same seg- ment, therefore, the two triangles, DEC and ABC are equiangular, and we have (th. 18, b. 2), AB\AC\\DE\DC (1) The two As, ADC and BEC are equiangular; for the angle DAC=EBC, both being in the same segment, are measured by half the same arc, i>(7; and the a,ng\e DCA=zECB; for DCE =BCA; and to each of these add the angle EGA, and DCA ^ECB; therefore (th. 18, b. 2), AD: AC:: BE: BC (2) By multiplying the extremes and means in these two proportions, and adding the equations together, we have, (ABXDC)-{'{ADxBC)=={DE-{-BE)XAO But, . . . DE-\-BEz=BD; therefore, {ABXDC)-\'{ADXBC)==BDXAC Q, E, D. c 76 GEOMETRY. Scholium. When two of the adjacent sides of the quadrilateral are equal, as AB=BC, then the resulting equation is, (ABXD0)+(ABXAI>)=BDXAC Or, . . ABX(DC-{-AJ))=BDxAC Or, . . . AB',AC:'.BD:{CD-\-AD) That is, as one of the equal sides of the quadnlateral, is to the adjoining diagonal, so is the transverse diagonal to tfie sum of the two unequal sides. THEOREM 22. If two chords intersect each other in a circle, at right angles, the sum of the squares of the four segments thu^ formed, is equal to the square of the diameter of the circle. Let AB and CD be two chords, intersect- ing each other at right angles. Draw BF parallel to ED, and join BF and AF. Now we are to show that AE'-\-EB^-\-E(P-\-ED^=AF', As BF is parallel to ED, ABF is a right angle, and therefore AF is a diameter (th. 9, b. 3). Also, because BF is parallel to CD, CB=DF (th. 13, b. 3). Because CEB is a right angle, . CE''-\-EB''=z CE'=DF' Because AED is a right angle, . AE^-\-EB''=AD'' Adding these two equations, we have, CF?4rEB'''\-AE''^ED''=DF^-\-AD^ But, as AF is a diameter, and -42>^ a right angle (th. 9, b. 3), Therefore . DF^-\-AD''^AF^ Hence, . . CE^^-EB^-^AE^-^ED^^AF^ Q. E. D. Scholium. If two chords intersect each other at riofht angles, 111 a circle, and their opposite extremities be joined, the two chords thus formed may make two sides of a right angled tripngle, of which the diameter of the circle is the hypotenuse. For AD is one of these chords, and CB is the other ; and we have shown that CB=DF; and AD and DF are two sides of a BOOK III. -n right angled triangle, of which AF is the hypotenuse ; therefore, AD and OB may be considered the two sides of a right angle, and AF \\& hypotenuse. THEOREM 23. If two secants intersect each other at rigid angles, the sinn of their squares, increased by the sum of the squares of the two parts without the circle, vnll he equal to the square of the diameter of the circle. Let AE and ED be two secants intersect- ing at right angles at the point E. From B^ draw BF parallel to CD, and join AF and AD. ybw we are to show that EA'-^ED'-{-EB'+EC^=AF^ Because BF is parallel to CD, ABF is a right angle, and consequently ^^ is a diameter, and BC—DF; and because AF is a diameter, ADFis a right angle. As AED is a right angle, AE^^ED^=AD^ Also, . . EB^'\-EC^=BC^=DF^ J3 _E \ Jl> By addition, AE'^-\-ED''+EB^+EC^=^AD'-^DF^^AF\ Q. E, D. 78 GEOMETRT. BOOK IV PROBLEMS. In this section, we shall, in most instances, merely show the construction of the problem, and refer to the theorem or theorems that the student may use, to prove that the object is attained by the construction. In obscure and difficult problems, however, we shall go through the demonstration as though it were a theorem. PROBLEM I . To bisect a given finite straight line. Let AB be the given line, and from its extremities, A and jB, with any radius greater than the half of -^^ (Post. 3), de- scribe arcs, cutting each other in n and m. Join n and m; and C, where it cuts AB wilj be the middle of the line required. Proof, (th. 15, b, 1, cor. 1 ). PROBLEM 2. To bisect a given angle. Let ABC be the given angle. With any radius, from the center B, describe the arc AC. From A and G, as centers, with a radius greater than the half of AC, de- scribe arcs, intersecting in n; and join Bn, it will bisect the given angle. Proof, (th. 19, b. 1). BOOK IV. 79 PROBLEM 3. From a given poirUf in a given line, to draw a perpendicular to thai line. Let AB be the given line, and C the given point. Take n and m equal distances on opposite sides of C; and from the points m and n, as centers, with any radius greater than nO or or mCj describe arcs cutting each other in S. Join SC, and it will be the per- pendicular required. Proof, (th. 15, b. 1, cor. ) The following is another method, which is preferable, when the given point, (7, is at or near the end of the line. Take any point, 0, which is manifestly one side of the perpendicular, and join OC; and with 0(7, as a radius, describe an arc, cutting AB in m and C. Join m 0, and produce it to meet the arc, again, in n; mn is then a diameter to the circle. Join Cn^ and it will be the perpendicular required. Proof, (th. 9, b. 3). PROBLEM 4. From a given point vnthotU a line, to draw a perpendicular to thai Ime. Let AB be the given line, and the given point. From 0, draw any oblique line, as Cn, Find the middle point of Cn by (problem 1), and from that point, as a center, describe a semicircle, having Cn as a diameter. From the point m, where this semicircle cuts AB, draw Cm, and it will be the perpendicular required. Proof, (th. 9, b. 3). eo GEOMETRY. PROBLEM 5. At a ^ven point in a line, to make an angle equal to anMer given angle. Let A be the given point in the line AB, and DOE the given angle. From (7 as a center, with any radius, CE, draw the arc ED. From ^, as a center, with the radius AF= CE, describe an indefinite arc ; and from F, as a center, with FG as a radius, equal to ED, describe an arc, cutting the other arc in G, and join AG; GAF will be the angle required. Proof, (th. 5, b. 3). PROBLEM 6. From a given point, to draw a line parallel to a given line. Let A be the given point, and CB the given line. Draw AB, making an angle, ABC; and from the given point. A, in the line AB, draw the angle BAD=ABC, by the last problem. AD and CB make the same angle with AB; they are, therefore, parallel. (Definition of parallel lines). PROBLEM 7. To divide a given line into any number of equal parts. LeiAB represent the given line, and let it be required to divide it into any number of equal parts, say five. From one end of the line A, draw AD, inde- finite in both length and position. Take any convenient distance in the dividers, as Aa, and set it oflf on the line AD; thus making the parts Aa, ab, be, , draw (problem 6) DE parallel to BC; AE will be the fourth proportional required. Proof, (th. 17, b. 2). PROBLEM 10. To find the middle, or mean proportional, between two given lines. Place AB and BC in one right line, and, on AC, as a diameter, describe a semicircle (postulate 3), and from the point B, draw BJ) at right angles to AC (problem 3); BD is the mean propor- tional required. Proof, (scholium to th. 17, b. 3). 6 82 GEOMETRY PROBLEM 11. To find the center of a given circle. Draw any two chords in the given circle, as AB and CD; and from the middle point, w, of ABf draw a perpendicular to AB; and from the middle point, m, draw a per- pendicular to CD; and where these two perpendiculars intersect will be the center of the circle. Proof, (th. 1, b. 3). PROBLEM 12. To draw a tangent to a given circle, from a given point, either in i^; therefore, ^^--Z»/' Hence, . . . AF—AB=AF^DF=AD=AE Therefore, . . AE : AB : : EB : AE By taking the extremes for the means, we have, AB:AE::AE:EB Q. E. D. PROBLEM 2S. To describe an isosceles triangle, having its two equal angles double of the third angle, and the equal sides of any given length. BOOK IV. 87 Let AB be one of the equal sides of the required triangle ; and from the point A, with AB radius, strike an arc, BD. Divide the line AB into extreme and mean ratio by the last problem, and suppose C the point of division, and A C the greater segment. From the point B, with AC, the greater segment, as radius, strike another arc, cutting the arc BJ) in D. Join BJ), DC, and DA, The triangle ABD is the triangle required. DEMONSTRATION. As AC=BDt by construction ; and as AB is to ud (7, as ^(7 is to BC, by the division of AB; therefore, AB:BD: : BD : BC Now, as the terms of this proportion are the sides of the two triangles about the common angje, B, it follows, from (th. 20, b. 2), that the two triangles, ABD and BDC, are equiangular ; but the triangle ABD is isosceles ; therefore, BDC is isosceles also, and BD=DC; hut BD= AC: hence, DC=AC (ax. 1), and the tri- angle A CD is isosceles, which gives the angle CDA=A. But the exterior angle, BCD=CDA-\-A, (th. 11, b. 1). Therefore, BCD, or its equal B=CDA-]'A; or the angle B=2A. Hence, the triangle ABD has each of its angles, at the base, double of the third angle. Q. K D. Scholium. As the two angles, at the base of the triangle ABD, are equal, and each double of the angle A, it follows that the sum of the three angles is Jive times the angle A. But as the three angles of every triangle always make two right angles, or 1 80 degrees, therefore, the angle A must be one-fifth of two right angles, or 36 degrees ; and BD is a chord of 36 degrees, when AB \s b. radius to the circle ; and ten such chords would extend exactly round the circle. PROBLEM 24. Within a given circle to inscribe a triangle, equiangular to a given triangle. 88 GEOMETRY. Let ABC be the circle, and ahc the given triangle. From any point, as A, draw the tangent EAD to the given circle (problem 1 2). From the point Ay in the line ADf make the angle DAC= the angle h, (problem 6), and the angle UAB= the angle c, and join BO. The triangle ABO is inscribed in the circle; it is equiangular to the triangle ahc, and is the triangle required. Proof, (th. 12, b. 3). PROBLEM 25, To describe an equilateral and equiangvlar pentagon in a given cirde. 1st. Describe an isosceles tri- angle, abc, having each of the equal ^ angles, b and c, double of the third angle, a, by problem 23. 2d. Inscribe the triangle AB C, in the given circle, equiangular to the triangle ahc, by problem 24; then each of the angles, B and O, is double of the angle A. 3d. Bisect the angles B and O by the hues BD and OJS, (problem 3), and join AU, EB, OD, DA, and the figure AEBOD is the pentagon required. DEMONSTRATION By construction, the angles BAO, ABD, DBO, BOE, EOA, are all equal; therefore, by scholium to th. 9, b. 3, the arc BO, AD, D 0, AE, and EB, are all equal ; and if the arcs are equal the chords AE, EB, cfec, are equal. Q. E. D, PROBLEM 26. To describe an equiangvlar and equilateral polygon, of six sides, in a circle. BOOK IV. 89 Draw any diameter of the circle, as AB, «nd from one extremity, B, draw BD equal Ui BC, the radius of the circle. The arc, BD will be one-sixth part of the whole cir- cumference, and the chord BD will be a side of the regular polygon of six sides. In the A CBD, as CB=CD, and BD ^=^CB, by construction the A is equilateral, and of course equiangular. But the sum of the three angles of every A, is equal to two right angles, or to 180 degrees; and when the three angles are equal to each other, each one of them must be 60 degrees ; but 60 degrees is a sixth parth of 360 degrees, the whole number of degrees in a circle ; therefore, the arc whose chord is equal to the radius, is a sixth part of the circumference ; and a polygon of six equal sides may be inscribed in a circle, with each side equal to the radius. Car. Hence, as BD, is the chord of 60 degrees, and equal to BO or CD, we say generally, /Ao/ the chord of 60 is equal to raditis. PROBLEM 27. To find the side of a regular polygon of fifteen sides, which may he inscribed in any given circle. Let CB be the radius of the given circle, and divide it into extreme and mean ratio (problem 22), and make BD equal to CE, the greater part; then BD will be a side of a regular polygon of ten sides (scholium to problem 23). Draw BA=^ to CB, and it will be a side of a polygon of six sides. Join DAy and that line must be the side of a polygon, which cor- responds to the arc of the circle expressed by \y less y^^, of the whole circumference ; or \ — yU=_*_=J_; that is, one-fifteenth of the whole circumference ; or, DA is a side of a regular polygon of 16 sides. GEOMETRY. BOOK V. ON THE PROPORTIONALITIES AND MEASUREMENT OF POLYGONS AND CIRCLES. THEOREM 1. The area of any circle is equal to the product of its radius into half of its circumference. Let (7-4 be the radius of the circle, and AB a very small portion of its circumference, and CAB will be a sector; and we may conceive the whole circle made up of a great number of such sectors ; and each sector may be as small as we please ; and when very small, ABy BJDy _ 2 ^^3 : 1 : : 1 : DB, or DB=^~^ But the whole perimeter of the circumscribing polygon is six 2 12 _ times DB; that is, six times —=, or, -r==4 /3=:6,9282032. 73 73 BOOK V. 83 Thus w§ have shown, that when the radius of a circle is 1, the perimeter of an inscribed polygon of six sides, is . 6.000000 And of a similar circumscribed polygon, is . . 6.9282032 But, if we call the diameter 1, the perimeter of the inscribed polygon of six equal sides will be, 3.0000000 And of the circumscribed, will be • . . . 3.4641016 As we would avoid all metaphysical verbiage in science, and come to the point at once, we lay it dovm as an axiom, that when the radius of a circle is 1, and of course the diameter 2, the circumference is greater than 6, and less than 6.9282032 ; and if the diameter is 1, the circumference must be greater than 3, and less than 3.4641016 ; and this we may call the first approximation to the ratio between the diameter and circumference of a circle. Scholium 3. As the area of a circle is numerically equal to the radius multiplied by half the circumference (th. 2, b. 5), therefore, if we represent the radius by B, and half the circumference by 7t, and the area of the circle by a, then we shall have this equation : JR7t=a If we now make jB=1, this equation gives 7i=a; that is, when the radius of a circle is \, the half circumference is numerically equal to the area. We will, therefore, seek the area of a circle whose radius is unity; and that area, if found, will be numerically the half circumference, and by inspecting the last figure, we perceive that it is perfectly axiomatic (the whole is greater than a part), that the area of the sector CbAd, is greater than the triangle Cbd, and less than the triangle CBD; and the area of the whole circle is greater than one polygon, and less than the other. Find- ing the AREA of a circle y or finding a square which shall be equal to a circle of given diameter ^ is known as the celebrated problem of squaring the circle, THEOREM 3. Given, the area of a regular inscribed polygon, and the area of a similar circumscribed polygon, to find the areas of a regular inscribed and circumscribed j>olygon of double the number of sides. 94 GEOMETRY. Let C be the center of tKe circle ; AB a side of the given inscribed polygon; EF parallel to AB, a side of the circumscribed polygon. If AM be joined, and AR and J?§ be drawn as tangents, at A and B^ AM will be a side of an inscribed polygon of double the number of sides; and AR=RM (scholium 2, th. 18, b. 3), BQ=QM, and AR-{'RM=RQ= the side of the circumscribed polygon of double the number of sides. The As ARO and RMC, are equal, for AC=CM. CR is common to both triangles, and AR=RM, tangents from the same point, R; therefore, OR bisects the angle EOM, Now, as the same construction, and the same reasoning would take place at every one of the equal sectors of the circle, it is suf- ficient to consider one of them, and whatever is true of that arc, would be true of every one, and true for the whole circle, and its polygons. To avoid confusion, let p represent the area of the given inscribed polygon, and P the area of the similar circumscribed polygon. Also let p' represent the area of an inscribed polygon of double the number of sides, and P' the circumscribed polygon of double the number of sides. As the As AGD and ACM have the common vertex A, they are to each other as their bases, CD to CM ; they are also to each other as the polygons of which they form a part. Hence, . . p : p' '. : CD \ CM (1) As AD and JSM are parallel, we have, CAiCEw CD. CM (2) But, because of the common vertex, J/", the two As, CAM and CEMy are to each other as CA to CE. But the As are like parts of the polygons p' and P ; we have, Therefore, . p' \P\\CA\CE (3) That is, . . p' ',P\\CD\ CM (4) (th. 6, b. 2) By comparing (1) and (4), we have, p' \P\\p\p\ or p'=: J PXp CM: CE CD : CA or CM CD : CM P 'P' BOOK V. 95 That is, the area ofp' is a mean proportional between P and p. The two As, BMC and EEC, having the same vertex, C, are to each other as their bases, MH to HE, But, because CH bisects the angle ECM, (th. 25, b. 2) MB: BE But, . . CM: CE That is, . BMC: EBC Or, . BMC : EBC By composition, (th. 8, b. 2), 2(BMC) : (BMC+EBC) : : 2p :p+p' But 2 times BMC is P\ and (BMC+EBC) is P Therefore, . . P' : P : : 2p :p+p' Or, . . , . P-?^ Now, P' is known, because 2pP is known ; and jo-f-jo' is also known, as p' has been previously determined. Hence, by means of P and^, we can determine P' and^'. Q. E. D, Scholium. By inspecting the figure in the scholium to theorem 2, we perceive, that if we double the number of sides of the inscribed polygon, we shall more nearly fill up the circle ; and if we double the number of sides of the circumscribed polygons, we shall more nearly pare them down to the surface of the circle. Hence, by continually increasing the sides of the polygons, as indi- cated by the last theorem, we can find two polygons which shall diffier from each other by the smallest conceivable quantity; but the surface of the circle is always between the two polygons ; and thus the sur face of the circle can be determined to any assignable degree of exactness. By taking the figure in the scholium to theorem 2, b. 6, we perceive that the area of an inscribed polygon of six sides, to radius unity must be . . CaXdaX^ Which is . . fV^> because da=^ And, . . . Ca2-|-rfa2==Cd'=l Or, ... Ca=i^Z Hence, . . . i>/3XiX6=fV3=p, which corresponda with pt in the last theorem. 96 GEOMETRY. The area of the circumscribing polygon is measured by CAX^AX6=6DA=3D5. But . . . Caidb'.'.CA: DB. (th. 17, b. 2.) >_ 2 That is, . . 1 J3 : 1 : : 1 : DB, or BD=-^ >/3 6 _ Therefore, . . ZDB= /"q~^ V^* which corresponds with the last theorem. Having, now, the area of an inscribed and circumscribed polyfron of six sides, by applying the last theorem we can readily determine the area of an inscribed and a circumscribed polygon of 12 sides. Thus, . . |,'=V7P=VIV3X2^-3=3 2pP 2X|VTX2 VT 18 12 _ P'-^P 3+fV3 3-1-IV3 2+^3 ^^ Now let p' and P' be the given polygons, and find others of double the number of sides, and thus continue until the inscribed and circum scribed so nearly coincide, as to determine a very approximate area of the circle. In this manner we formed the following table : Number of sides. Inscribed polygons. Circumscribed polygons 6 12 24 48 3= 6 : 2.69807621 3.0000000 ; 3.1058286 3.1326287 2^3= 12 =3.46410161 Q ni KonnA 2+V3" 3.1596602 3.1460863 V2+V-3 96 3.1393554 3.1427106 192 3.1410328 3.1418712 384 3.1414519 3.1416616 768 3.1415568 3.1416092 1536 3.1415829 3.1415963 3072 3.1415895 3.1415929 6144 3.1415912 3.1415927 Thus we have found, that when the radius of a circle is 1, the semi- circumference must be more than 3.1415912, and less than 3.1415927 ; and this is as accurate as can be determined with the small number of BOOK V. 97 given ike length of the three perpendiculars drawn from any point within, to the three sides, to determine the sides. PROBLEM 9. In a right angled triangle, having given the base (3), ajid the difference betioeen the hypotenuse and perpendiciUar (1), to find both these two sides. PROBLEM 10. In a right angled triangle, having given the hypotenuse (5), and the dif' ference between the base and perpendicular (1), to determine both these ttoo sides, PROBLEM 11. Having given, the area or measure of the space of a rectangle inscribed in a given triangle, to determine the sides of the rectangle. PROBLEM 12. In a triangle, having given the ratio of the two sides, together with both the segments of the base, made by a perpendicular from the vertical angle, to determine the sides of the triangle. PROBLEM 13. In a triangle, having given the base, the sum of the other two sides, and the length of a liv£ drawn from the vertical angle to the middle of the base, to find the sides of the triangle. PROBLEM 14. To determine a right angled triangle; having given the lengths of two lines dravmfrom the acute angles to the middle of the opposite sides. PROBLEM 15. To determine a right angled triangle; having given the perimeter, and the radius of its inscribed circle. PROBLEM 16. ? To determine a triangle; having given the base, the perpendicular, and the ratio of the two sides. PROBLEM 17. To determine a right angled triangle; having given the hypotenuse, and the side of the inscribed square. 108 GEOMETRY. PROBLEM 18. To determine the radii of three equal circles j inscribed in a given circle, to touch each other ^ and also the circumference of the given circle, PROBLEM 19. In a right angled triangle, having given the perimeter, or sum of aU the sides, and the perpendicular Ut fall from the right angle on the hypotenuse, to determine the triangle; that is, its sides, PROBLEM 20. To determine a right angled triangle; having given the hypotenuse and the difference of two lines, drawn from the two acute angles to the center of the inscribed circle, PROBLEM 21. To determine a triangle; having given tJte base, the perpendicular, and the difference of the two other sides. PROBLEM 22. To determine a triangle; having given the base, the perpendicular, and the rectangle, or product of the two sides, PROBLEM 23. To determine a triangle; having given the lengths of three lines drawn from the three angles to the middle of the opposite sides. PROBLEM 24. In a triangle, having given all the three sides, to find the radius of the inscribed circle. PROBLEM 25. To determine a right angled triangle; having given the side of the in- scribed square, and the radius of the inscribed circle, PROBLEM 26. To determine a triangle, and the radius of the inscribed circle; having given the lengths of three lines drawn from the three angles to the center of that circle. PROBLEM 27. To determine a right angled triangle; having given the hypotenuse, and thi radius of the inscribed circle. BOOK VI. 109 BOOK YI. ON THa IHTBRSBCTION OF PLJLNXS. DEFINITIONS. The 14tli definition of book t, defines a plane. It is a superfices, having length and breadth, but no thickness. The surface of still water, the side of a sheet of paper, may give a person some idea of a plane. A curved surface is not a plane ; although we sometimes say, ** the plane of the earth's surface." 1. If any two points he taken in a plane y and a straight line join the points, every point in that line is in the plane, 2. If any point in such a line should be either above or below the surface, such a surface would not be a plane. 3. A straight line is perpendicular to a plane, when it makes right angles with every straight line which it meets in that plane. 4. Two planes are perpendicular to each other when any straight line drawn in one of the planes, perpendicular to their common section, is perpendicular to the other plane. 5. If two planes cut each other, and from any point in the line of their common section, two straight lines be drawn, at right angles to that line, one in the one plane, and the other in the other plane, the angle contained by these two lines is the angle made by the planes. 6. A straight line is parallel to a plane when it does not meet the plane, though produced ever so far. 7. Planes are parallel to each other when they do not meet, though produced to any extent. 8. A solid angle is one which is formed by the meeting, in one point, of more than two plane angles, which are not in the same plane with each other. 110 GEOMETRY. THEOREM 1. If any three straight lines meet erne another ^ they are in one plane* For conceive a plane passing through BC to revolve about that line till it pass through the point E. Then because the points E and C are in that plane, the line EC is in it ; and for the same reason, the line £JB is in it; and BC is in it, by hypothesis. Hence the Unes AB, CD^ and BC are all in one plane. Cor. Any two straight lines which meet each other, are in one plane ; and any three points whatever, are in one plane. THEOREM 2. J^ two planes cut one another y the line of their common section is a straight line. For let B and D, any two points in the line of their common section, be joined by the straight line BD ; then because the points B and D are both in the plane AU, the whole line BD is in that plane ; and for the same reason BJ) is in the plane CF. The straight line BD is therefore common to both planes ; and it is therefore the line of their common section. PROPOSITION 3. THEOREM. ^ a straight line stand at right angles to each of two other straight lines at their point of intersection, it mil be at right angles to the plane of those lines. Let AB stand at right angles to BF and CD, at their point of intersection A. T/ien AB will be at rigid angles to any other line drawn through A in the plane, passing through FF, CD, and, of course, at right angles to the plane itself. (Def. 3.) Through A, draw any line, A Q, in the plane BOOK VI. Ill JEF CDy and from any point G, draw GH parallel to AD. Take HF=zAHy and join FG and produce it to D. Because HG is parallel to AD, we have FH'.HAwFG : GD But, in this proportion, the first couplet is a ratio of equality ; therefore the last couplet is also a ratio of equality, That is, FG= GD, or the line FD is bisected in G, 3om£D,BG,a.nd£F. Now, in the triangle AFD, as the base FD is bisected in G, we have, . AF^-\-AD'=2AG'-{-2GF' (1) (th. 39 b. 1.) Also, as DF is the base of the A BDFy we have by the same theorem, . BF^^-BD^^'iBG^^^GF^ (2) By subtracting (1) from (2) and observing that BF^-—AF^ :=AB^ because ^^i^ is a right angle ; and BD^^AD^=AB\ because BAD is a right angle, and we shall then have, AB'-\-AB'=2BG^—^AG'' Dividing by 2, and transposing AG^, and we have, AB^-\-AG^=BG^ This last equation shows that BA 6^ is a right angle. But A G .s any line drawn through Ay in the plane FF, CD, therefore AB is at right angles to any line in the plane, and, of course, at right angles to the plane itself. Q. E. D, PROPOSITION I. PROBLEM AND THEOREM. To draw a straight line perpendicvlar to a plane, from a given point above it. Let J/2Vbe the plane, and A the point above it. Take, DC, any line on the plane, and draw AC ^i right angles to it. From the point C, draw CB on the plane, at right angles to the line DC. Lastly, from Ay draw AB at right an- gles to the line BCy and join BD. ABC is a right angle hy construction, and now if we can prove that ABD is also a right angle, then AB is ai right angles to the plane, hy the last proposition. 112 GEOMETRY. Because ABC is a right angle, we have, To both members of this equation, add DC^ and we have, AB'-^{BC^'\-DC^)=AC'-]-J)C^ Because BCD is a right angle, BC^-\-DC^=BD^, and because ACD'\s2i right angle, AC'^-\-DC^=AD'^, and taking these latter values in the last equation, we have, AB'-^-BD'^^AI)^ ; which shows that ABD is a right angle, and proves our proposition. Q. E. D, PROPOSITION 5. THEOREM. Two straight lines, having the same inclination to a plane, whether perpendicular or oblique, are parallel to one another. This proposition is axiomatic from our definition of parallel lines ; for a stationary plane can have but one position, and the same in- clination from any fixed position, must, of course, give parallel lines ; but, for the sake of perspicuity, we will give the following as a demonstration. Let MN be a plane, and AB and CD lines having the same inclination to it. Then AB and CD are parallel. If the lines do not meet the plane, produce them until they do meet it in B and D. Join the points B and D, by the line BD, and produce it to II, The angle CDE=iABD, otherwise the two lines would not have the same inclination to the plane. But when one line, as BE, cuts two others, as AB CD, making the exterior angle, CDE, equal to the interior and opposite angle on the same side, ABE, then the two lines, AB and (7i>, are parallel. (Converse of th. 6, b. 1). Q. E, D, k PROPOSITION 6. THEOREM. If itffo straight lines he dravm in any position through parallel planes, they vjill he cut proportumaUy hy the planes. BOOK VI 113 Conceive three planes to be parallel, as represented in the figure, and take any points, A and JB, in the first and third planes, and join ABf which passes through the second plane at U, Also, take any other two points, as C and 2), in the first and third planes, and join CDf the line passing through the second plane at F. Join the two lines by the diagonal AD, which passes through the second plane at Q. Join BD, E0-, OF, and AO, We are now to show that, AE \ EB \\ GF \ FD For the sake of perspicuity, put A 0=X, and GD=i Y, As the planes are parallel, BD is parallel EO; then, in the two triangles ABD and AEG, we have, (th. 17 b. 2). AE:EB'.:X:Y Also, as the planes are parallel, GFh parallel to AC, and we have, . . CF:FD','.X:Y By comparing the proportions, and applying theorem 6, book 2, we have, . . AE \ EB : : CF : FD. Q. E. D. PROPOSITION 7. THEOREM If a straight line be perpendicular to a plane, all planes passing ikrotcgk thai line will be perpendicular to the first-mentioned plane. Let MN'hG a plane, and AB perpen- dicular to it. Let BC he any other plane, passing through AB ; this plane will be perpendicular to MX, Let BD be the common intersection of the two planes, and from the point B, draw BE at right angles to DB. Then, as -^4^ is perpendicular to the plane MX, it is perpendic- ular to every line in that plane, passing through B (def. 3, b. 6); therefore, ABE is a right angle. But the angle ABE (def. 5, b. 6), measures the inclination of the two planes ; therefore, the plane CB is perpendicular to the ulane MX, and thus we can show 8 / ^ or THE X f UNIVERSITY I r/si icnR °' H^ A -A-jA^' lt4 GEOMETRY. that any other plane, passing through ABy will be perpendicular to MN; therefore, t which is perpendicular to TD^ and, in like manner, OC on. PF; wherefore, the point will fall on the point P, and A will be equal to DP, But the triangles A OB, BPE, are right angled at and P; the hypotenuse AB=DEy and the side A 0=^DP; hence, those triangles are equal ; hence, the an- gle OAB^PDE. The angle OAB is the inclination of the two planes ASB, ASC; the angle PDE, is that of the two planes DTEy DTF; consequently, those two inclinations are equal to each other. Hence, If two solid angles are formed y <&c. Scholium. The angles which form the solid angles at S and T, may be of such relative magnitudes, that the perpendiculars, B and EP, may not fall within the bases, ASC and DTF; but they will always either fall on the bases or on the planes of the bases produced, and will have the same relative situation to Ay S, and (7, as P has to D, Ty and P. But, in case that and P fall on the planes of the bases produced, the angles BCO and EFP, would be obtuse angles ; but the demonstration of the problem would not be varied in the least. lis GEOMETRY. BOOK VII. SOLID GEOMETRY. The object of Solid Geometry is to estimate and compare the surfaces and magnitudes of solid bodies ; and, like Plane Geometry, it must rest on definitions and axioms. To the definitions already given, we add the following, as being exclusively applicable to Solid Geometry. Surfaces are measured by square units; so solids are measured by cube units. 1. A Cube is a solid, bounded by six equal square sur- WS^BL faces, forming eight equal solid angles. iRBH All other solids are referred to a imit of this figure JbSSm for measurement. 2. A Prism is a solid, whose ends are parallel, equal, and form equiangular plane figures ; and its sides, connecting these ends, are parallelograms. 3. A prism takes particular names according to the figure of its base or ends, whether triangular, square, rectangular, pentagonal, hexagonal, j5^ and BL, is equal to AC, hence, the solidity of these two prisms is expressed by (abh); and the parallelopipedon, and two prisms together, are measured by 2abk ; and, in addition to these, we have two equal pyramids of unknovm solidity; therefore, let each one be represented by x. Now, the whole pyramid, whose base is AQ, and vertex G, is expressed by (2a5/i-|-2ar). But the pyramid, whose base is FH, and vertex G, is expressed by(x). As these two pyramids are similar, they are to each other as the cubes of their like dimensions ; that is, they are to each other as the cube of GA to the cube of GF. But GA is the double of OF, by construction. Therefore, OA^ : OF^=S : 1 Hence, .... (2abk-{-2x) : x=8 : 1 Product of extremes and means gives, 8x=2abh-{'2x Therefore, x=^(abh) This last equation shows that the solidity of any pyramid is me- third of any rectangular solid of the same base and altitude. BOOK VII. 129 Cor, This measure of the pyramid is true, whatever be the figure of its base ; and when the base is a circle, the pyramid is called a cone ; hence, the solidity of a cone is one third of its cir- cumscribing cylinder. THEOREM 12. If a pyramid he cut hy a plane parallel to Us base, the solidity of the frustum that remains after the small pyramid is taken away, is equal to three pyramids of the same altitude as the frustum; one hav- ing for its base, the hose of the frustum; another, the upper base; and the third, a base which is the mean proportional between the upper and lower bases of the frustum. (The figure has been previously described in theorem 8.) Now, by the last theorem, the solidity of the whole pyramid is P( WF^\ n( JPF1 \ expressed by — ^^ — ^, and that of the small pyramid is ^ - The difference of these magnitudes measures the frustum ; That is, . . -^ ^~^ ^=the frustum. To make this expression cor- respond with the enumeration of this theorem, we must ban- ish FE and FH, and obtain their difference. By th. 8, book 7, we have, FF:FE=JP:J~p (1) From this proportion we have, FF=^ ^— , which, substituted in the above expression, (FH)PJp p(FH) ^ , gives, . ^ i-J±L, ^\ f = the frustum ; Zjp ^ 3 ^ Or, . (^^)t^/-^^>ZL)= the frustum. From proportion (1), FE—FH : FH=JP—Jp : ^p (2) But {FE—FH) is the altitude of the frustum, which we will designate by a. Then, from proportion (2), FH=i ^^ — 9 JP-JP 130 GEOMETRY This value of FH^ substituted in the last expression for the frustum, gives, e(^^^lf#) = the frustum. By actual division, we have, \(P-\- JPl>+p)== the frustum ; Or, . iaP-\-^aJFp-^iap= the frustum. Here we find expressions for three different pyramids, which, together, are equal to the frustum ; one has P for its base, another p, and the third JPp, which is the mean proportional between the two bases, P andj9; therefore, a frustum is equal, r 1 n^ H jKi^ THEOREM 14. jTAc convex surface of a frustum of a regular pyramid, is equal to the sum of the perimeter of the two bases multiplied by half the slant hight. Conceive a regular frustum of a pyramid to exist, as represented in the figure ; then each face will be a regular trapezoid, whose surface is measured by the half sum of its parallel sides (th. 31, b. 1), multiphed by the perpen- dicular distance between them, which is the slant hight of the frustum. Let S represent a side of the lower base, and s a side of the upper base, and a the slant hight ; then the surface of one face is measured by ia (S+^). There are just as many of these surfaces as the frustum has sides. Let m represent the number of sides ; then the whole sur- face must be ^a{mS-\-7ns). But [mS-\-ms), is the perimeter of the two bases ; and ^a is one-half of the slant hight. Therefore, &c. Q. E. D. Scholium. Let circles be described round the bases of the frustum, as represented in the last figure ; and conceive the number of sides to be indefinitely increased ; then S and s will be indefi- nitely small, and m indefinitely great ; but however small S and « may be (the corresponding number to m being as much in- creased), the expression [mS-{-ms) will still represent the perime- ters of the two bases. But, when S and s are indefinitely small, while OA, and DII, that is, the distances from the axis of the frustum from its edges being constant, the perimeter, mS, wil- become the perimeter of the circle of which OA is the radius; and ms will be the perimeter of the circle of which DII is the ra- dius ; that is, mS=^n{A 0), and ms=2ft(DB); and by addition. mS-{-ms=27t{AO-^J)H) But, in this case, ^ becomes -J^^i),. one-half the edge of the frustum ; and the frustum of the pyramid becomes the frustum of ft cone, and its surface is measured by ^ADX 2rt(A 0-\-DH); hence, 132 GEOMETRY. 7^ convex surface of a frustum of a cone, is equal to half its sides, multipled hy the sum of the circumferences of its two bases. The above expression is the same as If we take the middle point, P, between and E, and draw PM parallel to OA and ED, Then, . . . =^PM, which, substituted, gives .... ADX^TtPM That is, the convex surface of the frustum of a cone, is equal to its side, multiplied hy the circumference of a circle which is exactly midway between its two bases, THEOREM 15. Jf any regular semi-polygon be revolved abotd its axis, the surface thus described, will be measured by the product of its axis into the cir- cumference of its inscribed circle. If the semi-polygon, DABK, ^. By the scholium to the preceding theorem, the surface described by AB is measured by ABX cir. GI, which is equal to AT, or EL dr. GO, That is, . ELX^TtGC=ABx^^Gl The two triangles, ABT and CGI, are similar. As CG is per- dendicular to AB, the two angles CGI and IGA, are equal to a right angle. The acute angles of the A ABT are also equal to a right angle. That is, . jCGI-\-aIGA=jBAT-^aABT But, . . . JIGA= J ABT {th,5,h,l) By subtraction, . J CGI= J BAT BOOK VII. 133 Now, as these two triangles have each a right angle, they are equiangular and similar; Therefore, . CG: 01=: AB : AT^ffL Hence, . . HL-CG^AB^QI Multiplymg both members of this equation by 2;*, we have, HL-^tt CGh=^AB'^7t 01 Thus we find that the surface described by the side ABy is mea- sured by the product of EL into the circumference of the inscribed circle ; and in the same manner we may prove that the surface described by the side AD, is measured by BE into the circum- ference of the same circle, and so on of every other side ; and the surface described by all the sides taken together, is equal to (DE'-{-EL'\-ZC, (fee), multiplied into the circumference of the inscribed circle ; that is, the surface described by the whole poly- gon, is equal to EU, the axis of the polygon, into the circumference of its inscribed circle. Q, E. E. THEOREM 16. The convex surface of a sphere is equal to the ^product of its dia- meter into its circumference. The last theorem is true, whatever be the number of sides of the polygon ; and now suppose the number to be indefinitely great ; then the sides of the polygon will coincide with the circumference of the circle, and CO becomes CA, and the surface described by the sides of the polygon, is now the surface of the sphere, which is measured by the diameter EE, multiplied into the circumference of the circle 2nCA. Q. E. E, Cor. 1 . If we represent the radius of a sphere by R, its circum- ference is 2rti2, and its diameter 2i2; therefore, its convex surface is AhB?. The surface of a plane circle, whose radius is i?, is hR^; tlierefore, the surface of a sphere is 4 times a plane circle of the same diameter. Cor. 2. The surface of a segment is equal to the circumference |Hj| of the sphere, multiplied by the thickness of the segment. Cor. 3. In the same sphere, or in equal spheres, the surfaces of different segments are to each other as their altitudes. 134 GEOMETRY THEOREM 17 The solidity of a sphere is equal to the product of tts surface into a third of its radius. Let us suppose a sphere to be composed of a great multitude of regular pyramids, whose bases are portions of the surface of the sphere, and their common vertex the center of the sphere ; then the altitudes of all such pyramids is the radius of the sphere. The solidity of one of these pyramids is its base multiplied by •J of its altitude (th. 11, b. 7); and the solidity of all of these together, will be the sum of all the bases multiplied into ^ of the common altitude. But the sum of all the bases, is the surface of the sphere ; and the common altitude is the radius of the sphere ; therefore, the solidity of a sphere is equal to its surface multiplied by one third of its radius. Q. E. D. Let R = the radius of the sphere ; then (cor. 1, th. 16, b. 7), ^TiB} is its surface ; hence, its solidity must be Cor. If r represent the radius of any other sphere, its solidity will be l^r* ; and, by dividing by the constant factors, ^n, these two solids are to each other as jR' to r*, a result corresponding to theorem 7, book 7. THEOREM 18. The solidity of a sphere is two-thirds the solidity of its circumscrib- ing cylinder. Let i2 be the radius of the base of an upright cylinder ; then, hB^ will be the area of the base (th. 1, b. 6); but the altitude of a cylinder which will just inclose a sphere, must be 2i2; and the solidity of such a cylinder must be ^nl^ (def. 18, b. 7). By the last theorem, the solidity of a sphere, whose radius is It, is |rti?. Therefore, the cylinder is to the sphere as '^hR^ to ^kR^ Or, as 2 to f Or, as 1 to f Q. E, i>. BOOK VII. 135 We ^ve another method of demonstrating this truth, merely for the beauty of the demonstration. Let AK he the diameter of a semicircle, and also the side of a parallelogram whose width is the radius of the semicircle. M^KSHHSI Join the center of the semicircle to either ex- tremity of the parallelogram, as CJB, CL. Now Qg^^^^SJ conceive the parallelogram to revolve on AK, and it will describe a cylinder; the semicircle will describe a sphere, and the triangle ABQ will describe a cone. In A Cy take any point, 2), and draw DJU par- allel to AB, and join CO. Then, as CA=AB, CD=DB. In the right angled triangle CD 0, we have, CD^-\-DO^=CO^ (1) But, . . . BD^=:DE\ and CO^^DH^ Substituting these values in equation (1), and we have, CE^'\'DO^=DH^ (2) Multiply every term of this equation by n. Then, . . TtDE ^-\-rtD O^^^nDH^ Now, the first term of this equation, is the measure of the sur- face of a plane circle, whose radius is DE; the second term is the measure of a plane circle, whose radius is D 0; and the second member is the measure of the surface of a plane circle, whose radius is DH. Let each of these surfaces be conceived to be of the same extremely minute thickness ; then the first term is a section of a cone, the second term is a corresponding section of a sphere, and these two sections are, together, equal to the corresponding section of the cylinder; and this is true for all sections parallel to CR, which compose the cone, the sphere, and the cylinder ; therefore, the cone and sphere, together, are equal to the cylinder ; but the col^ described by the triangle ABC, is \ of the cylinder described by AR (th. 11, b. 7); therefore, the corresponding section of the sphere, is the remaining tioo-thirdsy and the whole sphere is two- thirds of the whole cylinder described by the parallelogram AL. Q, E. i>. 136 ELEMENTS OF ELEMENTARY PRINCIPLES OF PLANE TRIGONOMETRY. Trigonometrt in its literal and restricted sense, has for its object, tlie measure of triangles. When the triangles are on planes, it is plane trigonometry, and when the triangles are on, or conceived to be portions of a sphere, it is spherical trigonometry. In a more enlarged sense, however, this science is the application of the prin- ciples of geometry, and numerically connects one part of a magni- tude with another, or numerically compares dififerent magnitudes. As the sides and angles of triangles are quantities of different kinds, they cannot be compared with each other ; but the relation may be discovered by means of other complete triangles, to which the triangle under investigation can be compared. Such other triangles are numerically expressed in Table II, and all of them are conceived to have one common point, the center of a circle, and as all possible angles can be formed by two straight lines drawn from the center of a circle, no angle of a triangle can exist whose measure cannot be found in the table of trigonometrical lines. The measure of an angle is the arc of a circle, intercepted be- tween the two lines which form the angle — the center of the arc always being at the point where the two lines meet. The arc is measured by degrees^ mimttes, and seconds, there being 360 degrees to the whole circle, 60 minutes in one degree, and 60 seconds in one minute. Degrees, minutes, and seconds, are desig- nated by °, ', ". Thus 27° 14' 21", is read 27 degrees, 14 min- utes, and 21 seconds. All circles contain the same number of degrees, but the greater the radii the greater is the absolute length of a degree ; the cir- cumference of a carriage wheel, the circumference of the earth, or the still greater and indefinite circumference of the heavens, have the same number of degrees ; yet the same number of degrees in each and everv circle is precisely the same angle in amount or measure. PLANE TRIGONOMETRY. 137 As triangles do not contain circles, we can not measure triangles by circular arcs ; we must measure them by other trmngles, that is, by straight lines, drawn in and about a circle, from the center. Such straight lines are called trigonometrical lines, and take par- ticular names, as described by the following DEFINITIONS. 1. The sine of an angle, or an arc, is a line drawn from one end of an arc, perpendicular to a diameter drawn through the other end. Thus, JBF is the sine of the arc AB, and also of the arc BJDK BK is the sine of the arc BD, it is also the cosine of the arc AB, and BFy is the cosine of the arc BD. N. B. The complement of an arc is what it wants of 90° ; the supplement of an arc is what it what it wants of 180°. 2. The cosine of an arc is the perpendicu- lar distance from the center of the circle to the sine of the arc, or it is the same in mag- nitude as the sine of the complement of the arc. Thus, CF, is the cosine of the arc AB; but CF=KBt the sine of BD, 3. The tangent of an arc is a line touching the circle in one extremity of the arc, continued from thence, to meet a line drawn through the center and the other extremity. Thus, AH is the tangent to the arc AB, and DL is the tangent of the arc DB, or the cotangent of the arc AB. N. B. The CO, is but a contraction of the word complcTnent. 4. The secant of an arc, is a line drawn from the center of the circle to the extremity of its tangent. Thus, CH is the secant of the arc AB, or of its supplement BDE, 6. The cosecant of an arc, is the secant of the complement. Thus, CL, the secant of BD, is the cosecant of AB. 6. The versed sine of an arc is the difference betwefm the cosine and the radius ; that is, AF is the versed sine of the arc AB, and DK is the versed sine of the arc BD. For the sake of brevity these technical terms are contracted thus : for sine AB, we write sin.AB, for cosine AB, we write cos.AB, for tangent AB, we write tan.AB, &c. D LA a ,1 Fj '■/J .4- I /— ^ 1 •138 ELEMENTSOF From the preceding definitions we deduce the following obvious consequences : 1st, That when the arc AB, becomes so small as to call it nothing, its sine tangent and versed sine are also nothing, and its secant and cosine are each equal to radius. 2d, The sine and versed sine of a quadrant are each equal to thf radius ; its cosine is zero, and its secant and tangent are infinite. 3d, The chord of an arc is twice the sine of half the arc. Tims the chord BG, is double of the sine BF. 4th, The sine and cosine of any arc form the two sides of a right angled triangle, which has a radius for its hypotenuse. Thus, CF, and FBy are the two sides of the right angled triangle CFB, Also, the radius and the tangent always form the two sides of a right angled triangle which has the secant of the arc for its hypo- tenuse. This we observe from the right angled triangle CAR. To express these relations analytically, we write sin.2+cos.2=i22 (1) i^'-ftan.'rizsec.' (2) From the two equiangular triangles CFB, CAR, we have CF:FB=CA:AR That is, . cos. : sin.=i? : tan. tan.= ' (S) cos. ^ ' Also, . CF:CB=CA:CR That is, . cos : i?=i2 : sec. cos. sec.=i2* (4) The two equiangular triangles CAR, CDL. give CA'.AR=DL:DC That is, . i? : tan.=cot : i2 tan. cot.=i2^ (5) Also, . CF'.FB=DL:DC That is, . COS. : sin.=cot : i2 cos. i?=sin. cot. (6) By observing (4) and (5), we find that cos. sec.=tan. cot. (7) Or, . COS. :tan.=cot. :sec. The ratios between the various trigonometrical lines are always the same for the same arc, whatever be the length of the radius ; and therefore, we may assume radius of any length to suit our conven- ience ; and the preceding equations will be more concise, and more PLANE TRIGONOMETRY. 139- readily applied, by making radius equal unity. This supposition being made, the preceding becomes sin*^+cos.^=l (1) l+tan.2=sec.^ (2) taii.==?l^ (3) cos.= — (4) cos. ^ ' sec. ^ ' tan.= — - (6) cos. = sin. cot. (6) The center of the circle is considered the absolute zero point, and the different directions from this point are designated by the different signs + and — . On the right of C7, toward A, is commonly marked plus (+), then the other direction, toward E, is necessarily minus ( — ). Above AE\& called (+), below that line ( — ). If we conceive an arc to commence at Ay and increase contin- uously around the whole circle in the direction of ABD, then the following table will show the mutations of the signs. , sin. COS. tan. cot sec. cosec. vers. 1st quadrant. -{- + + 4" 4* -|- "h 2d" -I-—. — — — + -I- ,3d" ^ — + + — — + 4tJi « _ + _ — + — +/ PROPOSITION 1. The chiyrd of 60° and the tangent 45° are each equal to radius; the sine of 30° the versed sine of 60° and the cosine of 60° are each equal to half the radius. (The first truth is proved in problem 15, book 1). On C=, as radius, describe a quadrant ; take -4i>=45°, AB =60°, and ^^=90°, then i?j5'=30°. Join ABy CB, and draw Bn, perpendicular to CA, Draw Bm, parallel to AC. Make the angle C^^=90°, and draw CBH. In the A ABC, the angle A CB =60° by hypothesis ; therefore, the sum of the other two angles is (180— -60) =120°. But CB= CA, hence the angle CBA= the angle CAB, (th. 1 5 b. 1 ) , and as the sum of the two is 120°, each one must be 60°; therefore, each of the angles of triangle ABC, is 60° 140 ELEMENTS OF and the sides opposite to equal angles are equal • that is, AB, the chord of 60°, is equal to CA, the radius. In the A CAH, the angle CAH'is a right angle ; and by hypoth- esis, A CH. is half a right angle ; therefore, AHC, is also half a right angle ; consequently, AH=ACy the tangent of 45°= the radius. By th. 15, book 1, cor. Cn=^nA; that is, the cosine and versed sine of 60° are each equal to the half of the radius. As Bn and £!C are perpendicular to ^(7, they are parallel, and Bm is made parallel to Cn; therefore, Bm==Cn, or the sine 30°, is the half of radius. PROPOSITION 2. GHven the sine and cosine of two arcs to find the sine and cosine of the sum, and difference of the same arcs expressed hy the sines and co- sines of the separate arcs. Let O be the center of the circle, CD, the greater arc which we shall designate by a, and DFy a less arc, that we designate by h. Then by the definitions of sines and co- sines, 2>0=sin.a; 00=cos.a; FI=sm.b; GI=:cos.b. We are to find FM, which is =sm.(a-\-b); GM=cos.(a-{-b); I!P=sm.(a--b); OF=cos.(a—b). Because /iV is parallel to DO, the two As GI) 0, GIN", are equiangular and similar. Also, the A FBI, is similar to GIN; for the angle FIQ, is a right angle ; so is HIN; and, from these two equals take away the common angle HIL, leaving the angle FIH= GIN. The angles at H and N, are right angles ; therefore, the A FHI, is equiangular, and similar to the A GIN, and, of course, to the A GD 0; and the side HI, is homologous to IN, and D 0. Again, as FI=IE, and IK, parallel to FM, FH^IK, and HI=KE. By similar triangles we have GD'.DO=GI:IN, T> ' I T\T TUT sin.a cos .J jB : sm.a=cos.6 : iiv, or IN= That is. Also, B GD:GO=FI:FB PLANE TRIGONOMETRY. 141 That is. Also, That is. Also, That is. B:co3.a=sia.b:FB', or Fir= OD'.ao^Gi'.GJsr R : cos.a=cos.6 : (?i\^, or (?ir= OD:B0=FI'.IH It : sin.a=sin.5 : IE, or IH= cos.a sin.6 R cos.a cos. J M sin.a sin.J It Bj adding the first and second of these equations, we have m-{'FJI=FM=sm.(a+h) r^,^ , . • / , ,x sin.tt cos.J+cos.a sin.5 That IS, . sm. (a+b)— ~ — By subtracting the second from the first, we have , _. sin.a C0S.5— cos.a sin.J sm. (a— 5)= ^ By subtracting the fourth from the third, we have 02^— IR=OM= COS. (a-^h) for the first member, cos.a cos.5 — sin.a sin.b Hence, cos.(a+^)= It By adding the third and fourth, we have GK+IH= QX-{-NP= GP=:cos.{a—b) Hence, COS. (a — b) _cos.a COS. 5-4-siii-) :-..(^^) COS.^ COS sin.-4 — sin.^ cos.^-f-cos.-S \ 2 / sin.^ — sin.5 / A-^B \ cos.-B— cos.^ \ 2 / cos.-4+cos.-B cos.jB — C0S.-4' tan A—B r-^i (20) (21) (22) (23) (24) PLANE TRIGONOMETRY 143 These equations are all true, whatever be the value of the arcs designated by A and B; we may therefore, assign any possible value to either of them, and if in equations (20), (21) and (24), we make B=^ 0, we shall have, sin.^ ,-4 1 ,^^. =tan.-= r, (25) 1+C0S..4 2 cot.^-4 sin.-4 A 1 .^. .j=coV=— — . (26) 1— C0S.-4 2 tan.^-4 l-i-cos.-4 cot.-J^ 1 1 — cos.J^ tan.^A tan.^^. If we now turn back to formula (-4), and divide equation (7) by ), and (8) b; we shall have, l4-cos.^ ^cot4^_ 1^ _ 1— cos.^ tan4-4 tan.^^^ ^ ^ sin (9), and (8) by (10), observing at the same time, that — ^=tan, cos* , , ,. sin a cos.J+cos.a sin.6 tan.(a+ J) = t — ^ —, ^ ^ cos.a cos.o — sm.a sm.o , ,. sin.a cos.i — cos.a sin.5 tan.(a — 6)= rr-- HE ^ ^ cos.a cos.o-f-sm.a sm.o By dividing the numerators and denominators of the second members of these equations by (cos.a cos. 5), we find, sin.a cos.5 cos.a sin.5 , I T\ cos.a cos.6 cos.a C0S.6 tan.a+tan.J tan.(a+5)= j-—. ^~.=z — r ; — i. (28) ^ ^ cos.a cos.o sm.a sm.6 1 — tan.a tan.o ^ ' cos.a cos,b cos.a cos.6 sin.a C0S.5 cos.a sin.5 , ,. cos.a C0S.5 cos.a cos .5 tan.a — tan.J , ^. tan.(a— ^)= r — r— v- 7=r-r. ; — r (29) ^ - cos.a cos.o sm.a sm.o 1 -[-tan.a tan.o ^ ^ cos.a COS. 6 cos.a cos. 6 If in equation (11), formula (jB), we make a=b, we shall have, sin.2a=2sin.a cos.a (30) Making the same hypothesis in equation (13), gives, cos.2a4-l=2cos^a (31) The same hypothesis reduces equation (14), to 1 — cos.2a=2sin^a (32) The same hypothesis reduces equation (28), to 2tan.a ,„„. 144 ELEMENTS OF If we substitute a for 2a in (31) and (32), we shall have l-|-cos.a=2 cos.'^a. (34) and 1 — cos.a— 2 sin.^^a. (35) Recurring again to formula {B), we have, by transposing sin.(a+i)=2sin.a cos.i — sin.(a — b) sin.(a-|-i)=2cos.a sin.54-sin.(a — b) If, in the first of these expressions, we make a =30°, 2sin.a will equal radius, or unity; and if in the second we make «=60°, 2cos.a will also equal unity; these expressions then become, sin.(30°-|-5)=cos.5— sin.(30°— 6) (36) And . . sin.(60°+i)=sin.5-fsin.(60— °i) (37) The sines may be easily continued to 60°, by equation (36), when the sines and cosines of all arcs below 30° have been com- puted; then, by equation (37), the sines can be readily run up to 90°. The foregoing equations might have been obtained geometrically, but not so easily and concisely. ON THE CONSTRUCTION OF TABLES OF SINES, TANGENTS, &c. To explain this, we refer at once to Table II, which contains loga- rithmic sines, and tangents, and also natural sines and cosines. The natural sines are made to the radius of unity; and, of course, any par- ticular sine is a decimal fraction, expressed by natural numbers. The logarithm of any natural sine, with its index increased by 1 0, will give the logarithmic sine. Thus, the natural sine of 3° is .052336 The logarithm of this decimal is . . . —2.718800 To which add 10. The logarithmic sine of 3° is, therefore, . . 8.718800 In this manner we may find the logarithmic sine of any other arc, when we have the natural sine of the same arc. If the natural sines and logarithmic sines were on the same radius, the logarithm of the natural sine would be the logarithmic sine, at once, without any increase of the index. The radius for the logarithmic sines, is arbitrarily taken so large that the index of its logarithm is 10. It might have been more or less ; but, by common consent, it is settled at this value ; so that the sines of the smallest arcs ever used shall not have a negative index. In our preceding equations, sin.a, cos.a, &c., referred to natural sines; and by such equations we determine their values in natural num- bers ; and these numbers are put in the table, as seen in table 2, under the heads of not. sine, and nat. cosine. PLANE TRIGONOMETRY. 145 To commence computation, we must know the sine or cosine of some known arc ; and we do know the sine and cosine of 30°. The sine of 30° is ^ (prop. 1, trig.), and, hence, cos'. 30°=1 — i (eq. (1) trig.); or, COS. 30°=^<,y3. Now put 2a=30°, and equation (30) gives 2sin.l5o cos.l6°=0.6. (n) Eq. (1) gives . . cos.2i6°-f sin.2i5=l. (n) By adding (w) to (w), and extracting square root, we obtain, cos.l6°-f- sin.l5o=Vl-^=l-22474487. (p) By subtracting (m) from <7i), and extracting square root, C0S.160— sin.l5°==V0.6=0.707l0678 (q) Sub. (q) from (p) gives 2sin.l5°=0.5 1763709. Again, put 2a=15°, and in like manner apply equations (30) and (1), and we can have the sine and cosine of 7° 30', and thus we may bisect as many times as we please, but when we get down to any arc under 1', we can compute the sines by direct proportion. Also, by theorems 3 and 4, book 6, the semicircumference of a circle whose radius is unity, is 3.14169265; this, divided by 10800, the num- ber of minutes in 180°, will give .0002908882 for the length of the sine or arc of one minute. The logarithm of this number, with its index increased by 10, gives 6.463726, the log. sign of 1', which is found in the table. Having the sine and cosine of I', we can find the sine and cosine of 2' by equation (30); That is, . . sin.2a=2 sin.a cos.a Or, . . . sin.2'=2 sin.l'cos.l' For the sine of 3', and every succeeding minute, we apply equation (11), making a=2', and b=l'; That is, . . sin.3'=2 sin.2' cos.l — sin.l' Having the sine of 3', we obtain the sine of 4' by the application of the same equation ; that is, by making a=Z\ and b=l; Then, . . . sin.4'=2 sin. 3' cos.l — sin.2' sin.5'=2 sin. 4' cos.l — sin. 3' &c., &c. When the sine of any arc is known, its cosine is readily determined by the following formula, which is, in substance, equation (1), trigonometry. . . cos.=^(l-{-sin.)(l — sin.) When the sine and cosine of any arc are known, the sine and cosine of its double, are found from equation (30); and thus, from equations (30), (11), and (1), the sines and cosines of all arcs can be determined. When the sine and cosine of an archavebeen determined through a series of operations, the accuracy of the results should be tested by 10 146 ELEMENTS or equation (12) or (14), or by some other equation independent of former operations; and if the two results agree, they may be regarded as accurate. One independent method will be found by applying theorem 5, book 6. In that theorem we find the chord of 20° is .347296 ; the natural sine, then, of 10°, is .173648. Taken, the chord of 20°, and trisecting the arc by the same problem, we find the chord of 6° 40' to be .11628; and, of course, the natural sine of 3° 20' is .05814; and thus, by successive trisections we can obtain the sines, and of course the cosines of certain arcs ; and when we arrive at very small arcs, we can com- pute their increase or decrease by direct proportion.* Now, if the sine of an arc computed through successive trisections, agrees with the sine of the same arc computed through successive bisections, we must, of course, regard the result as accurate. When we have the sines and cosines of an arc, the tangent and co- tangent are found by (3) tan.= — ^^* (6) cot.= ^-; and the cos. sin. jfJ2 secant is found by equation (4) ; that is, sec.= COS. For example, the logarithmic sine of 6°, is 9.019235, and its cosine 9.997614. From these it is required to find the tangent, cotangent, and secant. JRsin. . . 19.019236 Cos. . subtract 9.997614 Tan. is jRcos. Sin. . Cotan. is JRMs Cos. . Secant is 9.021621 19.997614 subtract 9.01923^ 10.978379 20.000000 subtract 9.99767 4 10.002326 • Thus, from theorem 4, book 5, we find the chord of 28' 7" 30"' to be .008181208 ; and wishing to take away 7" 30'", we do it by proportion, as follows. The sine of 1' or 60" is .0002908882. Therefore, . 60 : 7J=.0002908882 Or, . . .8:1 =.0002908882 : .000036461 The chord of 28' 7" 30'" is . .008181208 of 7" 30'" is . .000036461 of SS' is , , .008144747 The natural sine of 14' is . .004072373 Now we may halve or doubU thli 8in« by equation (30). PLANE TRIGONOMETRY 147 The secants and cosecants of arcs are not given in our table, because they are very little used in practice ; and if any particular secant is required, it can be determined by subtracting the cosine from 20 ; and the cosecant can be found by subtracting the sine from 20. PROPOSITION 3. In any right angled plane triangle^ we may have the follomng proportions : 1st. As the hypoteniLse is to either side, so is the radius to the sine of the angle opposite to that side. 2d. As one side is to the other side, so is the radius to the tangent of the angle adjacent to the first-mentioned side. 3d. As one side is to the hypotenuse, so is radium to the secant of the angle adjacent to that side. Let CAB represent any right angled triangle, right angled at A. AB and AC are called the sides of the A, and CB is called the hypotenuse. (Here, and in all cases hereafter, we shall represent the angles of a triangle by the large letters A, B, C, and the sides opposite to them, by the small letters a, h, c.) From either acute angle, as C, take any distance, as CD, greater or less than CB, and describe the arc DE. This arc measures the angle C. From D, draw DF parallel to BA; and from E, draw EO, also parallel to BA or DF. By the definitions of sines, tangents, and secants, DF is the sine of the angle C; EQ- is the tangent, CQ the secant, and CF the cosine. Now, by proportional triangles we have, CB : BA=CD : DF or, a : c=B : sin. (7) CA :.AB=CE : EG or, b : c=E : tan. (7 V Q. E. D. CA : CB=^CE : CG or, h : az=zR : sec.Cj Scholium. If the hypotenuse of a triangle is made radius, one side is the sine of the angle opposite to it, and the other side is the cosine of the same angle. This is obvious from the triangle CDF. 148 ELEMENTS OF PROPOSITION 4. In any triangle f the sines of the angles are to erne another as the sides opposite to them. Let ABC be any tri- angle. From the points A and B, as centers, with any radius, describe the arcs measuring these an- gles, and drawjt?a, CD, and m», perpendicular to AB, Then, . . pa=^m.A, mn=Bm.B By the similar As, Apa and A CD, we have, B : sin.^=5 : CD; or, R{CD)=h Bm,A (1) By the sinular As Bmn and BCD, we have, R : sin.^=a : CD; or, R{CD)=asm.B (2) By equating the second members of equations (1) and (2). h sin.-4=a sin.^. . Hence, , sin.-4 : sin.-B=a :h \ O E D Or, . . o : 5=sin A : sin. B) Scholium 1. When either angle is 90°, its sine is radius. Scholium 2. When CB is less than A C, and the angle B^ acute, the triangle is represented by A CB, When the angle B becomes B' , it is obtuse, and the triangle is A CB' ; but the proportion is equally true with either triangle ; for the angle CB'D= CBA, and the sine of CB'D is the same as the sine of AB'C. In prac- tice we can determine which of these triangles is proposed by the side AB, being greater or less than A C; or, by the angle at the vertex C, being large as A CB, or small as A CB', In the solitary case in which A C, CB, and the angle A, are given, and CB less than A C, we can determine both of the As A CB and A CB'; and then we surely have the right one. PROPOSITION 5. J^ from any angle of a triangle, a perpendicular he let fall on the opposite side, or hose, the tangents of the segments of the angle are to one another as the segments of the base. PLANE TRIGONOMETRY. 149 Let ABC be the triangle. Let fall the perpendicular CD, on the side AB, Take any radius, as Criy and describe the arc which measures the angle C. From w, draw qnp parallel to AB. Then it is obvious that np is the tangent of the angle D CBy and nq is the tangent of the angle A CD. Now, by reason of the parallels AB and qp, we have, qn : np=^AD : DB That is, \An.ACD : ian.DCBz=AD : DB Q. E, D. PROPOSITION 6. If a perpendicular he let fall from any angle of a triangle to its op- *^site side or base, this base is to the sum of the other two sides, as the difference of the sides is to the difference of the segments of the base. (See figure to proposition 5.) Let AB be the base, and from (7, as a center, with the shorter side as radius, describe the circle, cutting AB in 0,AC in F, and produce AG to £J. It is obvious that AB is the sum of the sides A C and CB, and AF is their difference. Also, AD is one segment of the base made by the perpendicular, and BD=DG is the other; therefore, the difference of the seg- ments is A G. As .^ is a point without a circle, by theorem 18, book 3, we have, AFXAF=ABXAG Hence, . . AB : AF=AF : AG Q. K D. PROPOSITION 7. The sum of any two sides of a triangle, is to their difference, as the tangent of the half sum of the angles opposite to these sides, to the tangent of half their difference. Let ABC be any plane triangle. Then, by proposition 4, trigonometry, we have, CB :^(7=sin. A : sin.B Hence, CB-^-AC: CB—AC=sm.A+sm.B : sin.-4--sin.^ (th. 9 b. 2) 150 ELEMENTS OF But, tan. I — ~ — J : tan. I — - — J =sin.-4+sm.j5 : sin.^ — sin.B (eq. (19), trig.) Comparing the two latter proportions (th. 6, b. 2), we have, Ci5+^(7 : (7.5-^a=tan.(:^^):tan.(^-] Q. K D, PROPOSITION 8. Given the three sides of any plane triangle, to find some relation which tliey rmist hear to the sines and cosines of the respective angles. Let-4J5C7bethe triangle, and let the perpendicular fall either upon, or without the base, as shown in the figures ; and by recurring to theorem 38, book 1, we shall find C a jFJ X JJ CD: 2a (0 Kow, by proposition 3, trigonometry, we have, JB : COS. (7=6 : CD h COS. C Therefore, CD: R (2) ^ R{a''-\-b'—c^) COS. C7=— i — -— ^ 2ab Equating these two values of CDy and reducing, we have. In this expression we observe that the part of the numerator which has the minus sign, is the side opposite to the angle ; and that the denominator is twice the rectangle of the sides adjacent to the angle. From these observations we at once draw the fol- lowing expressions for the cosine -4, and cosine B. Thus, C0S.-4 cos 2bc 2ac (n) (P) PLANE TRIGONOMETRY. 151 As these expressions are not convenient for logarithmic compu- tation, we modify them as follows : If we put 2a=A, in equation (31), we hare, C0S.-4+ 1=2 cos.^ ^A In the preceding expression (»), if we consider radius, imitjr, and add 1 to both members, we shall have. cos.^+l=l+ 2bc Therefore, 2 cos.' |^r=?^f±^t^-Z:i 2bc Considering (b+c ) as one quantity, and observing that we have the diflference of two squares, therefore (J+c)2— a2=(6+c+a)(5+c-- a); but (6+c— a)=5-f c+a— 2a Hence, . 2 cos.' i^=(^+^lg±f±^=5L) Or, , . cos/ ^-4= — — By putting — - — =s, and extracting square root, the final result for radius unity, is 1 A Ks—a) eos.i^=^-A_^-^ For any other radius we must write, , . IRh(s—^ cos.J^=^-^ « . i. , T> lRh(s—h) By mference, cos4^= -yj ^ ^ Also, . . COS. iC=yJ -T-—^ In every triangle, the sum of the three angles must equal 180°; and if one of the angles is small, the other two must be comparatively large; if two of them are small, the third one must be large. The greater angle is always opposite the greater side ; hence, by merely inspecting the given sides, any person can decide at once which is the greater angle ; and of the three preceding equations, that one should be taken which applies to the greater angle, whether that be the par- ticular angle required or not; because the equations bring out the 152 ELEMENTS OF cosines to the angles ; and the cosines, to very small arcs vary so slowly, that it may be impossible to decide, with sufficient numerical accuracy to what particular arc the cosine belongs. For instance, the cosine 9.999999, carried to the table, applies to several arcs ; and, of course, we should not know which one to take ; but this difficulty does not exist when the angle is large ; therefore, compute the largest angle first, and then compute the other angles by proposition 4. But we can deduce an expression for the sine of any of the angles, as well as the cosine. It is done as follows : EQUATIONS FOR THE SINES OF THE ANGLES. Resuming equation (m), and considering radius, unity, we have, COS. C= — -—z 2aJb Subtracting each member of this equation from 1, gives '— ^=-(^-) (0 Making 2a=(7, in equation (32), then a=^0. And . . 1— cos.(7=2 sin.2^(7 (2) Equating the right hand members of (1) and (2), 2 sin.^^(7= 2al>—a*-^^-\-c^ 2ab 2ab (c-\-h — a)(c'{-a — b) 2ab / c-\-b — a \ / c-{-a — 'b \ . ,, ^ ^ 2 I I 2 / Or, . . . sm.>i(7= ^ But, . =— a and =_X_J j 2 2 2 2 Put . — - — =8, as before ; then. By taking equation (p), and operating in the same manner, we have . . . sm.i£=Jl^H^ ^ ^ ac From (n) . . 6m.iA=J ^'~^^['~^^ N CO sm.iC=yJ'- PLANE TRIGONOMETRY. 153 The preceding results are for radius unity; for any other radius, we must multiply by the number of units in such radius. For the radius of the tables, we write B; and if we put it under the radical sign, we must write H^; hence, for the sines corres- ponding with our logarithmic table, we must write the equations thus, . . . sin4^=^M^35^ 3in4i?=^/5S3^ r_ IR\s- a){s-b) ah A large angle should not be determined by these equations, for the same reason that a small angle should not be determined from an equation expressing the cosine. In practice, the equations for cosme are more generally used, because more easily applied. In the preceding pages we have gone over the whole ground of theoretical plane trigonometry, although several particulars might have been enlarged upon, and more equations in relation to the combinations of the trigonometrical lines, might have been given ; but enough has been given to solve every possible case that can arise in the practical application of the science ; but to show more clearly the beauty and spirit of this science, and to redeem a promise, we give the following geometrical dem(mstraiions of the truths expressed in some of the preceding equations. From C as the center, with CA as the radius, describe a circle. Take any arc, AB^ and call it A; AD a less arc, and call it B; then BD is the difference of the two arcs, and must be designated by (A—'B);AG=AB; therefore, DG=A-\-B; UG=sm,A; (See fig. p. 154.) Un=sm.B; Gn=sin.A-\-sin.B; Bn=sm.A — sin.^. Fm=mD=^CHz=cos.B; mn=cos.A; Therefore, Fm-{-mn=cos.A-\-cos.B=Fn; mD — mn=cos.B — cos.^=wi>/ Because . NF^AD; AB'\'NF^A-\-B; Therefore, . . 180°--(^-i-J5)=arc i^^; 154 ELEMENTSOF Or, . . . 90°— (^-)=JarcJ?'5; But the chord FB, is twice the sine of -J arc FB. That is, FB=^2sm, ( 90°— ^^^^ ] =2cos. ( ^^ \ The angle nGD=BFD, because both are measured by one half of the arc BD; that is, by [ — — - ] and the two triangles OnD, and FnB are similar. The angle GFuy is measured by In the triangle FBOy Fn is drawn from an angle perpendicular to the opposite side ; therefore, by Proposition 5, we have, Gn : n^=tan. GFn : i2LW.BFn That is, sin.-4+sin.-B : sin.-4 — sin.5=tan. ( ■ ] : tan. ( — — - ] This is equation (19). In the triangle GnDy we have sm.90*»:i)6^sin.ni>(7: Gn; sm.nI)G=co8.nGD That is, 1 : 2sin. ( — ^ ] =cos. ( — "^ ] ; sin.^+sin.^ Or, . sin.-44-sin.5=2sin. ( —- — ) cos. I -~— ) same as equation (16). In the triangle FnB, we have, sin.90 : FB=sm.BFn : Bn That is, 1 : 2cos. f — -— j =sin. [ -~ ) : sin.^—sin. J5 Or, . . sin.-4 — sm.^=2cos. [ — ^- ) sin. ( -~- ) same as equation (16). In the triangle FBn, we have, sin.90 : FB=cos.BFn : Fn That is, 1 : 2cos. / — g— ) =cos. ( -^- ):co&.A-\'Coa,B PLANE TRIGONOMETRY. 155 Or, cos.^4-cos.i?=2cos. ( T" ) cos. [ — - — J same as equa- tion (17). In the triangle OnD^ we have, sin.90° : GD=sm.nGD : nD That is, . 1 : 2sin. [ — - — j =sin. j — - — J : cos.^ — cos.^, same as equation (18). In the triangle FGn, we have, sin. OFn : 6^= cos. OFn : Fn A-^-B A-\-B That is, sin. — -— : sin.^+sin.jB=cos. — - — :cos.^4-cos.5 Or, (sin.-44-sin.^)cos. I — - — J =(cos.^+cos.^)sin. [ — — -- j A+B • ^ I • D sm Or. sm - sin.^H-sin.5_ ^ _. ( A-\-B \ ^.^-fcos.j5~ 3T^~ I ~2~" j COS. — - — 2 same as equation (20). We give a few more geometrical demonstrations from the follow- ing figure : Let the arc AD=A; then i>6^=sin.^; CO=cos.A; I>I=sm.^A; ^i>=2sin.^^; CI=cos.^A; CI=D 0; DB=2D (9=2cos.i^. The angle DBA, is measured by half AD; that is, by ^A. Also, . ADG==J)BA==iA. Now in the triangle BDO, we have, sm.I)BG:I)O=sm.90° :BD That is, sin.-J^^ : sin.^=l : 2cos.-J-^ Or, . sin.^=2sin.i[--4cos.^^ same as equation (30). In the same triangle sin.90° : BD=sm.BDG:BG; sm.BDG=cos.DBG; That is, . 1 : 2cos.^^=cos.^^ : l+cos.^ Or, . 2cos^^-4=sl-fco8.-4, same as equation (34). 156 KLEMENTSOF In the triangle D QA, we have, Bin.90° :^i>=sin.6?i>^ : OA That is, . 1 : 2sin4-4=6in4^ : 1 — cos.^ Or, . 2sin.^^^=l — cos.^, same as equation (35). By similar triangles, we have, BA\AD=zAD\AQ That is, . 2 : 2sin.^^=2sin.^-4 : versed sin.-4 Or, • versed sin.-4=2sin.^^^. APPLICATION OF THE PRINCIPLES OF TRIGONOMETRY. Every triangle consists of six parts; three sides, and three angles ; and to determine all the parts, three of them must be given, and at least (ym of these parts mtist be a side, because two triangles may have equal angles, and their sides be very different in respect to magnitude In right angled plane triangles, the right angle is always given ; and if two other parts, and one a side, be given, it will be sufficient for the complete determination of all the other parts. Before the invention of logarithms, the numerical computations for the parts of a triangle were all made by arithmetical proportion, as in the rule of three, through the help of natural sines and cosines ; but the operations, in many cases, were extremely laborious. For mere curiosity, we will use natural sines to solve the following triangle. Given, the hypotenuse of a right angled triangle, 840.4 feet, and one of the oblique angles, 38° 16', to find the other parts. The two oblique angles, together, make 90° (th. 11, b. 1, cor. 4); therefore, the other angle is 61° 44'. sin. 38° 16 As 1: 38° 16'=AC : CB But the natural sine of 38°, 16' is .61932 and AC=840.4. Therefore, 1 : .61932=840.4 : CB 840.4 247728 247728 49 5456 CJ?=520.476628 PLANE TRIGONOMETRY. 157 For the side AB, we have the following proportion : 1 :cos.38o 16'=AC :AB That is, . . 1 : ,78613=840.4 : AB 8404 314052 314052 628104 A5=659.823262 Before we go into logarithmic computation, it is important to say a word or two in relation to the nature of logarithms. Logarithms are exponential numbers ; and Algebra teaches us, that the addition of the exponents of like quantities multiplies the quantities, and the subtraction of the exponents divides the quantities. Hence, by logarithms, we perform miUtiplication by addition, and division hy subtraction. EXPLANATION OF THE TABLES. For the computation of logarithms, we refer at once to Algebra; here we shall point out the manner of finding them in the tables, and some of their uses. The logarithm of 1, is 0; of 10, is 1.00000; of 100, is 2.00000, &c. Hence, the logarithm of any number between 1 and 10, must be a decimal; between 10 and 100, must be 1 and a decimal; between 100 and 1000, must be 2 and a decimal. The whole number belonging to a logarithm, is called its index. The index is never put in the tables (except from 1 to 100, and need not be put there), because we always know what it is. It is always one less than the number of digits in the whole number. Thus, the number 3754 has 3 for the index to its logarithm, because the number consists of 4 digits ; that is, the logarithm is 3, and some decimal. The number 347.921 has 2 for the index of its logarithm, because the number is between 347 and 348, and 2 is the index for the loga- rithms of all numbers over 100, and less than 1000. All numbers consisting of the same figures, whether integral, frac- tional, or mixed, have logarithms consisting of the same decimal part. The logarithms would differ only in their indices. Thus, . the number 7956. has 3.900695 for its log. the number 795.6 has 2.900695 " the number 79.56 has 1.900695 " the number 7.956 has 0.900695 " the number .7956 has —1.900695 « the number .07966 has —2.900695 ** 158 ELEMENTS OF Prom this we perceive that we must take the logarithm out of the table for a mixed number or a decimal, the same as if the figures expressed an entire number ; and then, to prefix the index, we must consider the value of the number. The decimal part of a logarithm is always positive ; but the index becomes negative when the number is a decimal ; and the smaller the decimal, the greater the negative index. To prefix the index to a decimal, count the decimal point as 1, and every cipher as 1, up to the first significant figure, and this is the negative index. For example, find the logarithm of the decimal .0000831. Num. 0000831 log. —6.919601 The point is counted one, and each of the ciphers is counted one ; therefore the index is minus Jive. The smaller the decimal, the greater the negative index ; and when the decimal becomes 0, the logarithm is negatively infinite. Hence, the logarithmic sine of 0° is negatively infinite, however great the radius. The logarithm of any number consisting of four figures, or less, is taken out of the table directly, and without the least difficulty. Thus, to find the logarithm of the number 3725, we find 372, at the side of the table, and run down the column marked 6 at the top, and we find opposite the former, and under the latter, ,571126, for the deci- mal part of the logarithm. Hence, the logarithm of 3725 is 3.571126 the logarithm of 37250 is 4.571126 the logarithm of 37.25 is 1.571126, &c. Find the logarithm of the number 834785. This number is so large that we cannot find it in the table, but we can find the numbers 8347 and 8348. The logarithms of these num- bers are the same as the logarithms of the numbers 834700 and 834800, except the indices. 834700 log. 6.921530 834800 log. 5.921582 Diflference, . . 100 52 Now, our proposed number, 834785, is between the two preceding numbers ; and, of course, its logarithm lies between the two preceding logarithms ; and, without further comment, we may proportion to it thus, . . . 100 : 85=52 : 44.2 Or, . . . 1. : .86=62 : 44.2 PLANE TRIGONOMETRY. 159 To the logarithm . . . 6.921630 Add 44 Hence, the logarithm of 834785 is 6.921574 the logarithm of 8.34785 is 0.921574 From this we draw the following rule to find the log. of any number consisting of more than four places of figures. Rule. — TaJce out the logarithm of the four superior places, directly from the table, and take the difference between this logarithm and the next greater logarithm in the table. Multiply this difference by the inferior places of figures in the number, as a decimal. Example. Find the logarithm of 357.32514. " the logarithm of 3573. decimal part is .553033 The diflTerence between this and the next greater in the table, is 122. The figures not included in the above logarithm, are .2514 Multiply by . . . 122 6028 5028 2514 30.6708 This result shows that 31 should be added to the decimal part of the logarithm already found ; that is, the logarithm of the proposed number, 357.32514 is 2.653064 The logarithm of 357325.14 is 5.553064 We will now give the converse of this problem : that is, we give the decimal part of a logarithm, .553064, to find the figures corresponding. The next less logarithm in the table, is .663033, corresponding to the figure 3573. The difference between our given logarithm and the one next less in the table, is 31; and the difference between two con- secutive logarithms in this part of the table, is 122. Now divide 31 by 122, and write the quotient after the number 3573. That is, . . . 122)31. (254 244 660 610 500 488 The figures, then, are 3573254, which corresponds to the decima logarithm .663064 ; and the value of these figures will, of course, depend on the index to the logarithm. 160 ELEMENTSOF Prom this, we draw the following rule to find the number correspond- ing to a given logarithm. Rule. — If the given logarithm is not in the table, find the one next less, and take out the four figures corresponding; and if more than four figures are required, take the difference between the given logarithm and the next le.'^s in the table, and divide that difference by the difference of the two consecutive logarithms in the table, the one less, the other greater than the given loga- rithm; and the figures arising in the quotient, as many as may be required^ must be annexed to the former figures taken from the table. EXAMPLES. 1. Given, the logarithm 3.743210, to find its corresponding number true to three places of decimals. Ans. 6636.182 2. Given, the logarithm 2.633366, to find its corresponding number true to two places of decimals. Ans. 429.89 3. Given, the logarithm — 3.291746, to find its corresponding number. Ans. .0019677 TABLE II. This table contains logarithmic sines and tangents, and natural sines and cosines. We shall confine our explanations to the logarithmic sines and cosines. The sine of every degree and minute of the quadrant is given, directly, in the table, commencing at 0*^, and extending to 45°, at the head of the table ; and from 46° to 90°, at the foot of the table, increasing backward. The same column that is marked sine, at the top, is marked cosine at the bottom ; and the reason for this is apparent to any one who has examined the definitions of sines. The difference of two consecutive logarithms is given, corresponding to ten seconds. Removing the decimal point one figure, will give the difference for one second ; and if we multiply this diflTerence by any proposed number of seconds, we shall have a difference corresponding to that number of seconds, above the logarithm, corresponding to the preceding degree and minute. For example, find the sine of 19° 17' 22". The sine of 19° 17', taken directly from the table, is 9.618829 The difference for 10" is 60.2; for 1", is 6.02X22 . 133 Hence, 19° 17' 22" sine is 9.618952 Prom this it will be perceived that there is no difficulty in obtaining the sine or tangent, cosine or cotangent, of any angle greater than 30'. PLANE TRIGONOMETRY. 161 Conversely. Given the logarithmic sine 9.982412, to find its corres- ponding arc. The sine next less in the table, is 9.982404, and gives the arc 73° 48'. The difference between this and the given sine, is 8, and the difference for 1", is .61 ; therefore, the number of seconds cor- responding to 8, must be discovered by dividing 8 by the decimal .61, which gives 13. Hence, the arc sought is 73° 48' 13". These operations are too obvious to require a rule. When the arc is very small, such arcs as are sometimes required in astronomy, it is necessary to be very accurate ; and for that reason we omitted the difference for seconds for all arcs under 30'. Assuming that the sines and tangents of arcs under 30' vary in the same proportion as the arcs themselves, we can find the sine or tangent of any very small arc to great accuracy, as follows ; The sine of l', as expressed in the table, is . . 6.463726 Divide this by 60 ; that is, subtract logarithm . . 1.778151 The logarithmic sine of 1", therefore, is . . . 4.685575 Now, for the sine of 17", add the logarithm of 17 . 1.230449 Logarithmic sine of 17", is 6.916024 In the same manner we may find the sine of any other small arc. For example, find the sine of 14' 2li"; that is, 86r'5 To logarithmic sine of 1", is, 4.685575 Add logarithm of 861.5 2.935255 Logarithmic sine of 14' 21^" 7.620830 Without finder preliminaries, we may now preceed to practical EXAMPLES. 2. In a right angled triangle, ABC, given the base, AB, 1214, and the angle A, 61° 40' 30", to find the other parts. To find BC. As radius : tan.A 61° 40' 30' :: AB 1214 : BC 1535.8 . 10.000000 10.102119 3.084219 3.186338 N. B. When the first term of a logarithmic proportion is radius, Lhe resulting logarithm is found by adding the second and third loga- rithms, rejecting 10 in the index, which is dividing by the first term. In all cases we add the second and third logarithms together; which, in logarithms, is multiplying these terms together: and from that sum 11 162 ELEMENTS OF we subtract the first logarithm, whatever it may be, which is dividing by the first term. To find AC. As sin. C, or cos.A 61° 40' 30" . 9.792477 : AB 1214 . 3.084219 :: Radius . . 10.000000 : AC 1957.7 . 3.291742 To find this resulting logarithm, we subtracted the first logarithm from the second, conceiving its index to be 13. Let ABC represent any plane triangle, right angled at B. 1. Given AC 73.26, and the angle A 49° 12' 20"; required the other parts 1 Ans. The angle C 40° 47' 40", BC 65.46, and AB 47.87. 2. Given AB 469.34, and the angle A 61° 26 17", to find the other parts I Ans. The angle C 38° 33' 43", BC 588.7, and A C 752.9. 3. Given AB 493, and the angle C 20° 14'; required the remaining parts 1 Ans. The angle A 69° 46', BC 1338, and A C 1425. 4. Let AJ?=331, the angle A=49° 14'; what are the other parts ] Ans. AC 506.9, BC 383.9, and the angle C 40° 46'. 6. If AC:=46, and the angle C=yi^ 22', what are the remaining parts I Ans. AB2T.Zl,BC 35.76, and the angle A 52° 38' 6. Given A C 4264.3, and the angle A 66° 29' 13", to find the remain ing parts. Ans. AB 2354.4, BC 3555.4, and the angle C 33° 30' 47". 7. If A5=44.2, and the angle A=31° 12' 49", what are the other parts 1 Ans. AC 49.35, BC 25.57, and the angle C 58° 47' 11". 8. If AJ5=8372.1, and J5C=694.73, what are the other parts? Ans. AC 8400.9, the angle C 85° 15', and the angle A 4° 45'. 9. If AB be 63.4, and AC be 85.72, what are the other parts ] Ans. BCbl.n, the angle C 47° 42', and the angle A 42° 18' 10. Given AC 7269, and AB 3162, to find the other parts. Ans. BC 6546, the angle C 25° 47' 7", and the angle A 64° 12' 53". 11. Given AC 4824, and BC 2412, to find the other parts. Ans. The angle A 30° 00', the angle C 60° 00', and A5 4178 W or THE f UNIVERC \ OF PLANE TRIGONOMETlMfesss^^ 1G3 OBLIQUE AN&LED TRIGONOMETRY. EXAMPLE 1. In the triangle AJ5C, given AJ?=376, the angle A=48o 3', and the angle 5a=40° 14', to find the other parts. As the sum of the three angles of every triangle is always 180°, the third angle, C, must be 180<^--88° l7'=9lo 43'. To find AC. As 8in.9lo 43' : AjB376 . :: sin. B 40° 14' : AC 243 . 9.999805 2.675188 9.810167 12.385355 2.385550 Observe, that the sine of 91° 43' is the same as the cosine of 1° 43'. To find BC. As 8in.9l° 43' . 9.999805 : AB 376 . 2.575188 :: 8in.A48° 3' . 9.871414 BC 279.8 12.446 602 2.446797 EXAMPLE 2. In a plane triangle, given two sides, and an angle opposite one of them, to determine the other parts. Let AD=1751. feet, one of the given sides. The angle D=31° 17 19", and the side opposite, 1257.5. From these data, we are required to find the other side, and the other two angles. In this case we do not know whether A C or AE represents 1257.5, because AC=AE. If we take AC for the other given side, then DC is the other required side, and DAC is the vertical angle. If we take AE for the other given side, then DE is the required side, and DAE in the vertical angle ; but in such cases we determine both triangles. 164 (Prop. 4.) ELEMENTS OF To find the angle E=C. As AC=AJE;==1257.5 log. 3.099508 : D 31° ir 19" sin. 9.715460 :: AD 1751 log. 3.243286 12.958746 r : 46° 18' sin. 9.859238 E=C From 180° take 46° 18', and the remainder is the angle DCA = 133° 42'. The angle DAC=ACE—D {th. 11, b. 1); that is, DAC=46° 18'— 31° 17' 19"=15° 0' 41" The angles D and £, taken from 180°, give jDA£=102° 24' 41". To find DC. As sin.D 31° 17' 19' log. 9.715460 : AC 1257.5 . log. 3.099508 :: sin.Z)ACl5°0'4l"log. 9.413317 12.512825 : Z>C 626.86 2.797165 To find DE. \s sin.jD 31° 17' 17" 9.715460 : AE 1267.5 3.099508 :: sin.l02° 24' 41' . 9.989730 13.089238 : DjB 2364.7 . . 3.373778 N. B. To make the triangle possible, AC must not be less than ABi the sine of the angle 2>, when DA is made radius. EXAMPLE 3. In any plane. triangUj given two sides and the included angle, to find the other parts. Let Aj9=1751 (see last figure), Z)JEJ=2364.6, and the included angle Z)=31° 17' 19". We are required to find DE, the angle DAE, and angle E. Observe that the angle E must be less than the angle DAE, because it is opposite a less side. From .... 180° Takei? .... 31° 17' 19" Sum of the other two angles =148° 42' 41" (th. 11, b. 1) i sum . . . . = 74° 21' 20" By proposition 7, PLANE TRIGONOMETRY. 16& DE+DA : DE^DA=: tan.74° 21' 20 " : taii.i(I>A£— JB) That is, 4116.6 : 613.6= tan.740 21' 20" : i(DAE—E) Tan.74° 21' 20" . 10.662778 613.6 . . 2.787816 13.340693 4116.6 log. (sub.) 3.614423 i{DAE—E) tan.28o 1' 36" 9.726170 But the half sum and half difference of any two quantities are equal to the greater of the two ; and the half sum, less the half difference, is equal the less. Therefore, to 74° 21' 20" Add . 28 1 36 DA£=102° 22' 66" E= 46 19 44 Tojind AE. As sin.E A^ 19' 44" , 9.869323 : DA 1761 . . 3.243286 :: sin.2> 31° 17' 19" • 9.715460 12.968746 : AE 1267.2 . • 3.099423 EXAMPLE 4. Given the three sides of a plane triangle to find the angles. Given AC=1761, 05=1267.6, AjB=2364.6 If we take the formula for cosines, we will compute the greatest angle, which is C. To correspond with the formula. cos.iC=i/?!£^fiZ£j we must V ah take a— 1257.6 5=1761, and c=2364.6 The half sum of these is, 5=2686.6 • «— c=322 2^2 . . 20.000000 5=2686.6 . 3.429187 5— c=322 . 2.607866 Numerator, log. 26.937043 166 ELEMENTSOF R^ . . 20.000000 5=2686.5 . 3.429187 s— c=322 . 2.507856 Numerator, log. 25.937043 a 1257.6 3.099508 h 1751. 3.243286 Denominator, log. 6.342794 6.342794 2)19.594249 iC= 51° ir 10" COS. 9.797124 C=102 22 20 The remaining angles may now be found by problem 4. We give the following examples for practical exercises : Let ABC represent any oblique angled triangle. 1. Given AB 697, the angle A 81° 30' 10", and the angle B 40° 80' 44", to find the other parts. Ans, AC 534, BC 813, and the angle C 57° 59' 6". 2. If AC=720.8, the angle A=70° 5' 22", and the angle 5=59* 35' 36", required the other parts. Ans. AB 643.2, BC 785.8, and the angle C 50° 19' 2". 3. Given BC 980.1, the angle A 7° 6' 26", and the angle B 106* 2' 23", to find the other parts. Ans. AB 7284, AC 7613.3, and the angle C 66° 61' 11". 4. Given AB 896.2, BC 328.4, and the angle C 113° 45' 20", to find the other parts. Ans. AC 712, the angle A 19° 36' 48", and the angle B 46° 38' 62". 6. Given AC 4627, BC 6169, and the angle A 70° 26' 12", to find the other parts. Ans. AB 4328, the angle B 57° 29' 66", and the angle C 52° 4' 52". 6. Given AB 793.8, BC 481.6, and AC 600.0, to find the angles. Ans. The angle A 35° 16' 32", the angle B 36° 49' 18", and the angle C 107° 66' 10". 7. Given AB 100.3, BC 100.3, and AC 100.3, to find the angles. Ans. The angle A 60°, the angle B 60°, and the angle C 60°. 8. Given AB 92.6, BC 46.3, and AC 71.2, to find the angles. Ans. The angle A 29° 17' 22", the angle B 48° 47' 31", and the angle C 101° 66' 8". PLANE TRIGONOMETRY. 167 9. Given AB 4963, BC 6124, and AC 6621, to find the angles. Ans, The angle A 67° 30' 28", the angle B 67° 42' 36", and the angle C 64° 46' 66". 10. Given AB 728.1, BC 614.7, and AC 683.8, to find the angles. Ans. The angle A 64° 32' 62", the angle B 60° 40' 68", and the angle C 74° 46' 10". 11. Given AB 96.74, BC 83.29, and AC 111.42, to find the angles. Ans. The angle A 46° 30' 46", the ang*e B 76° 3' 46", and the angle C 67° 25' 30' . 12. Given AB 363.4, BC 148.4, and the angle JB 102° 18' 27", to find the other parts. Atw. The angle A 20° 9' 17", the side AC. = 420 8, and the angle C 67° 32' 16". 13. Given AB 632, BC 494, and the angle A 20° 16', to find the other parts, C being acute. Ans. The angle C 26° 18' 19", the angle B 133° 26' 41", aud AC 1035.86. 14. Given AB 63.9, AC 46 21 , and the angle B 68916,' to find the other parts. Ans. The angle A 38° 68', the angle C 82° 46, and BC 34,16. 16. Given AB 2163, BC 1672, and the angle C 112° 18' 22", to , find the other parts. Ans. AC 877.2, the angle B 22° 2' 16", and the angle A 45° 39' 22". 16. Given AB 496, BC 496, and the angle B 38° 16', to find the other parts. Ans. AC 326.1, the angle A 70° 62' and the angle C70° 62'. 17. Given AB 428, the angle C 49° 16', and (AC+jBC) 918, to find the other parts, the angle JS being obtuse. Ans. The angle A 38° 44' 48", the angle B 91° 59' 12", AC 664.49, and BC 353.5. 18. Given AC 126, the angle B 29° 46', and {AB—BC) 43, to find the other parts. Ans. The angle A 66° 61' 32", the angle C 94° 22' 28", A5 253.05, and BC 210^.54. 19. Given AB 1269, AC 1837, and the angle A 63° 16' 20", to find the other parts. Ans. The angle B 83° 23' 47", the angle C 43° 19' 63", and BC 1482.16. 168 EXE ME NTS OF APPLICATION OF TRIGONOMETRY TO MEA- SURING THE EIGHT AND DISTANCES OF VISIBLE OBJECTS. In this useful application of trigonometry, a base line is always sup- posed to be measured, or given in length ; and by means of a quadrant, sextant, circle, theodolite, or some other instrument for measuring angles, such angles are measured as connected with the base line, and the objects whose bights or distances it is proposed to determine, enable us to compute, from the principles of trigonometry, what those hights or distances are. Sometimes, particularly in marine surveying, horizontal angles are determined by the compass; but the varying effect of surrounding bodies on the needle, even in situations little removed from each other, and the general construction of the instrument itself, render it unfit to be applied in the determination of angles where anything like precision is required. The following examples present sufficient variety to guide the student in determining what will be the most eligible mode of proceeding in any case that is likely to occur in practice. EXAMPLE 1. Being desirous of finding the distance between two distant objects, C and X>, I measured a base ABj of 384 yards, on the same horizontal plane with the objects C and D. At A, I found the angle Z)AJS=48° 12', and CAJ5=890 18'; at B the angle ABC was 46° 14', and ABD 87° 4'. It is required from these data to compute the distance between C and D. From the angle CAB, take the angle DAB; the remainder, 41° 6', is the angle CAD. To the angle DBA, add the angle DAB, and 44° 44', the supple- ment of the sum, is the angle ADB, In the same way the angle A CB, which is the supplement of the sum of CAB and CBA, is found to be 44° 28'. Hence, in the triangles ABC and ABD, we have As sin. A CB 44° 28 . 9.846405 : AB 384 yards . . 2.684331 :: sin. ABC 46° 14' . 9.858635 12.442996 AC 395.9 yards . . 2.597561 PLANE TRIGONOMETRY. 109 As sin. ADB 44*^ 44' 9.847454 : AB 384 yards 2.584331 :: sin. ABD 87° 4' . 9.999431 12.583762 : AD 544.9 yards . . 2.7363.08 Then, in the triangle CAD, we have given the sides CA and AD, and the included angle CAD, to find CD; to compute virhich we proceed thus : The supplement of the angle CAD is the sum of the angles A CD, and ADC; Hence, . it =69° 27', and, by proportion we have. 2 AsAD+AC : AD^AC . :: tan. ^^^+^^^ 2 940.8 149 69° 27' 22 54 2.937497 2.173186 10.426108 . . A CD— ADC : tan. 2 12.599294 9.625797 the angle A CD sum the angle ADC . difF. As sin. ADC 46° 33' : A C 395.9 yards . :: sin. CAD 41° 6' : CD 358.5 yards . 92 21 46 33 9.860922 2.597585 9.817813 12.415398 2.554476 EXAMPLE 2. To determine the altitude of a lighthouse, I observed the elevation of its top above the level sand on the seashore, to be 15° 32' 18", and measuring directly from it, along the sand 638 yards, I then found its elevation to be 9° 56' 26"; required the hight of the lighthouse. Let CD represent the hight of the lighthouse above the level of the sand, and let B be the first station, and A the second ; then the angle CBD is 15° 32' 18", and the angle CAB is 9° 56' 26"; therefore, the angle ACB, which is the difference of the angles CBD and CAB, is 5° 35' 52". 170 ELEMENTS OF Hence, . As sin. ACB 6° 36' 52" . 8.989201 : AB 6Z8 2.804821 :: sin. angle A 9° 56' 26" 9.237 1U7 12.041928 : BC 1129.06 yards 3.052727 As radius .... 10.000000 : BC 1129.06 3.052727 :: sin.CjBD 15° 32' 18" . 9.427945 12.480672 : DC 302.46 yards 2.480672 EXAMPLE 3. Coming from sea, at the point D,l observed two neadlands. A and B, and inland, at C, a steeple, which appeared between the headlands. I found, from a map, that the headlands were 5.35 from each other ; that the distance from A to the steeple was 2.8 miles, and from B to the steeple 3.47 miles ; and I found with a sextant, that the angle ADC was 12° 15', and the angle BDC 15° 30'. Required my distance from each of the headlands, and from the steeple. CONSTRUCTION. The angle between the two headlands is the sum of 150 30' and 12° 15', or 27° 45'. Take the double, 55° 30'. Conceive AB to be the chord of a circle, and the segment on one side of it to be 55° 30 ; and, of course, the other will be 304° 30'. The point D will be somewhere in the circumference of this circle. Consider that point as determined, and join CD. In the triangle ABC we have all the sides, and, of course, we can find all the angles ; and if the angle ACB is less than (180°— (27° 45') )=152° 15', then the circle cuts the line CD, in a point E, and C is without the circle. Join AE, BE, AD, and DB. AEBD is a quadrilateral in a circle, and AEB-^ADB=180°. The angle ADE= the angle ABE, because both are measured by half the arc AE. Also, EDB=EAB, for a similar reason. Now, in the triangle AEB, its side AB, and all its angles, are known ; and from thence AE can be computed. Then, having the PLANE TRIGONOMETRY. 171 two sides AC and AE of the triangle AEC, and the included angle CAE, we can find the angle AEC, and, of course, its supplement, AED. Then, in the triangle AED we have the side AE, and the two angles AED and ADE, from which we can find AD. The computation, at length, is as follows : To find AE. 16° 30' As sin.AEB 162° 15' . 9.668027 12 15 : A5 5.35 . . . .728354 :: sin. ABE 12° 15' • 9.326700 angle EAB angle EBA 27 45 180 angle AEB 152 15 AE 2.438 To find the angle BA C. BC 3.47 AB 5.35 log. .728354 AC 2.80 log. .447158 2)11.62 BC 5.81 2.34 log. log. 1.175512 .764176 .369216 20 17° 41' 58" 2 angle angle BAC 35 23 56 EAB 15 30 angle EAC 19 53 56 180 2)160 6 4 80 3 2 22^133392 2)19.957880 cos. 9.978940 AEC-^ACE 2 10.355054 .387027 To find the angles AEC and ACE. As AC-\-AE 6.238 .719165 : AC—AE .362 AEC^ACE tan.- —80° 3' 2' AEC— ACE tan. 7i 21 30 12 -1.558709 10.755928 10.314637 9.596472 172 ELEMENTS OF angle AEC ACE 01 ACD 101*: >33'14' ' sum angle 68 32 50 diff. angle CDA 12 15 70 47 50 supplement 1 35 23 56 angle CAB 73 48 14 To find AD. Assin.ADC 12° 15' . 9.326700 : AC 2.8 . .447158 :: sin. A CD 58° 32' 50" 9.930985 10.378143 : AD 11.26 miles 1.051443 EXAMPLE 4. The elevation of a spire at one station was 23° 50' 17", and the horizontal angle at this station, between the spire and another station, was 93° 4' 20". The horizontal angle at the latter station, between the spire and the first station, was 54° 28' 36", and the distance between the two stations, 416 feet. Required the hight of the spire. Let CD be the spire, A the first station, and B the second ; then the vertical angle CAD is 23° 60' 17"; and as the horizontal angles CAB and CJ5A are 93° 4' 20", and 54° 28' 36" respectively, the angle A CB, the supplement of their sum, is 32° 27' 4". To find AC, As sin. 5 CA 32° 2T 3" : side AB 416 :: sin.AJ5C 54° 28' 36' 9.729634 2.619093 9.910560 12.529653 : side AC 631 2.800019 To find DC. \.s radius .... 10.000000 : side AC 631 2.800019 :: tan.DAC 23° 60' 17" . 9.645270 : DC 278.8 2.445289 PLANE TRIGONOMETRY. 173 By the application of the fourth example we can compute the dif- ferent elevations of different planes, provided the same object is visible from them. For example, let M be a promi- nent tree or rock near the top of a mountain, and by observations taken at Ay we can determine the perpendicular Mn. By like observations we can determine the perpendicular Mm. The difference between these two perpendiculars, is nm, or the difference in the elevation between the two points A and B. But if the distances between A and n, or B and m, are considerable, or more than two or three miles, corrections must be made for the convexity of the earth ; but for less distances such corrections are not necessary. EXAMPLES FOR EXERCISE. 1. Required the hight of a wall whose angle of elevation is observed, jit the distance of 463 feet, to be 16° 21' 1 A7is. 135.8 feet. 2. The angle of elevation of a hill is, near its bottom, 31° 18', and 214 yards further off, 26° 18'. Required the perpendicular hight of the hill, and the distance of the perpendicular from the first station. Ans. The hight of the hill is 665.2, and the distance of the perpen- dicular from the first station, is 929.6. 3. The wall of a tower which is 149.6 feet in hight, makes, with a line drawn from the top of it to a distant object on the horizontal plane, an angle of 57° 21'. What is the distance of the object from the bottom of the tower 1 Ans. 233.3 feet. 4. From the top of a tower, whose hight was 138 feet, I took the angles of depression of two objects which stood in a direct line from the bottom of the tower, and upon the same horizontal plane with it. The depression of the nearer object was found to be 48° 10', and that of the further, 18° 62'. What was the distance of each from the bottom of the tower 1 Ans. Distance of the nearer 123.6, and of the farther 403.8 feet. 6. Being on the side of a river, and wishing to know the distance of a house on the other side, I measured 312 yards in a right line by the side of the river, and then found that the two angles, one at each end of this line, subtended by the other end and the house, were 31^ 15' and 86° 27'. What was the distance between each end of the line and the house ? Ans. 351.7, and 182.8 yards. 174 ELEMENTSOF 6. Having measured a base of 260 yards in a straight line, close by one side of a river, I found that the two angles, one at each end of the line, subtended by the other end and a tree close to the opposite bank, were 40° and 80°. What was the breadth of the river 'i Ans. 190.1 yards. 7. From an eminence of 268 feet in perpendicular bight, the angle of depression of the top of a steeple which stood on the same hori- zontal plane, was found to be 40° 3', and of the bottom 66° 18'. What was the bight of the steeple ? Ans. 117.8 feet. 8. Wanting to know the distance between two objects which were separated by a morass, I measured the distance from each to a point where I could see them both ; the distances were 1840 and 1428 yards, and the angle which, at that point, the objects subtended, was 36° 18' 24". Required their distance. Ans. 1090.85 yards. 9. From the top of a mountain, three miles in hight, the visible hori- zon appeared depressed 2° 13' 27". Required the diameter of the earth, and the distance of the boundary of the visible horizon. Ans. Diameter of the earth 7958 miles, distance of the horizon 154.54 miles. lOr From a ship a headland, was seen bearing north, 39° 23' east. After sailing 20 miles north, 47° 49' west, the same headland was observed to bear north, 87° 11' east. Required the distance of the headland from the ship at each station '] Ans. The distance at the first station was 19.09, and at the second 26.96 miles. 11. The top of a tower, 100 feet above the level of the sea, was seen as on the surface of the sea, from the masthead of a ship, 90 feet above the water. The diameter of the earth being 7960 miles, what was the distance between the observer and the object 1 Ans. 23.9 plus -j^ for refraction = 25.7 miles. 12. From the top of a tower, by the seaside, of 143 feet high, it was observed that the angle of depression of a ship's bottom, then at anchor, measured 35°; what, then, was the ship's distance from the bottom of the wall 1 Ans. 204.22 feet. 1 3. Wanting to know the breadth of a river, I measured a base of 600 yards in a straight line close by one side of it ; and at each end of this line I found the angles subtended by the other end and a tree close to the bank on the other side of the river, to be 63° and 79° 12'. What then, was the perpendicular breadth of the river 1 Ans. 629.48 yards. 14. What is the perpendicular hight of a hill, its angle of elevation taken at the bottom of it, being 46°, and 200 yards further ofi*, on a level with the bottom, the angle was 31°1 Am. 286.28 yards. PLANE TRIGONOMETRY. 175 15. Wanting to know the hight of an inaccessible tower ; at the least distance from it, on the same horizontal plane, I took its angle of elevation equal to 68°; then going 300 feet directly from it, found the angle there to be only 32°; required its hight, and my distance from it at the first station. . } Hight 307.53. ^^' ^Distance 192.15. 16. Two ships of war, intending to cannonade a fort, are, by the shal- lowness of the water, kept so far from it, that they suspect their guns cannot reach it with eiSect. In order, therefore, to measure the dis- tance, they separate from each other a quarter of a mile, or 440 yards, then each ship observes and measures the angle which the other ship and fort subtends, which angles are 83° 45' and 85° 15'. What, then, is the distance between each ship and the fort 1 . S 2292.26 , ^^^- J 2298.05 y^'^"- 17. A point of land was observed by a ship, at sea, to bear east-by- south ;* and after sailing north-east 12 miles, it was found to bear south- east-by-east. It is required to determine the place of that headland, and the ship's distance from it at the last observation 1 Ans. 26.0728 miles. 18. Wanting to know my distance from an inaccessible object, 0, on the other side of a river ; and having no instrument for taking angles, but only a chain or chord for measuring distances ; from each of two stations, A and JB, which were taken at 500 yards asunder, I measured in a direct line from the object 0, 100 yards, viz., AC and BD, each equal to 100 yards ; also the diagonal AD measured 550 yards, and the diagonal BC 560. What, then, was the distance of the object from each station A and jBI A'» i ^^ 536.25. ^^- I BO 500.09. 19. A navigator found, by observation, that the vertex of a certain mountain, which he supposed to be 45 minutes of a degree distant, had an altitude above the sea horison of 31' 20". Now, on the supposition that the earth's radius is 3956 miles, and the observer's dip was 4' 15", what was the hight of the mountain I Ans. 3960 feet. N. B. This should be diminished by about its one-eleventh part foi the influence of horizontal refraction. • That is, one point south of east. A point of the compass is 11° 15. 176 . SPHERICAL SPHERICAL TRIGONOMETRY. Spherical Geometry is nothing more than the general principles of geometry applied to the various sections of a sphere ; and spher- ical trigonometry, is but the general principles of plane trigonome- try applied to triangles resting on a surface of a sphere, and the planes of the sides of the triangles passing through the center of the sphere. DEFINITIONS. 1. A sphere is a solid whose surface is equally convex in every part, and every point of the surface is equally distant from one point within, and this point is called the center. A sphere may be con- ceived to be generated by the revolution of a semicircle about its diameter. If the center of the semicircle rests at the same point, the posi- tion of the diameter may be in any direction or position, and the revolution of the semicircle will describe the same sphere. 2. Any plane that passes through the center of the sphere, di- vides the solid and the surface into two equal parts. 3. Any two planes that pass through the center of a sphere, in- tersect each other on the opposite points of the sphere, because the section of any two planes is a right line (th. 2, b. 6). 4. A great circle on a sphere, is one whose plane passes through the center of the sphere. 5. Every great circle has poles, two points on the sphere directly opposite to each other and equally distant from every point on the great circle. The distance from any pole to its equator in a»y direction, is one fourth of the whole distance round the sphere. 6. Any point on a sphere may be a pole to some ff recti circle. 7. A spherical triangle is formed by the intersection of three great circles on a sphere. Conceive three radii drawn from the three angular points to the center of the sphere, thence forming a solid angle. The angles of the three planes which form this solid angle at the center, are the three angles which measure the sides of the triangle, and the inclination of these planes to each other form the angles of the triangle. TRIGONOMETRY. 177 8. The complete measure of a spherical triangle, is but the complete measure of a solid angle at the center of a sphere ; and this solid angle is the same, whatever be the radius of the sphere. 9. Every great circle, or portion of a great circle on the surface of a sphere, has its poles ; conversely, every pole, or the point where two circles intersect, has its equator 90° distant, and the portion of this equator between the two sides, or the two sides produced, measures the spherical angle at the pole. The inclination of two tangents of two arcs formed at their point of intersection, also measures the spherical angle. (Def. 5, to b. 6). 10. We can always draw one, and only one great circle through any two points on the surface of a sphere ; for the two given points and the center of the sphere, give three points, and through three points only one plane can be made to pass (cor. th. 1, b. 6). PROPOSITION 1. Every section of a sphere hy a plane is a circle. If the plane passes through the center of the sphere, the section is evidently a circle, for every point on the surface of the sphere is equally distant from the center. These sections are great circles, and all great circles on the same sphere are equal to each other. Now let the cutting plane not pass through the center. From the center (7, let fall Cn perpendicular to the plane ; and when a line is perpen- dicular to a plane, it is perpendicular to all Hues that can be drawn in that plane (th. 3, b. 6); therefore, any line as nm in the plane, is at right angles to Cn. Hence nm= J Cm^ — Cn^. But nm is any line in the plane, from the point n to the surface of the sphere, and this value for nm is invariable, and it is the radius of a circle whose center is n, N. B. These circles are called small circles, and are greater or less, as they are nearer or more remote from the center C Small circles on a sphere, are never considered as sides of spheri- cal triangles. We again repeat, that sides of spherical triangles must be portions of ^reat circles, and each side must be less than 180°. 12 178 PHERICAL PROPOSITION 2. Any two sides of a spherical triangle are together greater than the third. Let A By AC, and BO, be the three sides of the triangle, and D the center of the sphere. The arcs AB, AO, and BC, are measured by the angles of the planes that form the solid angle at J). But any two of these angles are together greater than the third (th. 10, b. 6). Therefore, any two sides of the triangle are together, greater than the third. Q. K D. PROPOSITION 3. The sum of ths three sides of any spherical triangle is less than the circumference of a great circle. Let ABO be a triangle ; the two sides AB, AO, produced, will meet at the point on the sphere which is directly opposite to A; and the arcs ABD, and A OD, are together equal to a great circle. But by the last proposition, BO is less than the two arcs BJ) and DO. Therefore, AB, BO, and A (7, are together less than ABD-^-A OD; that is, less than a great circle. Q. E. D. PROPOSITION 4. Every right angled spherical triangle must have a complemental, supplemental, and four quadrarUal triangles in the same hemisphere. Let ABO, be a right angled spherical triangle, right angled at B. Produce the sides AB and AO, and they will meet at A', the opposite point on the sphere. Produce BO, both ways, 90° from the point B, to P and P', which are therefore, poles to the arc AB (def. 9, spherics). Through A, P, and the center of the sphere, pass a plane cutting the sphere into two equal parts, forming a great circle on the sphere, which great circle will be represented by the plane TRIGONOMETRY. 179 circle PAP' A on tlie paper. At right angles to this plane, pass another plane, cutting the sphere into two equal parts ; this great circle is represented on the paper, by the straight line P OP'. A and A'y are the poles to the great circle P OP', P and P', are the poles to the great circle ABA', As PCy PD and CD, are portions of great circles on a sphere, CPD is a spherical triangle, and it is crnnplemental to the given triangle ABC; because CD is the complement of AC, CP the complement of BC, and PD is the complement of D 0, or of the an- gle A. Again, the triangle A'BC, is supplemental to ABC, because A'=A; A' Ch the supplement of AG, and A'B is the supplement of AB. ACP is a spherical triangle, and one of its sides, AP, is a quadrant, and it is therefore called a quadrantal triangle. So also, are the triangles A'CP, ACP', and P'CA', quadrantal triangles. Cor. In every triangle there are six elements ; three sides and three angles, which are sometimes called parts. Now, if all the parts of the triangle ABC are known, the parts of the complemental triangle PCD, are also known, and the sup- plemental triangle A' BC, must be as completely known. When the triangle PCD is known, the triangles A CP and A' PC are also known, for the side PD, measures the angles PA C and PA C, and the angle CPD, added to the right angle A' PD, gives the angle A' PC, and CPA, is supplemental to this. Hence a solution of any right angled spherical triangle, is a solution to its complemental, supplemental, and all its quadrantal triangles. Definition. Every triangle, together with its supplemental tri- angle, form what is called a Lune. Thus, the triangles ABC and A'BC, form a lune ; PCD and P'CD, form a lune ; PAC and P'A C, also form a lune. It is obvious, that the surface of the lune PAP'B, is to the surface of the sphere, as the arc AB, is to the wliole circumference. PROPOSITION 5. If there he three arcs of great circles whose poles are the angular points of a sj^herical triangle, such arcs, if produced, will form, another triangle, whose sides will he supplemental to the angles of the first triangle, and the sides of the first tnangle will he supplemental to the angles of the second 180 SPHERICAL Let the arcs of the three great circles be QHy PQf KLy whose poles are respectively ,4, Bt and 0. Produce the three arcs until they meet in E^ i>, and F, We are now to show, that E is the pole to the great circle AG; D the pole of the great circle BC; F the pole to the great circle AB, Also, that the side EF^ is supplemental to the angle A; ED to the angle C; and DF to the angle B; and also, that, the side A (7, is supplemental to the angle E^ (fee. Any pole is 90° from any point on its great circle, and therefore, as A is the pale to the great circle GHy the point Ai is 90° from the point E. As Q is the pole of the great circle LKy C is 90° from any point in that great circle ; therefore, G is 90° from the point Ey and Ey being 90° from both A and G, it is the pole of the arc A G. In the same manner, we may prove that D is the pole of BGy and i^the pole of AB, Because A is the pole of the arc GHy the arc OH measures the angle A (def. 9 spherics); for the same reason, PQ measures the angle By and XJT measures the angle G. Because ^is the pole of the arc AGy E 11=90° Or, ... . EG-\-OJI=90° For a like reason, . . FH-\-GH=90° Adding these two equations, and obseiTing that GH=^Ay and afterward transposing one Ay we have, EG^GH^FH=\ZO°--A. Or, EF=\ZO°—A ^ In like manner, .... i^i>=180°— i? I (a) And, jE'i>=180°— a J But the arc (180° — ^), is a supplemental arc to A, by the de- finition of arcs ; therefore, the three sides of the triangle EDF, are supplements of the angles Ay By (7, of the triangle ABC. Again, as Ey is the pole of the arc A C, the whole angle E, is measured by the whole arc LIT. But, .... AG-\-GB=90° Also, .... AG-\-AL=90° By addition, . . A G+A (7-f Cir-\-AL= 1 80° TRIGONOMETRY. 181 By transposition, . A C+ CII+AL= 1 80°—^ C That is, .... Zff, or JS^ISO^-^AO ^ In the same manner, . . F^ISO^'—AB \ (b) And, J9=180°— ^(7 J That is, the sides of the first triangle, are supplemental to the angles of the second triangle. Q. E, D. PROPOSI TION 6. The sum of the three angles of any spherical triangle, is greater than two right angles, and less than six right angles. Turn to equations (a), of the last proposition, and add them to- gether. The first member of the equation so formed will be the sum of three sides of a spherical triangle, which sum we may des- ignate by aS'. The other member will be 6 right angles (there being 2 right angles in each 180°) less the three angles A, B, and 0. That is, . . iSr=6 right angles— (^+^+(7) By proposition 3, the sum S, is less than 4 right angles; there- fore, to it add s, a sufiicient quantity to make 4 right angles. Then, 4 right angles=6 right angles — (A-{-B-{-C)-\-s Drop 4 right angles from both members, and transpose (A-\-B-\- C) Then, . A-\-B-]-C=2 right angles+5 That is, the three angles of a spherical triangle, make a greater sum than two right angles by the indefinite quantity s, which quan- tity is called the spherical excess, and is greater or less according to the size of the triangle. Again the sum of the angles is less than 6 right angles. There are but three angles to any triangle, and no one of them can come up to 180°, or 2 right angles. For an angle is the inclination of two lines or two planes ; and when two planes incline by 1 80°, the planes are parallel, or are in one and the same plane ; therefore, as neither angle can equal 2 right angles, the three can never equal 6 right angles. Q. K D. Scholium. By merely inspecting the figure to proposition 4, we perceive that the triangle PAB, has two right angles ; one at A, the other at B, besides the third angle APB. The triangle P'A'O, has 3 right angles. The triangle A'P'Q^ has two of its angles, each greater than a right angle. 182 SPHERICAL PROPOSITION. 7. Wkh the sines of the sides, and the tangent of one side of any right angled spherical triangle, two plane triangles can he f(yrmed thai will be similar, and similarly situated. Let ABO, be a spherical triangle, right angled at JB; and let D be the center of the sphere. Because the angle CBA, is a right angle, the plane CDB, is perpendicular to the plane DBA. From 0, let fall CII, perpendic- ular to the plane DBA, and as the plane CBD is perpendicular to the plane DBA, QH will lie in the plane CBD, and be perpendicular to the line DB, and perpendicular to all lines that can be drawn in the plane DBA, from the point ^(th. 3, b. 6). Draw HO- perpendicular to DA, and join GC; GC will lie wholly in the plane CD A (def. of planes), and CHG is a right angled triangle, right angled at R. We mil now demonstrate that the angle DGC, is a right angle. The right angled A CHG, gives CH''-^HG''= GG'' ( 1 ) The right angled A DGH, gives DG''-\-HG''=DFP (2) By subtraction, . . . CH''—DG''=^CG'—DIP{^) By transposition, . . . CH^-{-DH''=CG''^DG^'\) But the first member of the equation (4), is equal to CD^; be- cause CDH, is a right angled triangle ; Therefore, CD'^^^GC-VDG'' Hence, CD, is the hypotenuse to the right angled triangle DGC (th. 36, b. 1). From the point B, draw BE at right angles to DA, and BF at right angles to DB, in the plane CDB extended ; the point F being in the line DC. Join EF, and as F is in the plane CD A, and E is in the same plane, the line EF, is in the plane CD A. Now we are to show, that the triangle CHG is similar, and similarly situated to the triangle BEF. As HG and BE are both at right angles to DA, they are paral- lel ; and as GH and BF are both at right angles to DB, they are parallel; and by reason of the parallels, the angles GHG and EBF, are equal ; but GEO is a right angle ; therefore, EBF is also a right angle. TRIGONOMETRY. 183 Now as QH and BE are parallel, and CH and BF parallel, we have, DH '. DB=HG : BE And, . . . DH:DB=HC'.BF Therefore, . . HO : BE=^HC : BF (th. 6, b. 2) Or, . , , HQ '. HC=BE : BF Here, then, are two triangles, having an angle in the one equal to an angle in the other, and the sides about the equal angles pro- portional ; the two triangles are therefore equiangular (th. 20, b. 2); and they are similarly situated, for their sides make equal angles at H and B with the same line, DB. Q. E. D. Scholium. By the definition of sines, cosines, and tangents, we perceive, that CH is the sine of the arc BOy DH is its cosine, and BF its tangent ; CO is the sine of the arc A C, and D its cosine. Also, BE is the sine of the arc AB, and DE is the cosine of the same arc. With this figure we are prepared to demonstrate the following theorems. PROPOSITION 8. THEOREM 1. In any right angled spherical triangle^ the sine of one side is to the tangent of the other side, as radius is to the tangent of the angle adjacent to the first-mentioned side. Or, as the sine of one side is to the tangent of the other side, so is the cotangent of the angle, adjacent to the first-mentioned side, to the raditis. In the right angled plane triangle EBF, we have, EB : BF=E : tan.BEF That is, . sin. c : tan.a=B : tan.-4 Q. E. D, A modification of this proposition demonstrates the latter part of the theorem. By reference to equation (5), plane trigonometry, B^ we shall find that, tan.-4. cot.A=^JR^; therefore, tan.^= ■- cot.A Substituting this value for tangent A, in the preceding proposi- tion, and dividing the last couplet by B, we shall have. sm. c : tan.a= 1 : r cot.-d Or, . . sin. c : tan.a=cot.-4 : B Q, E, J), Or, . . . jB sin. c =tan.a cot.^ (1) 184 SPHERICAL Cor. By changing the construction, drawing the tangent to A B, in place of the tangent to BC, and proceeding in a similar manner, we have, i2 sin.a=tan.c cot. (7 (2) PROPOSITION 9. THEOREM. 2. In any right angled spherical triangle, the sine of the right angle is to the sine of the hypotemise, as the sine of either of the other angles to tlie sine of the side opposite to that angle. N. B. For the sake of perspicuity, if not of brevity, we will repre- sent the angles of the triangle, by A, By C, and of the sides or arcs opposite to these angles by a, b, c; that is, a opposite A, &,c. The sine of 90°, or radius, is designated by R. In the plane triangle CHO, we have, sm.CIIO: CO=:sm.Oair: CH That is, . . E : sin.5=sin.^ : sin.a Q. JS, D, Or, . . . ' i2sin.a=sm.d sin.^ (3) Cor. By a change in the construction of the figure, drawing a tangent to AB, &c., we shall have, H : sin. 5=sin. C : sin.c Q. E. D. Or, . . . i2sin.c=sin.5 sin. (7 (4) Scholium, Collecting the four preceding equations drawn from theorems 1 and 2, we have, ( 1 ) a sin.c=tan.a cot.^ (2) i2 8in.a=tan.c cot. (7 (3) i2sin.a=sin.5sin.^ (4) JR sin.c=sin.6 sin. C These equations refer to the right angled triangle ABC; but the principles are true for any right angled spherical triangle. Let us apply them to the right angled tri- angle PDC, the complemental triangle to^^a this application, equation (1) becomes, R sin. CD=iQii.PD cot. C (n) (2) becomes i2 sin.Pi>=tan. (7i> cot.P {m) (3) becomes i2sin.Pi)=sin.P(7sin.C7 (o) (4) becomes i2sin.CZ>=sin.P(7sin.P (p) TRIGONOMETRY. 185 By observing that sin. CD=cos.A C=cos.b, And that . . tan.Pi>=cot.i) 0=cotAy &c ; and by running equations (»), (m), (o), and (p), back into the triangle ABCf and we shall have, (6) Rcos.h=coi.A cot. 07- (6) i2cos.u4=cot.5 tan.c (7) i2cos.^=cos.a sin.(7 (8) R cos.5=cos.a cos.c By observing equation (6), we find that the second member refers to sides adjacent to the angle A. The same relation holds in respect to the angle C, and gives, (9) R cos.C=cot.i tan.a Making the same observations on (7), we infer, (10) i2 cos.(7=cos.c sin.^ Observation 1 . Several of these equations can be deduced geo- metrically without the least difficulty. For example, take the fig- ure to proposition 7. Observe the parallels in the plane DBAf which give, DB : DH=DE : DO That is, . . R \ cos.a=cos.c : cos.5 ' A result identical with equation (8), and in words is expressed thus : As radius is to cosine of one side, so is the cosine of the other side, to the cosine of the hypotenuse. Observation 2. Equations numbered from (1) to (10), cover every possible case that can occur in right angled spherical trig- onometry, but the combinations are too various to be remembered, and readily applied to practical use. We can remedy this inconvenience, by taking the complement of the hypotenuse, and the complements oi the two oblique angles, in place of the arcs themselves. Thus h is the hypotenuse, and let h' be its complement. Then, 5-f-5'=90° ; or, 5=90°— 5'; and, sin.5=cos.6', cos.6=sin.6'; tan.5=cot.5'. In the same manner if -4' is the complement to A, Then, . sin.^=cos.^V cos.-4=sin.J[V and, tan.-4=cot.-4V and similarly, sin. C= cos. CV cos. (7= sin. C", and tan. (7= cot. C. 186 SPHERIC A I, Substituting these values for J, A, and C, in the foregoing ten equations (a and c remaining the same), we have, Napier's circular parts. U) Ji sin.c=tan.a tan.^' I Omitting the consid- 12) i2sin.a=tan.ctan.(7' i eratlon of the right augle ^e*\ n ' jf J, i there are five parts. — 13) Jt sm.a= COS. cos.A i r, , . . , ' iLach part taken as a 14) i2sin.C=COS.5'cOS.(7' i middle part, is connect- 15) jRsin.6'=tan.^' tan.C" ed to its adjacent parts 16) i? sin.^'=tan.6' tan.C by one equation, and 17) i2 sin.^' = COS.a cos. (7' ^o its extreme parts by io\ D • r/ another equation; and 18) ic sin.o=cos.a cos.c i . „ ( therefore, ten equations 19) i?sm.(7'=tan.5'tan.« are required for the com- 20) jR sin. (7'=C0S.C COS.^' ' binations of all the parts. These equations are very remarkable, because the first members are all composed of radius into some sine, and the second members are all composed of the product of two tangents, or two cosines. To condense these equations in'o words, for the purpose of assist- ing the memory, we will refer them, any one of them, directly to the right angled triangle ABC, in the last figure. When the right angle is left out of the question, a right angled triangle consists oi five parts — three sides, and two angles. Let any one of these parts be called a middle j)art, then two otlier parts will lie adjacent to this part, and two opposite to it, that is, separated from it by two other parts. For instance, take equation (11), and call c the middle part, then A' and a will be adjacent parts, and C and h' opposite parts. Again, take a as d^ middle part, then c and C'will be adjacent parts, and A' and b' will be opposite parts ; and thus we may go round the triangle. Take any equation from (11) to (20), and consider the middle part in the first member of the equation, and we shall find that they correspond to these two invariable and comprehensive rules. 1 . The radius into the sine of the middle part equals the product of the tangents of the adjacent parts, 2. The radius into the sine of the middle part equals the product of the cosines of the opposite parts. TRIGONOMETRY. 187 These rules are known as Napier's Rules, because they were first brought forth by that distinguished mathematician, who was also the inventor of logarithms. We caution the pupil to be very particular to take the complements of the hypotenuse, and the complements of the oblique angles. OBLIQUE ANGLED SPHERICAL TRIGONOMETRY. The preceding investigations have had reference to right angled spherical trigonometry only; but the application of these prin- ciples cover oblique angled trigonometry also, for every oblique angled spherical triangle may be considered as made up of the sum or difference of two right angled spherical triangles. With this explanatory remark, we give, PROPOSITION 9. THEOREM. 3. In all spherical triangles, the sines of the sides are to each other , as the sines of the angles opposite to them. This was proved in relation to right angled triangles in theorem 2, and we now apply the principle to oblique angled triangles. Let ABC, be the triangle, and let CD be perpendicular to AB, or to AB pro- duced as represented in the margin. Then by theorem 2, we have, R : sin.^ (7=sin. J : sin. CD Also, . sm.CB : i2=sin. CD: sm.B. By multiplying these two proportions term by term, and leaving out the com- mon factor jR, in the first couplet, and the common factor sin. CD, in the second, we have, sm.CB : sin.^ C=sin.^ : sin.5. Q. E. D. C(yr, From the truth of this theorum, it follows, that the angles at the base of an isosceles triangle are equal, and that in every spherical triangle the greater angle is opposite the greater side. 188 SPHERICAL PROPOSITION 10. THEOREM I. In any spherical triangle, if an arc he let fall frma any angle to the opi^osUe side aa a hose, or to the base produced, the cosines of the other two sides mil be to each other as the cosines of the segments of the base. By the application of equation (8) to the last figure, we have, R COS. A C=s cos, AD cos. DC Similarly, . R cos.BC^=cos.DC cos.BD Dividing one of these equations by the other, omitting common factors in numeratprs and denominators, we have, cos.^ C COS. AD cos.BO^cos.BD Or, . cos.^C : cos.BO=cos.AD : cos.BD. Q. E. D, PROPOSITION!!. THEOREM 5. J^ from any angle of a spherical triangle, a perpendictdar he let fall on the base, or on the base produced, the tangents of the segments of the base will be to each other reciprocally proportional to the cotan- gents of the segments of the angle. By the application of equation (2) to the last figure, we have, R sm.CD=iB.n,AD cot. A CD Similarly, . R sm.CD=i^xi.BD cot.BCD Therefore, by equality, tan.^Z> cot.J[(7i?=tan.^i> cot.BCD Or, . tan.^i> : t&ji.BD=cot.BCD : cot.ACD. Q. E. D, PROPOSITION !2. THEOREM 6. The same construction remaining , the cosines of the angles cU the extremities of the segments of the base, are to each other as the sines of the segments of the opposite angle. Equation (7) applied to the triangle A CD, gives R cos.^=cos. CD sin.A CD (s) Also, . . i2cos.^=cos.Ci>sin.-B(7i> (t) TRIGONOMETRY. 189 Dividing equation (s) by (t), gives cos.^ sin.^ CJ) cos^~smlBa5 Or, . . cos.i? : cos.^=sin.J5 C?i> :sin.^(7i). Q. JS, D, PROPOSITION 13. THEOREM 7. The same constniciion remaining, the sines of the segments of the dose, are to each other as the cotangents of the adjacent angles. Equation ( 1 ), applied to the triangle A CD, gives B Bm.AD= tan. CD cot. A (s) Similarly, . B sm.£D= tan.OZ) cot.^ (t) Dividing (s) by (t), gives sin.^J9 cot.-4 mnJBD~''cotJB Or, . 8m.BD : sin.^i>=cot.5 : cot.-4. Q. E. D, PROPOSITION 14. THEOREM 8. The same construction remaining, the cotangents of the two sides are to each other as the cosineb of the segments of the angle. Equation (9), applied to the triangle ACD, gives B COS.A CD= Qoi.A C tan. CD {a) Similarly, . i2 cos.^(7i>=cot.J5(7tan.C7i> {t) Dividing («) by {t), gives coB.AC D _coi.AC cos.BGD'^cotBC Or, . cot AC : cot.J5C7= cos.^C7i> : cos.B CD. Q. E. D, Remark. The preceding theorems enable us to solve any spherical triangle, right angled or oblique, when any three of the *M? parts are given. But oblique angled spherical triangles we have thus far considered as composed of two right angled triangles ; and it is sometimes a little troublesome to select the theorems or equations which apply to the case in question. To remedy this 190 SPHERICAL inconvenience, we will at once seek a relation between the cosines and sines of an angle of any spherical triangle, and the sines and cosines of its sides. Therefore, we investigate the following propositions. PROPOSITION 15. PROBLEM. Investigate, snd show the relation between the cos we of an angle of a spherical triangle, and the sines and cosines of its sides. Let ABC be a spherical triangle, and CD a perpendicular from the angle C on to the side AB, or on to the side AB produced. Then, by proposition 10, th. 4, cos.^C : COS. (7-5=cos.^i> : cos.BD (1) When CD falls within the triangle, BD=(AB—AD) When CD falls without the triangle, BD=(AD—AB) Hence, . cos.-Bi>=cos.(-4i) — AB) Now, cos.(-4^ — AD)=cos.(AD — AB), because each of them is equal to cos. AB cos.-4i>4-sin.^jB sin. AD. (Plane trig. eq. 10.) This value of cos.BD, put in proportion (1), gives cos.AC : COS. OB=^cos. AD : cos.-4^ cos.^2)-j-sin.^^ sin.^i) (2) Dividing the last couplet of proportion (2) bycos.^J9, observing that . . . — ^7-=r=tan.-4i>, and we have cos.AD cos.AC : cos.CB=l : cos.^^+sin.^^ tan. AD (3) By applying equation (6) to the triangle A CD, taking the radius as unity, we have cos.-4=cot.-4C7tan.^i> (k) But, . tan.-4Ccot.^(7=l (eq. 5, plane trig.) (I) Multiply equation (k) hy tan. AC, observing equation (/), and we have . tan.^(7cos.^=tan.^2) Substituting this value of tan. AD, in proportion (3), we have cos. .4 C : cos. CB= 1 : cos.-4.5+ sin.-4jB tan.-4 C cos. A ( 4 ) TRIGONOMETRY. 191 Multiplying extremes and means, gives COS. CB=coa.A C cos.^-S+sin.^jB(cos.^ C tan. A C)co8.A But, . . tan. AC=: — '-t-?^* or, cos.^(7tan.^(7=sin.J[(7 cos.^ u Therefore, . cos.(75=cos.^(7cos.-4^+sin.-4^sin.-4(7cos.-4 „ . cos.CB — cos.^C7cos.^5 ,_^. ^ , .^ Hence, . cos.-4= :: — --jr—. — -—: (F) final result.* By processes perfectly similar, like theorems may be deduced for the angles B and C. If the sides opposite the angles A, B, and 0, be respectively represented by a, b, and c, the formula will be expressed thus : COS. a — cos.5 cos.c^ cos.^= cos.jB= sin.6 sin.c C0S.5 — cos. a cos.c COS. C- sm.a sm.c COS.C — COS. a C0S.5 sin.a sin.6 {S) * As this equation has been denominated" The fundamental formida of Spherical Trigonometry,''^ and as it is susceptible of a more geome- trical demonstration, we give the following, which we beUeve will be very acceptable to every lover of mathematical science. Let ABC be a spherical triangle, and O the center of the sphere. From the angle A, draw AD tangent to the arc AB, and AE tangent to the arc A C. OD and OE, drawn from the center of the sphere to the extremities of the tangents, are, of course, secants. OD is the secant of AB, and OE the secant of the arc AC. Because AD is a tangent, it is perpendicular to the radius OA. For the same reason AE is perpendicular to the same radius OA. But OA is the common intersection of the two planes A 0J5 and AOCy and hence, by definition 5, book 6, the angle DAE is the inclination of the two planes AOB and AOC, and is, therefore, equal to the spherical angle A. As is customary, let the side opposite to A be designated by a, opposite B by b, opposite C by c. 192 • SPHERICAL These formulas are not adapted to the use of logarithms ; and the use of natural sines and cosines would lead to tedious operations; we must, therefore, make some advantageous mutations, or the equations will be useless ; hence the following investigations : . In equation (35), plane trigonometry, we find 1 + cos .-4 = 2cos^^^ COS. a — cos.h cos.c Therefore, 2 cos.2|-4= 1 -f sin.b sin. c (sm.b sin.c — cos.b cos.c)4-cos.a , —i r—^- '- (m) sm.6 sm.c ^ But, . cos.(^-f-c)=cos.J cos.c — sin.c sm.b (9), plane trigonometry. By comparing this last equation with the second member of equation (m), we perceive that equation (m) is readily reduced to cos. a — cos(5-|-c) ^ sm.b sm.c Then, AD= tan.c, AE= tan.6, 0D= sec.c, 0E= sec.b. Designate DE by x, and observe that the angle BOC is measured by the arc JBC=a. Now, to the two plane triangles ODE and ADE, if we apply equa- tion (?n), proposition 8, plane trigonometry, we shall have sec.2 c-l-sec.2 b — x^ cos.a= ! 2 sec.^ sec.o . tan.2c4-tan.2 6 — x^ cos.A= ■— — 2 tan.c tan.o Clearing these two equations of fractions, and subtracting the latter from the former, and observing, that for any arc, sec.^ — tan.^=R~ ; and if R is unity, as it is in this case, we shall have, 2 sec.c sec.Z> cos. a — 2 tan.c tan.& cos.A=2 Dividing by 2, and substituting the values of the secants and tan- gents from equations (4) and (5), plane trigonometry, Namely, . sec.= — , tan.=^* , we shall then have, cos. cos. cos. a sin.c sin.& cos.A ^ , S.C cos.b cos.c C0S.6 TRIGONOMETRY. 198 Considering (b+c) as one arc, and then making application of equation (18), plane trigonometry, we have, 2 C0S^:i--4 = ; r— : ^ sm.o sm.c But, . — - — = — a; and if we put S to rep- resent — - — , we shall have ^A sm,S sm.(S — a) cos^— = r-^-4 2 sm.o sm.c ^ ^ Isin.S sm.(S — a) Or, . . COS.— =-J . , : 2 ^ sm.6 sm.c The right hand member of this equation gives the value of the Clearing of fractions, transposing, and changing signs, will give BUi.o sin.& cos.A=cos.a — cos.<7 cos.& _, . . co%.a — cos.^ cos.ft Therefore, . . . cos.A= ; r-i 8m.6^ sm.o For the sake of the mathematical exercise, I will suppose we have the three sides of a spherical triangle, as follows : a=70° 4' 18", fe=59° 16' 23", and c=63° 21' 27", from which we require the angle A, and we have no other formula except the above equation, amA logarithms are not yet invented. From the table of natural sines and cosines, we find cos.«=0.34090 cos.6=0.61l91 sin.5=0.8791 cos.c=0.44840 sin.c=:0.8938 By the multiplication of decimals, retaining only five places, we find, COS.& cos.c=0.22953, and sin.& sin.c=0.76786 From cos.o . . 0.34890 Take cos.6 cos.c . 0.22953 0.76786)0. 1 1 137(0. 14506=cos. A By comparing this decimal with the table, we find it very nearly corresponds to 81° 40'. The true value of A is 81° 38' 20" 13 194 SPHERICAL cosine when the radius is unity. To a greater radius, the cosine would be greater ; and in just the same proportion as the radius increases, all the trigonometrical lines increase ; therefore, to adapt the above equation to our tables where the radius is R, we must write R in the second member, as a factor; and if we put it und r the radical sign, we must write R"^. For the other angles we shall have precisely similar equations ; That is A IRhm.Ssin.iS—a) COS.— =x/ r ^ / COS B_ jRhm.S sm.{ S—b) |-^/ sm.a sm.c Rhin.S sm.{S—c) sin.a sin.b {T) Formulas, for the sines of the angles, are obtained as follows : From equation (32), plane trigonometry, we obtain 2 sin.^^^=l — C0S.-4. Substituting the value of cos.-4, taken from equation {S)^ and COS. a — cos. 6 cos.c we have 2 sin.2i^=l. sin.5 sin.c (sin.5 sin.c+cos.fi cos.c) — cos.a sin. 6 sin.c But, cos.(ficrc)=sin.J.sin.c-|-cos.6 cos.c ( (10) plane trig.) This equation reduces the preceding one to cos.(5cr c) — cos.a sin .6 sin.c 2sm.2|^=- Considering {h c) as a single arc, and applying equation ( 1 8), plane trigonometry, we have ^ . /a+5 — c\ . ( a-\-c — b\ 2 sm?^A=^ But, Also, a-\-h — c__a-|-5+c 2 ~ 2 " 2 2 sin.6 sin.c c= S — c, if we put S- a-\-b-{'C : _ TRIGONOMETRY. 195 Dividing the preceding equation by 2, and making these sub- stitutions, we have, , . sm.(S — c)sin.(AS^ — h) , j* • -x [.\A= ^ . / . — ^ -f when radius is unity. Sin. sin.6 sm.c When radius is i2, we have sm.^A=^j ^ s in.(/S— c)sin.(/S— 6) sin.5 sin.c «. .1 , . , T, /i22sin.(>S'--a)sin.(AS^— c) Similarly, sin4^=^ — ^ ^ ^ ^ sin.a sin.c And, '^ ^ sin.a sin.6 sin («7) To apply to our tables, H? must be put under the radical sign. We shall show the application of these formulas, and those in equations {S), hereafter. From (30), plane trigonometry, we have sin.^=2 sin.^^ cos.-J^ Squaring, . sin.^u4=4 sin.^-J-^ cos.^-J^^ {t) Square the first equation in {T), and multiply it by the square of the first equation in ( U)y and four times their product is 4 sin.»M COS.' iA= ^ "'"'"^ Bin.( ^-a)sin.( ^-i)sin.( ,S^ ) sin.^6 sin.^c Comparing the first member with equation (^), we have 8in. sin.^6 sin.^c By operating in the same manner with the several equations in {T) and ( ?7), we have _ 4 i?*sin.iS^sin.(AS' — a)sin.(^ — h)&m.(S — c) , . £— V n . i ^ (V) oi-r* « n c•^■n * /» » ' Sin. sm.' a sin." c The numerators of the second members of (w) and (v), are the same ; and if we divide (w) by (v), and extract the square root, we shall have sin.-4 sin.a sin.jB sin.6 Or, . . sin.^ : sin.u4=sin.J : sin.a, a truth that was demonstrated in proposition 9, spherical trigonometry. 196 • SPHERICAL We have again demonstrated it in this manner, to show that equation (J^), from which all the preceding equations arose, is really the fundamental equation of spherical trigonometry. A spherical triangle consists of six parts ; three sides, and three angles ; and there are certain relations existing between them ; but the combinations of these relations have their limits ; and when we have gone through these relations, if we continue to combine equations, we shall only fall on truths previously demonstrated, and this is exemphfied by our last operations. APPLICATION. SOLUTION OF RIGHT ANGLED SPHERICAL TRIANGLES. 1. At a certain time the sun's longitude was 40° 29' 30", and the obliquity of the ecliptic 23° 27' 32". What was the declination 1 Ans. 14° 68' 62". This example presents a right angled spherical triangle, A5C. The hypotenuse, AC=40° 29' 30", and the angle A=23° 27' 32", and the side, CB, is required. By our system of notation, AC=b, BC=a. This can be solved by equation (3) or (13), which are essentially the same ; that is. JR 8in.a=8in.2> sin. A 8in.&=sin.40° 29' 30" . 9.812470 8in.A =sin.230 27' 32" . 9 .599985 Ans, sin.a=sin.l4° 68' 62" . 9.412455 Rejecting 10 in the index, is the same as dividing by the radius, as the equation requires. 2. At a certain time, the difference between the longitude of the sun and moon, was 76° 10' 20", and the moon's latitude, at the same time, was 5° 9' 12" north. What was the true angular distance between the centers of the sun and moon ? Ans. 76° 13' 45". This problem presents a right angled spherical triangle, whose base A5=76o 10' 20", and perpendicular 5C=5° 9' 12". The hypotenuse AC, is required. Equation (8) or (18) solves it. c=76° 10' 20" COS. . 9.378406 a = 5° 9' 12" COS. . 9.998241 6=76° 13' 45" COS. . 9.376647 TRIGONOMETRY. 197 3. An astronomer observed the sun to pass his meridian on a certain day when his astronomical clock gave 2 h. 9 min. 33 sec. for the sideriai time, and the altitude was such as to give the declination of 13° 6' 6" north. What was the sun's longitude, and what was the obliquity of the ecliptic ] Ans. Lon. 34° 39' 46". Obliq. eclip. 23° 27' 26". This problem presents a right angled spherical triangle, giving its base and perpendicular, and demanding the hypotenuse, and the angle at the base. 2 h. 9 m. 33 s.=c=32° 23 16 cos. . 9.926671 a=13 6 6 COS. . 9. 988576 i>=34 39 46 COS. . 9.916146 To find A, we apply equation (3) or (13), as they are one and the same. Rsm.a . . . 19.364869 sin.ft (subtract) . 9. 754918 . A=23°27'26" . . 9.699951 At a certain time the sun's longitude will be 160° 33' 20", and the obliquity of the ecliptic 23° 27' 29". Required its right ascension and declination. Ans. R. A. 162° 37' 28"; Dec. \\^ 17' 7"N. Observation. This problem presents a right angled spherical triangle, whose base and hypotenuse are each greater than 90° ; and in cases of this kind, let the pupil ob- serve, that the base is greater than the A^o- ie7tw.se, and the oblique angle opposite the base, is greater than a right angle. In all cases, a triangle and its supplemental triangle, make a lun£. It is 180° from one pole to its opposite, whatever great circle be traversed. It is 180° along the equator AjBA', and also 180° along the ecliptic ACA'; and the lune always gives two triangles; and when the sides of one of them are greater than 90°, we take its supplemental triangle, as in this case we operate on the triangle A'CB. But A'C is greater than A'B; therefore, AB is greater than AC. The angle A'CB is less than 90°; therefore, ACB is greater than 90°, because the two angles together make two right angles. These facts are technically expressed, by saying, that the sides and opposite angles are of the same affection*; and if the two sides of a right angled spherical triangle are of the same affection, the hypotenuse * Same affection : that is, both greater, or both less tlian 90*^. Vffe cut i^ection : the one greater, the other less than 90°. 196 SPHERICAL will be less than 90°; and of different affection, the hypotenuse will be greater than 90°. If, in every instance, we make a natural construction of the figure and use common judgment, it will be impossible to doubt whether an arc must be taken greater or less than 90°. We now solve the triangle A'CB, A'C=29° 26' 40". To find BC. Eq. (3) or (13). h sin. 29° 26' 40' A sin. 23° 27' 29" a sin. 11° IT T To find A'Bf we use equation (1) or (11), thus : ten. 11° 17' 7" . 9.300016 10.362674 9.691594 ^.699984 9.291578 cot. 23° 27' 29' c sin. 27° 22' 32' 180 9.662590 A5=152° 37' 28' We select the following examples to exercise the pupils in right Angled spherical trigonometry: 1. In the right angled spherical triangle AJ5C, given AB 118° 21' 4", and the angle A 23° 40' 12", to find the other parts. Am, AC 116° 17' 55", the angle C 100° 69 26", and 5C 21° 6' 42". 2. In the right angled spherical triangle ABC, given AB 53° 14' 20", and the angle A 91° 25' 63", to find the other parts. Ans. AC 91° 4' 9", the angle C 53° 15' 8", and J5C 91° 47' 11". 3. In the right angled spherical triangle ABC, given AB 102° 50' 25", and the angle A 113° 14' 37", to find the other parts. Ans. AC 84° 51' 36", the angle C 101° 46' 57", and BC 113° 46' 27". 4. In the right angled shpherical triangle ABC, given AB 48° 24' 16", and BC 59° 38' 27", to find the other parts. Ans. AC 70° 23' 42", the angle A 66° 20' 40", and the angle C 52° 32' 55". 6. In the right angled spherical triangle ABC, given AB 151° 23' 9", and J5C 16° 35' 14" to find the other parts. Ans. AC 147° 16' 61", the angle C 117° 37' 21", and the angle A 31° 52' 60". TRIGONOMETRY. 199 6. In the right angled spherical triangle ABC, given AB 73° 4' 31", and AC 86° 12' 15", to find the other parts. Ans. BC 76° 61' 20", the angle A 77° 24' 23", and the angle C 73° 29' 40". 7. In the right angled spherical triangle ABC, given AC 118° 32' 12", and AB 47° 26' 35", to find the other parts. Ans. BC 134° 66' 20", the angle A 126° 19' 2", and the angle C 66° 68' 44". 8. In the right angled spherical triangle ABC, given AB 40° 18' 23", and AC 100° 3' 7", to find the other parts. Ans. The angle A 98° 38' 53", the angle C 41° 4' 6", and 5C 103° 13' 52". 9. In the right angled spherical triangle ABC, given A C 61° 3' 22", and the angle A 49° 28' 12", to find the other parts. Ans. AB 49° 36' 6", the angle C 60° 29' 19", and BC 41° 41' 32". 10 In the right angled spherical triangle ABC, given AB 29° 12' 60", and the angle C 37° 26' 21", to find the other parts ] Ans. Ambiguous ; the angle A 65° 27' 58" or its supplement, A C 63° 24' 13" or its supplement, BC 46° 55' 2" or its supplement. 11. In the right angled spherical triangle ABC, given AB 100° lO' 3", and the angle C 90° 14' 20", to find the other parts. Ans. Ambiguous ; AC 100° 9' 55" or its supplement, BC 1° 19' 53" of its supplement, and the angle A 1° 21' 8" or its supplement. 12. In the right angled spherical triangle ABC, given AB 54° 21' 35", and the angle C 61° 2' 15", to find the other parts. Ans. Ambiguous; BC 129° 28' 28" or its supplement, AC 111° 44' 34" or its supplement, and the angle A 123° 47' 44" or its supplement. 13. In the right angled spherical triangle ABC, gWen AB 121° 26' 25", and the angle C 111° 14' 37", to find the other parts. Ans. Ambiguous; the angle A 136° 0' 3 'or its supplement, A C 66°' 15' 38" or its supplement, and BC 140° 30' 56" or its supplement. The solution of right angled spherical tri- angles includes, also, the solution of quad- rantal triangles, as may be seen by inspecting the adjoining figure. WTwn we have one quadrantal triangle, we have four, which JiU up the whole hemisphere. To effect the solution of either of the four quadrantal triangles APC, AP'C, A' PC, or 200 SPHERICAL A'P' C, it is sufficient to solve the small right angled spherical triangle ABC. To the half lune AP'B, we add the triangle ABC, and we have the quadrantal triangle AP' C; and by subtracting the same from the equal half lune APB, we have the quadrantal triangle PA C. When we have the side, A C, of the same triangle, we have its sup- plement, A' C, which is a side of the triangle A' PC, and of A'P'C. When we have the side, CB, of the small triangle, by adding it to 90°, we have P' C, a side of the triangle A'P' C; and subtracting it from 90°, we have PC, a side of the triangle APC, and A' PC. EXAMPLES. \. Ina quadrantal triangle, there are given the quadrantal side, 90°, a side adjacent, 42° 21', and the angle opposite this last side, equal to 36° 31'. Required the other parts. By this enumeration we cannot decide whether the triangle APC or AP'C, is the one required, for A C=42° 21' belongs equally to both triangles. The angle APC=AP'C=Z6° Z1'=AB. We operate wholly on the triangle ABC. To find the angle A, call it the middle part. Then, R cos.( CAB)=R sin.PA C=cot.A C.t&n.AB cot.AC= 42°21' . 10.040231 tan.A5= 36 31 . 9.869473 cos.CAJ?= 35 40 61 9.909704 90 PAC= 64 19 9 P'AC=125 40 61 To find the angle C, call it the middle part. R COS. A CjB=sin. CAB cos. AS sm.CAB= 35° 40 51" 9.766869 cos.AjB= 36 31 . 9. 905085 cos.ACB= 62 2 45 9.670954 180 A CP=A'CP'=:1 17 67 15 TRIGONOMETRY. ml To find the side J?C, call it the middle part, R sm.BC=tSin.AB cot. A CB, tan.AJ?= 36° 31' 0" 9.869473 cot.ACB= 62 2' 45" 9.724835 sm,BC= 23 8' 11" 9.694308 90 PC= 66 61' 49" P'C=113 8' 11" We now have all the sides, and all the angles of the four triangles in question. 2. In a quadrantal spherical triangle, having given the quadrantal side, 90°, an adjacent side, 115°, 09', and the included angle, 116° 66', to find the other parts. This enunciation clearly points out the particular triangle A' PC. A'P'=90°; and conceive A'C=115° 09'. Then the angle P'A'C=115°65'=P'i). From the angle P'A'C take 90° or P'A'B, and the remainder is the angle OA'D=BA C =25° 55'. ,We here again operate on the triangle ABC. A'C taken from 180°, gives . . 64°6l'=AC To find BC, we call it the middle part. R sin. jBC=sin. AC sin.^AC. sin.AC= 64° 61' . 9.956744 sin.5AC= 25 55' . 9.640544 sin.jBC= 23 18' 19" 8.597288 90 P'C=113 18' 19" To find AB we call it the middle part. R sin.AJB=tan.BC cot.BAC. tan.J?C= 23° 18' 19" cot.5AC= 26 65' . Bm.AB= 62 26' 8' 180 9.634251 9.313 423 9.947674 A'B=ll1 33' 62"=the angle A'P'C 202 SPHERICAL To find the angle C, we call it the middle part. R cos.C= =cot.AC tam.BC cot. A C= : 64° 61' . 9.671634 t3Ln.BC= : 23 18' 19" . 9.634251 COS. C= : 78 180 19' 63" 9.306885 P'CA'= :101 40' 7" Thus we have found the side P'C= 11 3° 18' 19" ) The angle A'P'C= 117° 33' 62" } Ans. " P'CA'=10lo 40' 7" > 3. In a quadrantal triangle, given the quadrantal side, 90°, a side adjacent, 67° 3', and the included angle, 49° 18', to find the other parts. Ans. The remaining side is 63° 6' 46", the angle opposite the quad- rantal side, 108° 32' 27", and the remaining angle, 60° 48' 54". 4. In a quadrantal triangle, given the quadrantal side, 90°, one angle adjacent, 118° 40' 36", and the side opposite this last mentioned angle, 113° 2' 28", to find the other parts. Ans. The remaining side is 64° 38' 67", the angle opposite, 61° 2' 36", and the angle opposite the quadrantal side is 72° 26' 21". 5. In a quadrantal triangle, given the quadrantal side, 90, and the two adjacent angles, one 69° 13' 46", the other 72° 12' 4", to find the other parts. Ans. One of the remaining sides is 70° 8' 39", the other is 73° 17' 29", and the angle opposite the quadrantal side is 96° 13' 23". 6. In a quadrantal triangle, given the quadrantal side, 90°, one adja- cent side, 86° 14' 40", and the angle opposite to that side, 37° 12' 20", to find the other parts. Ans. The remaining side is 4° 43' 2", the angle opposite, 2° 61' 23", and the angle opposite the quadrantal side, 142° 42' 2". 7. In a quadrantal triangle, given the quadrantal side, 90°, and the other two sides, one 118° 32' 16", the other 67° 48' 40", to find the other parts — the three angles. Ans. The angles are 64° 32' 21", 121° 3' 40", and 77° 11' 6"; the greater angle opposite the greater side, of course. 8. In a quadrantal triangle, given the quadrantal side, 90°, the angle opposite, 104° 41' 17", and one adjacent side, 73° 21' 6", to find the other parts. Ans. The remaining side is 49° 42' 18", and the remaining angles are 47° 32' 39", and 67° 66' 13". TRIGONOMETRY. 203 OBLiaUE ANGLED SPHERICAL TRIGONOMETRY. All cases of oblique angled spherical trigonometry may be solved by right angled trigonometry, except two ; because every oblique angled spherical triangle is composed of the sum or difference of two right angled spherical triangles. When a side and two of the angles, or an angle and two of the sides are given, to find the other parts, conform to the following directions : Let a perpendicular be drawn from an extremity of a given side, and opposite a given angle or its supplement ; this will form two right angled spherical triangles ; and one of them will have its hypotenuse and one of its adjacent angles given, from which all its other parts can be computed ; and some of these parts will become as known parts to the other triangle, from which all its parts can be computed. To facilitate these computations, we here give a summary of the practical truths demonstrated in the foregoing propositions. 1. The sines of the sides of spherical triangles are proportional to the sines of their opposite angles. 2. The sines of the segments of the base, made by a perpendicular from the opposite angle, are proportional to the cotangents of their adjacent angles. 3. The cosines of the segments of the base are proportional to the cosines of the adjacent sides of the triangle. 4. The tangents of the segments of the base are proportional to the the tangents of the opposite segments of the vertical angles. 6. The cosines of the angles at the base, are proportional to the sines of the corresponding segments of the vertical angles. 6. The cosines of the segments of the vertical angles are propoi tional to the cotangents of the adjoining sides of the triangle. The two cases in which right angled triangles are not used, are, 1st. When the three sides are given to find the angles ; and, 2d. When the three angles are given to find the sides. The first of these cases is the most important of all, and for that reason great attention has been given to it, and two series of equations, (T) and (I/), have been deduced to facilitate its solution. We now apply the following equation to find the angle A, of the triangle AJB C, whose sides are a, b, c. a==70° 4' 18". b=6Z° 21' 27". 00:590 iQ' 23". a is opposite A, b is opposite B, and c is opposite C« 204 * SPHERICAL IR^ sin.S sin.(iS— a) ^ N sm.o sm.c We write the second member of this equation thus : showing four distinct logarithms. R The logarithm corresponding to -; — r is the sin.ft subtracted from R 10 : and -: — is the sin.c subtracted from 10, which we call sin.c 8tn.complement. BC=a= 70° 4' 18" AB==c= 59° 16' 23" sin.com. 0.065697' AC=b= 63° 21' 27" sin.com. 0.048749 2)192 42 8 5:= 96 21 4" sin. 9.997326 8—a= 26 16 46 sin. 9.646158 2 )19.757930 iA= 40 49 10 cos. 9.878965 2_ A= 81 38 20 When we apply the equation to find the angle A, we write a first, at the top of the column ; when we apply the equation to find the angle B, we write b at the top of the column. Thus, To find tlie angle B fl22 sm.S 8m.i&—b) sin.fl sm.c .i5=V- h= 63° 21' 27" c= 59 16 23 sin.com. . .065697 a= 70 4 18 sin.com. . .026857 2)192 42 8 S= 96 21 4 sin. . . 9.997326 S—l= 32 59 37 sin. . . 9.736034 2) 19.825874 iB= 35 4 49 cos. . 9.912937 2 B= 70 9 38 TRIGONOMETRY. 205 By the other equation in formula (T), we can find the angle C; but, for the sake of variety, we will find the angle C by the application of the third equation in formula (t/). /jR2 sin.(S— 6) sin.(S— a) sin. '^ — — Mt-^V 8in.i6in.a =^/(sL)(sL)^-(*-^) «-(*-: c— 590 16' 23" , 0= 70 4 18 sin.com. .026817 6= 63 21 27 sin.com. .048479 2)192 42 8 8= 96 21 4 8— a— 26 16 46 sin. . 9.646168 /S— 5= 32 69 37 sin. . 9.736034 2)19.467768 JC= 32° 23' 17" sin. 2 . 9.778879 C= 64 46 34 To show the harmony and practical utility of these two sets of equations, we will find the angle A, from the equation sin -iW(sl)(i.c)^-(^>^-(^ a= 70 4' 18" 6= 63 21 27 sin.com. c— 69 16 23 sin.com. .048749 .066697 2)192 42 8 «= 96 21 4 8—b= 32 69 37 sin. . iS— c= 37 4 41 sin. . . 9.736034 . 9.780247 2)19.630727 iA= 40° 49' 10" sin. . 2 . 9.816363 A= 81 38 20 2. In a spherical triangle ABC, given the angle A, 38° 19' 18", the angle J5, 48° 0' 10", and the angle C, 121° 8' 6", to find the sides a, b, c. Apply proposition 5, spherics. 206 SPHERICAL i4= 38° 19' 18" supplement 141° 40' 42" B= 48 10 supplement 131 69 50 C=121 8 6 supplement 58 51 64 We now find the angles to the spherical triangle, whose sides are these supplements. Thus, UP 40' 42" 131 59 50 sin.com. * .128909 58 61 54 sin.com. .067651 2)332 32 26 166 16 13 sin. 9.375376 24 36 31 sin. 9.619253 2)19.191088 66° 47' 37i' ' cos. . 9.695543 2 angle =133 35 15 supp. = 46 24 45=a of the original triangle. In the same manner we find 6=60° 14' 25" c=89° 1' 14" EXAMPLES EOK EXEECISE. 1. In any triangle, ABC, whose sides are a, b, c, given 6=118^' 14", c=120° 18' 33", and the included angle A=27° 22' 34", to find tlie other parts. A71S. a=23° 67' 13", angle B—dl^ 26' 44", and C=102° 6' 54". 2. Given A=81° 38' 17", 5=70° 9' 38", and C=64° 46' 32", to find the sides a, b, and c. Ans, a=70° 4' 18", 6=63° 21' 27", and c=59° 16' 23". 3. Given the three sides a=93° 27' 34", 6=100° 4' 26", and c=96° 14' 50", to find the angles A, B, and C. Ans. A=94° 39' 4", 5=100° 32' 19", and C=96° 68' 36". 4. Given two sides, 6=84° 16', c=81° 12', and the angle C=80° 28', to find the other parts. Ans. The result is ambiguous, for we may consider the angle B as acute or obtuse. If the angle B is acute, then A=97° 13' 45", 5=83° 11' 24", and a=96° 13' 33". If B is obtuse, then A=21° 16' 44", 5=96° 48' 36", and a=21° 19' 29" * The sine complement of 131^ 59' 50", is the same as the sine complement of 48° 0' 10". TRIGONOMETRY. 207 6. Given one side, cz=64° 26', and the angles adjacent, A=49°, and 5=52°, to find the other parts. Ans. 6=45° 56' 46", a=43° 29' 49", and C=98° 28' 5". 6 Given the three sides, a=90°, 6=90°, c=90°, to find the angles A, B, and C. Ans, A=90°, J5=90°, and C=90°. 7. Given the two sides, a=77° 25' 11", and c=128° 13' 47", and the angle C, 131° 11' 12" to find the other parts. Ans. 6=84° 29' 24", A=69° 14', and 5=72° 28' 46'. 8. Given the three sides, a, 6, c, a=68° 34' 13", 6=59° 21' 18, and c=112° 16' 32", to find the angles A, J3, and C. Ans. A=45° 26' 12", 5=41° 11' 6", C=134° 54' 27" APPLICATION. Spherical trigononometry becomes a science of incalculable importance in its connection with geography, navigation, and astronomy; for neither of these subjects can be understood without it ; and to stimulate the student to a study of the science, we here attempt to give him a glimpse at some of its points of application. Let the lines in the an- nexed figure represent cir- cles in the heavens above and' around us. Let Z be the zenith, or the point just overhead, Hch the horizon, PZII the meri- dian in the heavens, P the pole of the earth's equator ; then Ph is the latitude of the observer, and PZ is the co.latitude. Qcq is a portion of the equator, and the dotted, curved line, m!S' S, parallel to the equator, is the parallel of the sun's declination at some particular time ; and in this figure the sun's declination is supposed to be north. By the revolution of the earth on its axis, the sun is apparently brought from the horizon, at Sy to the meridian, at m : and from thence it is carried down on the same curve, on the other side of the meridian ; and this apparent motion of the sun (or any other celestial body) makes angles at the pole P, which are in direct proportion to their times of description. 208 SPHERICAL The apparent straight line, Zc, is what is denominated, in astro- nomy, ^e prime vertical; that is, the east and west line through the zenith, passing through the east and west points in the horizon. When the latitude of the place is north, and the declination is also north, as is represented in this figure, the sun rises and seta on the horizon to the north of the east and west points, and the distance is measured by the arc cS, on the horizon. This arc can be found by means of the right angled spherical triangle cqSy right angled at q. Sq is the sun's declination, and the angle Scq is equal to the codatitvde of the place ; for the angle Pch is the latitude, and the angle Scq is its complement. The side cq^ a portion of the equator, measures the angle cPq^ the time of the sun's rising or setting before or after six, apparent time. Thus we perceive that this little triangle cSq, is a very important one. When the sun is exactly east or west, it can be determined by the triangle ZPS'; the side PZ is known, being the co.latitude; the angle PZS' is a right angle, and the side PS' is the sun's polar distance. Here, then, is the hypotenuse and side of a right angled spherical triangle given, from which the other parts can be computed. The angle ZPS' is the time from noon, and the side ZS' is the sun's zenith distance at that time. FORMULA FOR TIME. The most important problem in navigation is that of finding the time from the altitude of the sun, when the sun's declination and the latitude of the observer are given. This problem will be un- derstood by the triangle PZS. When the sun is on the meridian, it is then ap- parent noon. When not on the meridian, we can de- termine the interval from noon by means of the tri- angle PZS; for we can know all its sides ; and the angle at Py changed into time at the rate of 15° to TRIGONOMETRY. 209 one hour, will give the time from apparent noon, when any par- ticular altitude, as TS^ may have been observed. FS is known by the sun's declination at about the time ; and PZ is known, if the observer knows his latitude. Having these three sides, we can always find the sought angle at the pole, by the equations already given in formulas (T), or {U)\ but these formulas require the use of the co.latitude and the CO. altitude, and the practical navigator is very averse to taking the trouble of finding the complements of arcs, when he is quite certain that formulas can be made, which comprise but the arcs themselves. The practical man, also, very properly demands the most concise practical results. No matter how much labor is spent in theoriz- ing, provided we arrive at practical brevity ; and for the especial accommodation of seamen, the following formula for finding time has been deduced. From the fundamental equation of spherical trigonometry, taken from page 191 we have, cos.ZS—cos.PZ coa.PS cos.P=- sin.PZ sin.P/S' Now, in place of cos. ZS, we take sm. ST, which is, in fact, the same thing, and in place of cos. PZ, we take sin.lat., which is also the same. In short, let A=: the altitude of the sun, L= the latitude of the observer, and D= the sun's polar distance. r^, _. sin.^ — sin.Z cos.D ihen, . . cos.P= ~ — ; — — cos./y sm. JJ But, . 2 sin.2 iP=\ — cos.P (See eq. 32, page 143.) sin.^ — sin.Z cos. D Therefore, 2 sin.^ iP=l cos.L siu.D (cos.Z sin.Z>-}-sin.Z cos.2))— -sin.-4 cos.Z sin.D _sin. (L-]rD) — sin.^ cos.Z sin.i> 14 210 SPHERICAL Considering (L-^-D) as a single arc, and applying equation (16), plane trigonometry, we have, after dividing by 2, sm.2 iF=z r-^-n COS.// sm.lJ L+D-A L+D+A ' , ., J)ut, • = A and if we assume ^_L-\-D-\-A, we shall have. 2 . „,„ cos-zS^sinY/S' — A) sm.« iP= ^~\— — -^ ■* cos.// sm.x> ^ -ID hos.Ssm(S — A) Or, sm. iP=\ -~i — =—^ ^ N />ns /^ sm 7) cos.Z sin.i> This is the final result, when the radius is unity, and when the radius is greater by B, then the sin. -JP, will be greater by M; and, therefore, the value of this sine, corresponding to our tables is. sm •*^=>/(3l7-)(s-i^)«°^-'^^'"(^-^) This equation is known as the sailor's formula for time, and a very concise and beautiful formula it is ; it is used by thousands who have little knowledge of how it is obtained, or who know little of the amount of science there is wrapt up in it. When the observer has logarithmic tables that contain secants and cosecants, the above equation can be modified. Because, sec.Z= =: andcosec.i>=-^ — ^ cos.// sm.X' (See equations, plane trigonometry, page 138.) Therefore, sin.|P=^( '-^ ] ( -^^ ) cos. ^ sin. (^S'— J) Here, then, we have /o«r distinct logarithms to be added together and divided by 2, which is extracting^^quare root. TRIGONOMETRY. 211 The first logarithm is the secant of the latitude, diminished by the index 10 ; the second is the cosecant of the polar distance, diminished by the index 10 ; the third is the cosine of the half sum of altitude, latitude, and polar distance ; and the fourth is the sine of an arc, found by diminishing this half sum by the altitude. Navigators retain this formula in memory by the following words : Altitude — latitude — polar distance — half sum — remainder; secant — cosecant — cosine — sine. EXAMPLE. In latitude 39° 6' 20" north, when the sun's declination was 12° 3' 10", north, the true altitude* of the sun's center was observed to be 30° 10' 40", rising. What was the apparent time ? Alt. 30° 10' 30" . Lat. 39 6 20 cos.com. .110146 P.D. 11 bQ 60 sin.c !om. .009680 2)147 13 40 S= 73 36 60 cos. . 9.460416 (S—A)^ 43 26 20 sin. . 9.837299 2)19.407541 30 22 6 2 sin. 9.703770 P= 60 44 10 This angle, converted into time, at the rate of 15° to one hour, or 4 minutes to 1°, gives 4h. 2m. 56s. from apparent noon ; and as the sun was rising, it was before noon, or 7h. 67m. 4s. A. M If to this the equation of time were given and applied, we should have the mean time ; and if such time were compared to a cluck or watch, we could determine its error. A good observer, with a good instrument, can, in this manner, determine the local time within 4 or 6 seconds. * The instrument used, the manner of taking the altitude, its correction for refraction, seraidiameter, and other practical or circumstantial details, do not belong to a work of this kind, but to a work on practical astronomy of navigation. 212 SPHERICAL The great importance of determining the exact time, at sea, is to determine the longitude, which is but the difference of the local time between the observer's meridian and the assumed prime meridian. A timepiece, of nice and delicate construction, called a chrono- meter, by its rate of motion and adjustment, will show the time at Greenwich, or at any other known meridian to which it refers ; and this time, compared with an observation on the sun, will determine the amount of difiference in local times, which is, in substance, longitude. The same triangle, FZS, gives the bearing of the sun, which is is called its azimuth ; that is, the angle FZS is the azimuth from the north, and its supplement, RZS, is its azimuth from the south. This is the true bearing ; and if the bearing per compass is the same, then the compass has no variation ; if different, the amount of difference gives the amount of the variation of the compass. HOW TO MANAGE A LOCAL SOLAR ECLIPSE. We shall touch this subject only so far as to show the applica- tion and utility of spherical trigonometry. The angular semidiameter of the sun is about 1 5', and that of the moon, about the same ; and, of course, when an eclipse of the sun commences or ends, the apparent distance between the sun and moon cannot be greater than about 32', or a little more than half a degree. The nautical almanac, or the astronomical tables, will give us the time when the sun and moon fall into line on the same meridian of riff hi ascension, and give us, also, their dift'erence in declinations, at the same time, together with all the other necessary elements, such as semidiameters, horizontal parallax, hourly motions, (fee. Now let us take the time when the moon is in conjunction with the sun in right ascension, and demand the apparent distance between the centers of the sun and moon, as seen from any particular locality. By the time as given in the nautical almanac, we know the sun's distance from the local meridian, either east or west. TRIGONOMETRY. 21& Look at the last figure. Let S represent the position of the Eun*s center, F the pole, and Z the zenith of the observer. Then, in the triangle ZFS, we know the two sides, ZF and FS; and from the apparent time, we know their included angle, ZFS. The declination of both sun and moon is also given in the nau- tical almanac, corresponding to this time ; and their diflference gives the space which we represent by Sm, on our figure. From the triangle FZm (two sides and angle inclnded), compute Zm and the angle ZmF, The efiFect of parallax is to depress the body in a vertical direc- tion ; and if m is its true place, as seen from the center of the earth, n may represent its apparent place, as seen by the observer, whose zenith is Z. The arc mn is computed from the horizontal parallax, by the following proportion, p representing the lunar horizontal parallax. Had. : cos. 3) app.altitude =p : mn. The angle Smn=ZmFy and the angle ZmF is computed from the triangle FZm. Now, the triangle Smn is always very small ; the sides are never more than a degree in length, and are generally much less ; and it therefore may be regarded as a plane triangle, with two sides, Sm and mn, and the angle Smn, between them, given. From these data we can compute the distance between S and n; and if that distance is less than the sum of the semidiame- ters of the sun and moon, the sun must then be in an eclipse — otherwise it is not. But whether the distance between ;S^ and n is less, equal, or greater than the semidiameters of the sun and moon, by it the computer can assume an approximate time for the beginning or end of the eclipse, as the case may be. In case the computer wishes to compute the apparent distance between sun and mC=2cos.4 AI>=2, and BC=2 sin.(a— 5) These values substituted above, and we have 2 sin.a 2 cos.5=4 sin.(a — b)-]-2 sin.b 2 cos.o Dividing by 4, transposing, &c.. And sin.(a — 5)=sin.a cos.5 — sin.6 cos.a Again, let the arc AC=2a, the arc CB—2b; then the arc ACB=2(a-\-b), And the chord AB=2 s]n.{aA-b) ) A C'=2 sin a i BD=2 cos.{a-\-b) ] J)C=2 cos.a ) AD=2, and ^(7=2 sin.J Substituting these values in equation (2), we have, 2 cos.a 2 sin.(a-|-6)=4sin.6-|-2 sin.a 2 cos.(a-fi) Dividing by 4, cos.a sin.(a+5)=sin.6-(-sin.a cos.(a-|-^) To demonstrate the truth of equation (10), we use the last figure, conceiving the arc ^C to be 2a, the arc BD to be 2b. TRIGONOMETRY. 219 Then the arc BC will be measured by (180°— 2(a+5) ); its half will therefore be measured by 90° — {a-\-b). But, . 2 sin.(90°— a4-i)=2 cos.(a+i)=^C On this hypothesis, The chord ^(7=2 sin.a ) *, i>^=2 sin.J ) C7i>=2 cos.a f ^^^^' AB=:^ cos.b f AI)=2, and 5(7=2 cos.(a4-i) Substituting these values in equation (2), we have 2 cos.J 2 cos.a=4 cos.(a+^)+2sin.a 2sin.d Dividing and transposing, cos. (a4-i)= cos.a cos .6 — sin.a sin. J To demonstrate equation (10). Draw the diameter AD, and on one side of it take the arc AB=2a, and on the other side take the arc DU=2b. Join BD, AE, and BE. From B, draw BCF through the center of the circle ; then the arc DEF = 'the arc AB, and EF is the difference of the arcs AB and DE; it is therefore measured by 2(a — 6). Now, in the quadrilateral ABDE, we have AJD'BE=AB'DE+DB'AE AB=2 sin.a ) . , J)E=2 sin.b ) BD=2 cos.a ] ^^^°' AE=2 cos.b \ AD=2, and BE=2 cos.(a—b) These values, substituted in the last equation, will give 4 cos.(a — 6)=2 sin.a 2 sin.5-|-2 cos.a 2 cos.b cos. (a — 6)=sin.a sin.J+cos.a cos.6 PROBLEMS FOR EXERCISE. A 1. Show, yeome/nca/fy, that rad.*(rad.4'COS.w4)=2 cos^ — ; that rad.*(rad — cos.A)=2 sin^«-; that rad**sin.2-4s=2 sin.-4»cos.<4/ 220 APPENDIX TO 2. Prove that tan.-4+tan.J5= — ^ ^ {, , radius beinff unity. cos.^'cos.^ ^ ^ 3. Demonstrate, geometrically^ that rad.'sec.2^=tan..4 tan. 2^ 4. Show that in any plane triangle, the base is to the sum of the other two sides, as the sine of half the vertical angle is to the cosine of half the difference of the angles at the base. 5. Show that the base of a plane triangle is to the difference of the other two sides, as the cosine of half the vertical angle is to the sine of half the difference of the angles at the base. 6. The difference of two sides of a triangle, is to the difference of the segments of a third side, made by a perpendicular from the opposite angle, as the sine of half the vertical angle is to the cosine of half the difference of the angles at the base ; required the proof. NOTE. When we give our attention to the relations existing between the arc of a circle and its sine, cosine, and tangent, it becomes very desir- able to find some law which will invariably and unconditionally nume- rically connect the arc with its trigonometrical lines ; and the object has been accomplished, though not in as elementary a manner as is desirable for a work like this. In the calculus the process is clear and simple ; but simple as it may be, the reader must first understand the calculus before it can he even comprehensible to him. We give the following investigation, independent of the calculus, taken from the French works of Legendre, with our own modifica- tions and illustrations. By a little careful study, any one can thoroughly comprehend it, who is familiar with algebraic equations, and under- stands the binomial theorem, LEMMA. If there he an algdrraic equation in which the members consist of quan- tities, part real and part imaginary, then the real quantities in the two members are equal, and the imaginary quantities are equal. N. B. Imaginary quantities contain the factor J — 1, and such quantities are, emphatically, imaginary; they have no real existence. TRIGONOMETRY. 221 Suppose we have an equation in which the sum of the real quantities in the first member is represented by A ; and the sum of the like quantities in the second member by B. Also, the sum of the imagi- nary quantities in the first member, suppose represented by iSV — 1» and the sum of the like quantities in the second member by TJ — 1; that is, suppose the following equation to exist. Then, . A=J5, and SJ^l=zTJ^^ If A is not equal to B, one must be greater than the other ; and as they are supposed to be real and definite quantities, their difference must be real and definite ; and, therefore, we can represent it by the definite quantity D. That is, suppose A greater than B by D; then the equation becomes B+D-]-S^~—l=B-\-Tj^l Strike out B from both members, and transpose S^ — 1 Then, D=Tj^—Sj'^=iT—S)J^l That is, a real quantity equal to an imaginary one — a perfect absurdity; and this absurdity is in consequence of supposing A not equal to B; therefore, we must admit that A=B, It necessarily follows that Let a represent any arc, the radius unity; then, cos.2a-j-sin.2a=l Conceive the first member as composed of the two factors, co8.a-{-h sin. a, and cos. a — h sin.o The product of these two factors, is cos.^a — h^ sin.'a; and, by hypothesis, this product must equal the first member of the equation ; that is, Dropping cos. 'a from both members, there remains — A^ sin.2a=sin.'a 222 APPENDIX TO Dividing by sin. 'a, and changing signs, we have h^= — 1, or h:=-\-tJ — 1, which shows that the coefficient, A, is imaginary.* The different powers of h are h=+lj^h A2=— 1, A5=—l V^, h' =+1, A5=+J^l, A«=— 1, and so on. Observe that all the even powers of h are rational quan- tities ; in short, units, with the signs pliis and minus alternating. Thus, . A2=— 1, A^=+l, ;i6=— 1, A8=-|-l, and so on. All the odd powers are imaginary, and the signs alternating. If we multiply the two similar factors, cos.a-(-A sin.a And, . . . cos.b-\-h sin .6 Product will be, cos.a cos.&-{-(sin.a cos.&-{-cos.a sm.h^h-^-Ti^Bm.a sin.ft Now let A=^ — 1, and ^'= — 1; then this product is (cos. a C0S.& — sin.a sin.Z>)-|-(sin.acos.6-|-cos.asin.&)>7 — 1 Comparing this expression with equations (9) and (7), page 141, we perceive that it is the same as coB.(ji-\-l)-\-Bm.{a-\-l)^ — 1 ; Hence, (cos.a-f-^ sin.fl)(cos.J-|-A sin.a)=cos.(a-f-^)+^ sin.(a-|-^) In case we give to h its particular imaginary value, »J — 1 It is very remarkable that the product of these factors can be found by simply adding the arcs, which is a property analagous to logarithms. If we make a=J in the preceding equation, we have (cos.a-f-^ sin.fl)(cos.a-|-^ sin.a)=cos.2o-|-^ sin.2a (1) (cos.a-f-A sin.a)(cos.2a-f-^ sin.2a)=cos.3a-|-'^ sin. 3a (2) (cos.a-j-A sin.a)(cos.3a-j-^ sin.3a)=cos.4a-4-A sin.4a (3) and so on. The first member of equation (1), is (cos.a-|-^ sin.a)2 * This investigation shows, also, that the sum of any two squares may be regarded as the product of two binomial factors. Thus, . . . a:24-y2=(;e_|_y^-Zj:)(a;-y^~I) TRIGONOMETRY. 233 The first member of equation (2), is (cos.fl-j-A sin.a)*, and so on. Therefore, in general, if n is taken to represent any entire number whatever, we shall have, cos.na+A sin.«ass=(cos.a-|-A sin.a)" But, • (cos.a+A sin.a)°=cos."«(l4-A tan.a)" sin.a Because, • . . =tan.a cos.a Hence, . co8.na-{-h sm.na=co8.''a(l-\-h t&n.ay (4) Expanding the binomial in the second member, we have n — I n — 1 n — ^2 Substituting the expanded binomial in equation (4), it becomes cos.Tia-j-^ sin.7wi= n — 1 n — 1 n — 2 cos.°a(l-4-TOA tsLn.a-\-n—^h^ tan.'a-j-TO— r ^-h^ tan.^a, &c.) Calling to mind the principles explained in the preceding lemma, and recollecting that all the terms containing the odd powers of h must be imaginary, and all the other terms real, therefore, we may put cos.Tia equal to all the real quantities in the series, multiplied by the fa6tor cos."a; and the imaginary quantity h sin.Twi, must be put equal to all the terms in the series containing the odd powers of h, and the whole multiplied by the factor cos."a. But as every term of this equation will contain A, we can divide by h, and thus convert every odd power into an even power, and change the equation from imaginary terms to real terms. Thus, by equating the parts of the preceding equation, we have cos.na= , . ^ — 1 , o « . n — 1 n — 2 71—3 , cos°a(l+^ -2~ t&nM-\-n —z ^ j- h* tan.''a+ &c.) n — 1 n — 2 , „ , n — 1 n — 2 n — 3 sin.na=cos."a(7i tan.a-j-7i — r o~ ^ tan.^a-j-Ti — — - n — 4 — r— h* tan.*a-|" ^') X Put x=.na. Then %=-. Also observe that h'^=z — l, and A*=l, and so on, alternately. Making these substitutions, the preceding equations become APPENDIX TO x'x^-a tan'a , a?(a?— a)(ar— 2a)(a?— 3a) tan. to revolve along in the loop, holding the threads F'D and FD at equal tension; and 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'+^F'A'. Therefore, . FF'-\-'2,FA=FF'-^^F'A' Hence, .... FA^F'A' From the expression FF'^'^.FAy take away FA, and add F'A\ and the sum will not be changed, and we have FF'+9,FA=:A'F''\-FF'+FA=^A'A Hence, . . ,F'D+FD=A'A Q. E. D. PROPOSITION 2. THEOREM. The distance from either focus to the extremity of the minor axis, is equal to half the major axis. As F'Cz=CF (see last %ure), and CD is at right angles to F'F, therefore, . . . F'D=FD. But, . . ,F'D-{-FD=A'A Or, ... . ^D=AA Or, . . . . FD= half A' A, or CA. Q. E, D, Scholium, Half the minor axis is a mean proportional between the distance from either focus to the principal vertices. In the right angled triangled FCD we have CD^=FI)^~FC^ But, .... FD=AC 230 Therefore, Or. CONIC SECTIONS. =( J C-\-FC){AC—FC) =AF'XAF AF: CD=CD:FA' PROPOSITION 3. THEOREM Fvery diameter is bisected in the center. Let D be any point in the curve, and C the center. Join DC, and produce it. From F' draw F'D' parallel to FD; and from F draw FD' parallel to F'D. The figure DFD'F ' is a par- allelogram by construction ; and there- fore its opposite sides are equal. Hence, the sum of the two sides F'D' and D'F is equal to F'D and DF; therefore, by definition 1, the point D' is in the ellipse. But the two diagonals of a parallelogram bisect each other ; there- fore, DG=^ CD', and the diameter DD' is bisected at the center, C, and DD' represents any diameter. Therefore, &c. Q. E. D. PROPOSITION 4. THEOREM. A tangent to the ellipse makes equal angles with the two straight lines drawn from the point of contact to the foci. Let i^ and i^' be the foci, and D any point in the curve. Join F'D and FD, and produce F'D to JI, making DH=DF, and join FH. Bisect FH in T. Join TD and produce it to t. Now by theorem 15, book 1, the angle FDT= the angle HDT, and HDT=^ its opposite vertical angle, F'Dt. Therefore, . . FDT=^F'Dt It now remains to be shown that 71 is a tangent, and only meets the curve at the point D. THBELLIP&E. 231 If possible, let it meet the curve in some other point, as t, and join Ft, tJI, and F't. By theorem 15, book 1, Ft=tH To each of these add F't; Then, . . F't-^-tH^^F't^Fi But F't-\-tH are, together, greater thanJ^^, because a straight line is the shortest distance between two points ; that is, F't-\-Fiy the two lines from the foci, are, together, greater than FH, or greater than F'DArFD; 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. q. E. D. Cor. The tangents at the vertices of either axis are perpendicular to that axis ; and as the ordinates are parallel to the tangents, it follows that all ordinates to the major or minor axis must cut one axis at right angles, and be parallel to the other axis. Scholium, Axij 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, when they approach to, are reflected off, from any surface at equal angles; that is, any and every single ray makes the angle of reflection equal to the angle of incidence. Therefore, if a light is placed at one focus of an ellipse, and the sides a reflecting surface, the reflections will concentrate at the other focus. If the sides of a room be elliptical, and a stove is placed at one focus, it will concentrate heat 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 5. THEOREM. Tangents to the ellipse, at the vertices of the diameter, are parallel to one another. 232 CONIC SECTIONS. ^ I)' Let DB' be the diameter, and F' and F the foci. Join F'D, F'D', FD, and FD\ Draw the tangents, 7^ and Ss, one through the point i>, the other through the point D', These tangents will be parallel. By proposition 3, 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 angles of the parallelogram, we have 8D'F'-^SD'F'=tDF'-\-TDF, But, by proposition 4, sD'F'=SD'F; therefore, their sum is double of either one of them, and the above equation may be changed to . . ^SD'F='itDF Or, . . . SD'F=tDF' But BF' and B'Fare parallel ; therefore, SB'F and tDF' are, in effect, alternate angles, showing that Tt and Ss are parallel. Q. F. D, Cor. If tangents be drawn through the vertices of any two conjugate diameters, they will form a parallelogram circumscribing the ellipse. PROPOSITIONS THEOREM. If y from the vertex of any diameter, straight lines are drawn through thefody meeting the conjugate diameter y the part intercepted by the conjugatey is eqvxd to half the vnajor axis. Let DD' be the diameter, and Tt the tangent. Draw FF' parallel to Tt. Join F'D and DF, and produce J)F to F; and from F draw FO parallel to FF' or Tt. Now, by reason of the parallels. THE ELLIPSE. 233 we have the following equations among the angles. TDF=DFa 5 ^ ' TDF=^DKH S But, by proposition 4, tD 0= TDF Therefore, by equality, DQF^BFQ And, . . . DHK=DKH Hence, the triangle DQF is isosceles; also, the triangle DHK is isosceles. Whence, . DG=DF, and Dff=DK Because ffO is parallel to FO, and F'C=CF, Therefore, . . F'H=^HQ Add . . DF=DQ F'H-{-DF=DH But the sum of the lines in both members of this equation is F'D-^DF, which is equal to the major axis of the ellipse ; therefore, either member is half the major axis ; that is, DJI, or its equal, DIT, is each equal to half the major axis. Q. F. D. PROPOSITION 7. THEOREM. Perpendiculars from the foci of an ellipse upon a tangent, meet the tangent in the circumference of a circle, whose diameter is the major axis. Let F'F be the foci, G the center, and D a point in the ellipse, through which passes the tangent Tt. Join F'D and FD, and produce F'D to H, making DH—FD, and produce FD to (7, making DQ^F'B, Then F'H and FQ are each equal to the major axis, A' A, Join FHy meeting the tangent in Ty and join F' O, meeting it in t. Draw the dotted lines, GT and Gt. By proposition 4, the angle FDT= the angle F'Dt; and observ- ing that opposite vertical angles are equal, therefore, the four angles formed by lines crossing at i>, are all equal. The triangles DF' G and DHF are isosceles by construction, and as their vertical angles at D are bisected by the line Tt, therefore, F't^tG, and FT= TH. 234 CONIC SECTIONS. Comparing the triangles F'OF and F'Ct, we find FC equals the half of F'F, and F't the half of FCf; therefore, Ct is the half of FG. But A'A=FO; hence, Ct=iA'A=OA, Comparing the triangles FF'H and FCT, we find the sides FH and FF' cut proportionally in T and 0; therefore, they are equiangular and similar, and CT is parallel to F'H, and equal to half of it. That is, CT is, equal to CA; and CAy CT, and Ct, are all equal ; and hence a circle described from the center, (7, at the distance of CA, will pass through the points T and t. Therefore, perpendiculars, &c. Q. E. D. PROPOSITION 8. THEOREM. The product of the perpendicvlars from the foci upon a tangent, is equal to tJte square of half the minor aocis. Produce TC and GF' (see figure to the last proposition), and they will meet in the circle, at S; for FT and F't are both per- pendicular to the same line, Tt; they are, therefore, parallel ; and the two triangles CFT and CF' S, having a side, FC, of the one, equal to CF', of the other, and their respective angles equal, therefore CS=CT, and S is in ihe circle, and SF'=FT. Now, as A' A and St are two lines that intersect each other in a circle, therefore, (th.l7, b. 3) SF'XF't=-A'F'XF'A FTXF't=A'F'XF'A But, by the scholium to proposition 2, it is shown that A'F'XF'A=^ the square of half the minor axis. Hence, . . FTX F't-= the square of half the minor axis. Therefore, the product, &c. Q. E. D. Cot, The two triangles, FTD and F'tB, 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. THE ELLIPSE 235 PROPOSITION 9. PROBLEM. Oiven the major aans and the distance between the fod of any ellipse^ to find the relation between an abscissa of the major axis and its cor- responding ordinate. Let F' and F be the foci, C the center, and put CF', or CF=c, and CA=A. Then F'J)=A, and in the triangle F' DO or FDO, if the hypo- tenuse FJ) and FC are both known, then DC is known; therefore, we may put CD=B, and consider A, By and c, known quantities. Take any point on the major axis, as t, and draw tP at right angles to A' A, Measuring from the point A', A't is the abscissa, and tP is the corresponding ordinate. The problem requires us to find the mathematical relation between these two hues. We can find it by the aid of the two right angled triangles F'tP and FtP, Put . . A't=x, and tP=y Then . . F't=A't—A'F'=x--(A—c)= =x+c^A And . . Fi=:A't-^A'F=x^(A+c)= -x c A Put . . F'P=r,SindF'P=r' Then, . F'P+FP=r'+r=2A 0) In the triangle FPt we have (x-[-c^Ay+f==r'' (2) In the triangle FPt we have (x^^Ay+y'=r^ (3) By subtracting (3) from (2), expanding and reducing, we 4ca^— 4c^=r'^— r' (4) Or, . . . 4c(a:— ^)=(/-i-r)(/— r)(6) 236 CONIC SECTIONS. But tlie first factor in the second member of equation (6) is equal to 2A; hence we have r'^^±.(^-.A) (6) But, . . . r'+r=2A (7) By adding (6) and (7), then dividing by 2, and then subtracting (6) from (7), and dividing by 2, we have the two following equations : f=A+^(x-A) (8) r=^-j{'-^) (9) It should be observed that equations (8) and (9) are expressions for lines, one of which is called radius rector in astronomy. By squaring equation (9), and comparing it with equation (3), equating the two values of r^ we shall then have ar2+c^^^2— 2car— 2^a;+ 9.cA+y^= A^—Zc{x--A)'^~(x—AY Or, . ar^+c*— 2^+y'=^'(^— 2a:^+^') Or, A''x^-\'c'A^—2A^x+Ahf ^=:=(^x'—2c'xA+c'A^ Or, . Ay+(A'—c'y=(A^--c')2Ax Observing that A^ — c^=B\ the square of the semi minor axis, and substituting this value, the preceding equation becomes u4y4-^a^=2^^2a; Hence, . • . . f=:~^{2Ax—x') (10) Or .... y^^^i-^J^Ax—x" (11) ^ We cannot reduce this equation to lower terms, or condense it to a more simple form ; and, therefore, it must rest as the final result; and, in the language of analytical geometry, it is called the equation of the ellipse. THE ELLIPSE. 237 Any definite value may be assigned to ic, not greater than 2^, and when any particular value is assigned, the equation will give the corresponding value of the ordinate, y, and as y has the double sign, it shows that y may be drawn both above and below A' A, or shows that the curve is symmetrical on both sides of A' A. Now let us examine the result when particular values are given to X. At the point A' x=0; and this value of x put in the equa- tion, gives y=0; obviously the proper result. Again, suppose x^=2Af and this value of x put in the equation, gives y=±^ V4Z2^T'=±^ X That is, y=0, for that point, also. If we suppose x=3A, y will come out imaginary; showing that there is no real value to y beyond the point A; and in this way imaginary equations have real practical utility. If we suppose a;=-4, then y will become CD=^B. If we make AF'-=.x, then x=^A — c; and this value put in the equation, gives . y = dz-j J ( %A — x) ( A — c) By the definition, the double ordinate from either focus, is called the 'parameter; and we perceive by this equation that the semi parameter is the third proportional to the major and minor axes ; For, . . A\ B=^B : y; a proportion that gives the preceding equation. It is sometimes most convenient to take C, the center of the ellipse, for the zero point, in place of the point A\ one extremity of the major axis. If we make this change, it will cause no changes in the ordinate y, but X, in the equation for the ellipse, must be diminished by A; and X, a measure from that point, can never be greater than A, but it can have the double sign plus or minus. At the point A\ x will be equal to minus A, and at the other extremity of the major axis, X will be equal U>plus A. To change the equation y^=^-^^{^Ax — a?) into its equivalent 238 • CONIC SECTIONS. expression, when the origin of x is changed from A' to (7, we must put X — A=x'. Hence, x and x' designate the sanie point on the axis ; and if a; is less than A, then x' is negative. If . X — A=zx', then x=^A-i-z' (2Ax'-^^)=(^A'^x)x=(A--x')(A'\-x')=A^—^'^ Hence, f^±lA-^x-)=B^-^^-^ Or, . . AY-\~Bx'^^A''I^ We may omit the accent of x, for x, or x' , is only a different symbol for any point on the major axis corresponding to the ordinate y. The accent was only taken to avoid confusion while changing the zero point ; therefore, the following equation is the equation for the ellipse, the zero point being the center. ' Ay-\'B'x'=A^J3' In case -4s=J5, the ellipse becomes a circle, and the equation becomes • . Ah^-\-A^3^=A* Or, . . . . f+a^=A* This last equation is obviously the equation of the circle, y being the sine of any arc, x its cosine, and A the radius. The change in the zero point from the vertex of the major axis to the center, changes equations (8) and (9) into r^A r ex ex Or, without the accent, r'=^+-j , and r=A — j PROPOSITION 10. THEOREM. The squares of the ordinates of the major axis are to each other ob the rectangles of their corresponding abscissas. THE ELLIPSE. Let y be any ordinate, and x its corres- ponding abscissa. Then, by the last pro- position, we shall have Let y' be any other ordinate, and a;' its corresponding abscissa, 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 have y^ _(2^ — x)x Hence, . y* : y'^=(^A—x)x : {%A—x)x' By simply inspecting the figure, we cannot fail to perceive that (2-4 — x), and Xy are the abscissas corresponding to the ordinate y, and {^A — x') and x'y are the two corresponding to y'. Therefore, the squares of the ordinates, &c. Q. E. D. PROPOSITION 11. THEOREM. J^ a circle he described on the major axis of an ellipse, and any ordinate be dravm common to both the circle arul the ellipse, the ordinate corresponding to the circle is to the part corresponding to the ellipse as the major axis of the ellipse is to its minor axis. On A' A (see figure to last proposition), as a diameter, describe a circle. Draw any ordinate, as OH. The part DH is y, of the last proposition. The proportion in the last proposition is true, and y and y' may be any two ordinates, whatever. And now suppose y' represents the semi minor axis ; then x' will equal A, and 9. A — x'=^A, Taking this hypothesis, the proportion referred to becomes f : B''={'')^ and (/')', must be in proportion to • 1 1 ,1 ;, and 57' 24" 9' 58' 6" 6' 59' 22" 9 Or as the numbers, ., and 3444.9' 3486.6* 3562.9* Multiply by 3562.9 and the proportion will not be changed, and we may put / 3562,9 \ 4 , / 3562.9 \ A ^ „ ^ '•={3444:9) • '•-(3486-:6) ' ^"'^'•='- By the aid of logarithms, we soon find r=1.016982 r'= 1.010867 and r"=l. Hence, r-—r'=cf=0.006 125, r— /'=<;' =0.0 169 8 2 158° 37' 46" 196° 48' 4" 128 47 31 128 47 31 m= 29 50 15 w= 68 33 fm CONIC SECTIONS. To correspond with the formulas, we must take the natural sine and cosine of m and n, m=29° 60' 15" sin. .497542 . cosine .867440 «=68 33 sin. .927238 . cosine .374472 ^'cos.m=a=0.140172 'cos.w=5=0.642510 aA'B; that is, A'F^A'B But A'Bj being the hypotenuse of the right angled triangle A'B'B, it is greater than A'B\ But A'F is greater than A'B; much more then is A'F greater than A'B'; therefore, any point, (fee. Q. E. D. PROPOSITION 3. THEOREM. The line which bisects the angle which is formed by the two lines drawn from any 'point in the curve , one to the focus, the other perpen- dicular to the directrix, is a tangent to the curve at that point. Let P be any point in the curve. Draw PF to the focus, and PB per- pendicular to the directrix. Let PT be so drawn as to bisect the angle BPF. Then PT will touch the para- bola at the point P, and be tangent to the curve. Join BF, and PBF is an isosceles triangle ; therefore, the angle PBI= the angle PFL The angle BPI= the angle FPl, by hypothesis ; hence, the two triangles BPI and PIF, being equi- JB' JL \ y ^w )/, // y 4n/ ' z,_ A yi- T C i\f ID THEFARABOLA. 251 angular, and having PI common, are in all respects equal, and PI is perpendicular to BF, and BI=FL It now remains to be shown that any other point than P, in the line APT, is without the curve. Take any other point in the line TP, as A, and draw the dotted lines -4i^ and ^jB. They are equal. (Th. 15, b. 1, scholium.) But AB being the hypotenuse of the right angled triangle AB'B it is greater than AB'; that is, AF is greater than AB' ; conse- quently A is without the curve, as proved by the last proposition. In the same manner it may be proved that any other point in the line -4^ is without the curve, except the point P. AT is, therefore, a tangent to the curve at the point P. Q. E, D. Cor. 1 . A line of light, parallel to the axis, striking the point of the parabola at P, will be reflected to F; because the angle of incidence is equal to the angle of reflection ; and the same will be true at every point of 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 ; hence, in certain situations, parabolic mirrors have been made for lighthouses, for the purpose of throw- ing all the light seaward. Cor. 2. The angle BPF continually increases, as the 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, 3. Since an ordinate to any diameter is parallel to the tangent at the vertex, an ordinate to the axis is perpendicular to the axis. PROPOSITION 4. THEOREM. If a tangent he dravmfrom any point m the curve to the axis pro- duced, the extremities of the tangent are equally distant from the focus. Let PT (see figure to the last proposition) be a tangent, meet- ing the curve at jP, and the axis at T. Then we are to prove that PF==FT 253 CONIC SECTIONS. PB is parallel to FT; therefore, the angle BPT= the angle PTF. But BPT=TPF. (Prop. 3.) Hence, the angle PTF= the angle TPF; consequently, the triangle TFP is isosceles, and PF=TF, Q. K J). PROPOSITION 5. THEOREM. The suhtangent to the axis is bisected hy (lie vertex. From the point P (see last figure) draw Pi>, an ordinate to the axis. i>7^ is a subtangent, and it is bisected at F. As PD is parallel to BC^ and PB parallel to (72), PBCD\s a parallelogram. Therefore, . PB=CD But, . PB=PF, by the definition of the curve And, . . PF=FT. (Prop. 5.) Therefore, . . CD=^FT That is, . J)V-\-VC=-TV+VF But, VC=VF By subtraction, J)V=TV Q.E.D. C(yr. Hence, to draw a tangent to any point P, draw the ordinate PD, and take VT= VD, and join TP; it will be a tangent at P. PROPOSITION 6. THEOREM. i/^, from any point in a parabola, a tangent and a normal be drawn^ both terminated in the axis, these two lines wUl be chords of a circle, of which the focus is the center, and the distance to the point P, the radius. Let P be the point, F the focus, and TVC the axis. Draw PD perpen- dicular to the axis, and take TV= VD (cor. to last prop.) and join TP, which is the tangent from P. From P draw PC, at right angles to TP; then PC, is the normal. (Def. 11.) Draw PF. By proposition 4, PF—FT. Now, if FP be made radius, and a semicircle described, the points T, P, and C, will be in the circumference, and TC will be the diameter. THE PARABOLA. 253 Hence T PC is & right angle, and FP=FC, and TF and PC, are chords to this circle ; therefore, if from any point &c. Q. K J). PROPOSITION 7. THEOREM. The svhnormal is equal to half the latus rectum. Take the figure to the last proposition. By the definition of the curve. FP=D F-f VF=FD-\-2 VF Or, . ^VF^FP--FD (1) CD=FC—FD (2) By subtracting (2) from (1), and observing that FP=iFCy we have, ^VF^CD=0 Or, . . CD=2VF But CD is the subnormal, and 2 VF is half the lotus rectum ; therefore, the subnormal =y, and let 2p repre- sent the parameter. As TPC, is a right angled triangle, right angled at P, PD is a mean proportional between TJ) and DC. (Scho. to th. 17, b. 3). But, TD=2x (Prop. 5). And, J)C=:p (Prop. 7). Therefore by multiplication, TD'DC=f=2px By taking the square root, y=dzj2px, the double sign shows two equal values to y, the one above, the other below the axis ; hence, the curve is symmetrical in respect to its focus and axis. PROPOSITION 10. THEOREM. The sqitares of ordinates to the axis are to one another, as their corresponding abscissas. By the last proposition, any ordinate represented by y, and its THE PARABOLA. 255 corresponding abscissa represented by «, are connected together by the following equation. y'^zlpx (1) Any other ordinate represented by y', and its corresponding ab- scissa represented by x\ have a like connection. That is, . . y'^'^^px' (2) Dividing (2) by (1), omitting the common factor 2p, and we hare Or, r'2 : y'^x' : x Q. K D. PROPOSITION 11. THEOREM. As the parameter of the axis is to the sum of any two ordinaies, so is the difference of those ordinates to the difference of their abscissas. Let CV^ be a portion of a parabola, V the vertex, VD the axis, VB and VD ab- scissas, and FB and UD their correspond- ing ordinates. Put VB=:x, VD=x', PB^y, And ED=y' Then, AR=x'—x, RE=y'-{-y, and CR=y'—y From Proposition 10. y'^=i :^pX By subtraction, y'^ — y^=2p(x' — x) Or, . . (y'+y)(y'—y)=2p(x'-^x) Or, . . . 2p Or, . . • 2p I Q. K D. 9:RU=GB:AR J 25G CONIC SECTIONS, Chr. Take the product of the extremes and means of this last proportion and we have (2p)x'=y^ (Prop. 10). But, . Bj division. AB CR'EE X- y AR_ CR'RE . VD : AR=DU^ : CR'RU That is, any abscissa of the axis, is to any other diamater, so is the square of the ordinate to the rectangle of the segments of the double ordinate. Or, Or, PROPOSITION 12, THEOREM. J^ a tangent be draton/rom any point of a parabola, and from any point in the tangent a line be drawn parallel to the axis, and termi- nated in the double ordinate, this line vMl be cut by the curve in the same proportion as the line cuts the double ordinate. Let CT be a tangent for the point C, V the vertex, VD the axis, and CU the double ordi- nate CI>=y VD=x Take any point /, in the tangent, and draw IR parallel to VD, cutting the curve at A. Then we are to show That . . IA'.AR=CR\ RE Produce i> F to T, and observe, that DV=VT, Or, ... . By similar As, By eq. of the curve By equality, . Proposition 11, DT=^SlDV CR ', RI=^CD : DT =y : 2a; 2p : 9,y=y : 2a; CR : RI=^2p : {2y)CE 2» : RE=^ CR : AR (Prop. 5). Prod, term, by term, 2^?- CR ; RI-RE^^p' CR : CE-AR THE PARABOLA. 257 In this last proportion the antecedents are equal ; therefore, the tonsequents are equal. Hence, . RI»RE=CE'AR Or, . . BI\AR^CE:RE By division, (RI-^AR) : AR=( CE—RE) : RE That is, . IA:AR=CR:RE Q. E. D. Cor. The same is true, if a line be drawn from any other point of the tangent. Therefore, . EP : PO=CG : GE PROPOSITION 13. THEOREM. ^ am points be taken on a tangent, and from thence lines be drawn 'parallel to the axis to meet the curve, the length of such lines will be to each other as the squares of the distances of the points from the point of contact measured on the tangent. Let CJI be a tangent to a parabola, and / and JT" any points taken upon it. Let D F be the axis produced to T. Draw IR parallel to VD, meeting the curve at A; and also, draw EG par- allel to VE, meeting the curve at P. We are now to prove, that lA :HP=Cr : CH^ By the last proposition, we have lA :AR=CR:RE Multiplying the last couplet by CR, and substituting the value of CR*RE taken from corollary to Proposition 11, and AR'CE^ lA :AR=CR': VD Dividing the second and fourth terms by AR, and afterward multiplying the same terms by VD, observing that VD= VT, then we have IA\ VT=zCR^: CD^ 17 258 CONIC SECTIONS But by similar triangles. Therefore, by equality, lA : TV=:Cr : CT^ In the same manner, we may prove that ffP: TV=CH^: CT^ Dividing one of these proportions by the other, terra by term. And, • . _.i=-^^.l Or, . . . lA: HP=Cn : CH^ Q. E. D. Application, Conceive CH to be the direction of a projectile, and undisturbed by the resistance of the air, or the force of gravity, it would move along the line CH^ passing over equal distances in equal times. Now let gravity act in the direction of IR, and as bodies fall in proportion to the squares of the times of descent, therefore, lAy TV^ HPy &c., must be to each other, as the squares CP, CT^, CH^, &c; that is the real path of a projectile un- disturbed by atmospheric resistance must have the same property, as just demonstrated in this proposition. In other words, the path of a projectile is some parahola, more or less curved according to the direction and intensity of the projectile force. PROPOSITION 14. THEOREM. Tlic abscissas of any diameter are to each other as the sqtiares of tlieir corresponding ordinates. By the definition of a diameter, it must be the axis, or parallel to the axis ; and ordinates to any diameter must be parallel to the tangent drawn through the vertex of that diameter. Hence, if CS is a diameter, and CP a tan- gent, and ly Ty and 0, any points on the tan- gent, and from thence Hues drawn parallel to the axis to meet the curve, and from thence lines parallel to the tangent to meet the diameter, the figures so formed will be parallelograms, and their opposite sides equal. THE PARABOLA. 259 By the last proposition, lE^ TA, =& : y That is, the parameter is a third proportional to the major and minor axes. There are many other properties of the hyperbola not here demon- strated, but being of little or no practical importance, we omit them. LOGARITHMIC TABLES; ALSO A TABLE OF NATURAL AND LOGARITHMIC SINES, COSINES, AND TANGENTS, TO EVERY MINUTE OF THE QUADRANT. LOGARITHMS OF NUMBERS TROM 1 TO 10000. N. - Log. N. Log. N. Log. N. Log. 1 000000 26 1 414973 51 1 707570 76 1 880814 2 301030 27 1 431364 52 1 716003 "77 1 886491 3 477121 28 1 447168 53 1 724276 78 1 892095 4 602060 29 1 462398 54 1 732394 79 1 897627 5 698970 30 1 477121 55 1 740363 80 1 903090 6 778151 31 1 491362 56 1 748188 81 1 908485 7 845098 32 1 505150 57 1 755875 82 1 913814 8 903090 33 1 518514 58 1 763428 83 1 919078 9 954243 34 1 531479 59 1 770852 84 1 924279 10 1 000000 35 1 544068 60 1 778151 85 1 929419 11 1 041393 36 1 556303 61 1 785330 86 1 934498 12 1 079181 37 1 568202 62 1 792392 87 1 939519 13 1 113943 38 1 579784 63 1 799341 88 1 944483 14 1 146128 39 1 591065 64 1 806180 89 1 949390 IB 1 176091 40 1 602060 65 1 812913 90 1 954243 16 1 204120 41 1 612784 66 1 819544 91 1 959041 17 1 230449 42 1 623249 67 1 826075 92 1 963788 18 1 255273 43 1 633468 68 1 832509 93 1 968483 19 1 278754 44 1 643453 69 1 838849 94 1 973128 20 1 301030 45 1 653213 70 1 845098 95 1 977724 21 1 322219 46 1 662578 71 1 851258 96 1 982271 22 1 342423 47 1 672098 72 1 857333 97 1 986772 23 1 361728 48 1 681241 73 1 863323 98 1 991226 24 1 380211 49 1 690196 74 1 869232 99 1 995635 25 1 397940 50 1 698970 75 1 875061 100 2 000000 N. B. In the following table, in the last ni ne columns of each p age, where the first or leading figures change from 9'^ 5 to O's, points or do ts are now introduced instead of the O's through the r est of the line, to cat( ih. he eye, and to indicate that from thence the corr Bsponding natural n umbers in the first column stands in the next lower Zirte, and its annexe d first two figures of the Logarithms in the second cc lumn. LOGARITHMS OF NUMBERS. 3 N. I 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 10-2 8600 9026 94^ 9876 .300 .7^4, 1147 1570 1993 2415 103 012837 3259 3680 4100 4521 4940 5360 5779 6197 6616 104 7033 7451 7868 8284 8700 9116 9532 9947 .361 .775 105 021189 1603 2016 2428 2841 3252 3664 4075 4486 4896 103 5306 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 2576 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 060698 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 5182 119 5547 5912 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 6216 6562 125 6910 7257 7604 7951 8298 8644 8990 9335 9681 1026 126 100371 0715 1059 1403 1747 2091 2434 2777 3119 3462 127 3804 4146 4487 4828 5169 5510 5861 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 3943 4277 4611 4944 5278 5611 5943 6276 6608 6940 131 7271 7603 7934 8265 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 9G90 ..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 5196 5507 5818 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 3205 3510 3815 4120 4424 4728 5032 143 5336 5640 5943 6246 6549 6852 7154 7457 7759 8061 144 8362 8664 8965 9266 9567 9868 .168 .469 .769 1068 145 161368 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 3769 4060 4351 4641 4932 5222 5512 5802 18 4 LOGARITHMS N. 1 2 3 4 5 6 7 8 9 160 176091 6381 6670 6959 7248 7536 7826 8113 8401 8689 161 8977 9264 9552 9839 .126 .413 .699 .986 1272 1558 152 181844 2129 2415 2700 2985 3270 3555 3839 4123 4407 163 4691 4975 5259 5542 5826 6108 6391 (it)74 6956 7239 164 7521 7803 8084 8366 8647 8928 9209 9490 9771 .,51 155 190332 0612 0892 1171 1451 1730 2010 2289 2567 2846 166 3125 3403 3681 3959 4237 4514 4792 5069 5346 5623 157 5899 6176 6453 6729 7005 7281 ';556 7832 8107 8382 158 8667 8932 9206 9481 9755 ..29 .303 .577 .850 1124 159 201397 1670 1943 2216 2488 2761 3033 3305 3577 3848 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 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 4633 4792 5051 168 5309 5568 5S26 6084 6342 6600 6868 7115 7372 7630 169 7887 8144 8400 8657 8913 9170 9426 9682 9938 .193 170 230449 0704 0960 1215 1470 1724 1979 2234 2488 2742 171 2996 3250 3604 3757 4011 4264 4517 4770 6023 5276 172 5528 5781 6033 6285 6537 6789 7041 7292 7644 7795 173 8046 8297 8548 8799 9049 9299 9550 9800 ..50 .300 174 240549 0799 1048 1297 1546 1795 2044 2293 2541 2790 175 3038 3285 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 4648 4790 5031 180 5273 5514 5755 5996 6237 6477 6718 6958 7198 7439 181 7679 7918 8168 8398 8637 8877 9116 9355 9594 9833 182 260071 0310 0548 0787 1025 1263 1601 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 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 3690 3927 188 4158 4389 4620 4850 5081 5311 5542 5772 6002 6232 189 6462 6692 6921 7151 7380 7609 7838 8067 8296 8526 190 8754 8982 9211 9439 9667 9895 .123 .351 .578 .806 191 281033 1261 1488 1715 1942 2169 2396 2622 2840 3075 192 3301 3527 3753 3979 4205 4431 4656 4882 6107 5332 193 5557 5782 6007 6232 6456 6681 6905 7130 7354 7578 194 7802 8026 8249 8473 8696 8920 9143 9360 958^ 9812 195 290036 0257 0480 0702 0925 1147 1369 1591 1813 2034 196 2256 2478 2699 2920 3141 3363 3584 3804 4025 4246 197 4466 4687 4907 6127 6347 5667 6787 6007 6226 6446 198 6665 6884 7104 7323 7642 7761 7979 8198 8416 8635 199 8853 9071 9289 9507 9725 9943 .161 .378 .595 ,8.3 OF NUMBERS. 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 70ii8 7282 203 7496 7710 7924 8137 8351 8564 8778 8991 9204 9417 204 9030 9843 ..56 .268 .481 .693 .906 1118 1330 1642 205 311754 1966 2177 2389 2600 2812 3023 3234 3445 3666 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 0662 0769 0977 1184 1391 1598 1805 2012 210 2219 2426 2633 2839 3046 3252 3458 3655 3871 4077 211 4282 4488 4694 4899 5105 6310 5516 5721 5926 6131 212 6336 6541 6745 6960 7155 7359 7563 7767 7972 8176 213 8380 8583 8787 8991 9194 9398 9601 9805 ...8 .211 214 330414 0617 0819 1022 1225 1427 1630 1832 2034 2236 215 2438 2640 2842 3044 3246 3447 3649 3850 4051 4253 216 4454 4655 4856 5057 5257 5458 5658 5859 6059 6260 217 6460 6660 6860 7060 7260 7459 7659 7858 8058 8257 218 8456 8656 8855 9054 9253 9451 9650 9849 ..47 .246 219 340444 0642 0841 1039 1237 1435 1632 1830 2028 2225 220 2423 2620 2817 3014 3212 3409 3606 3802 3999 4196 221 4392 4589 4785 4981 5178 5374 5570 576() 5952 6157 222 6353 6549 6744 6939 7135 7330 7525 7720 7915 8110 223 8305 8500 8694 8889 9083 9278 9472 9666 9860 .54 224 350248 0442 0636 0829 1023 1216 1410 1603 1796 1989 225 2183 2375 2568 2761 2954 3147 3339 3532 3724 3916 226 4108 4301 4493 4685 4876 5068 5260 5452 5643 5834 227 6026 6217 6408 6599 6790 6981 7172 7363 7554 7744 228 7935 8125 8316 8506 8696 8886 9076 9266 9456 9646 229 9836 ..25 .215 .404 .593 .783 .972 1161 1350 1639 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 ( 983 . 7169 233 7356 7542 7729 7915 8101 8287 8473 8659 8845 : 9030 234 9216 9401 9587 9772 9958 .143 .328 .513 .698 ! .883 235 371068 1253 1437 1622 1806 1991 2175 23)0 2544 ' 2 28 236 2912 3096 3280 3464 3647 3831 4015 4198 4382 4565 237 4748 4932 5115 5298 5481 5664 5846 6029 6212 or 94 238 6577 6759 6942 7124 7306 7488 7670 7852 80^4 8216 239 8398 8580 8761 8943 9124 9306 9487 9668 9849 . .30 240 380211 0392 0573 0754 0934 1115 1296 1476 1656 1837 241 2017 2197 2377 2557 2737 2917 3097 3277 3456 3636 242 3815 3995 4174 4353 4533 4712 4891 5070 6249 5428 243 5606 5785 5964 6142 6321 6499 6677 6856 '■<034 7212 244 7390 7568 7746 7923 8101 8279 8456 8634 8811 8989 1 245 9166 9343 9520 9698 9875 ..51 .228 .405 .582 .759 246 390335 1112 1288 1464 1641 1817 1993 2169 2345 2521 247 2697 2873 3048 3224 3400 3576 3751 3926 4101 4277 248 4452 4627 4802 4977 5152 5326 5501 5676 5850 6025 249 6199 6374 6548 6722 6896 7071 7246 7419 7592 . 7766 6 LOGARITHMS N. 397940 1 2 3 4 6 6 7 8 9 9501 250 8114 8287 8461 8634 8808 8981 9154 9328 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 46b3 254 4834 5005 5176 5346 5517 6688 5858 0029 6199 6370 256 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 3636 3803 3970 4137 4305 4472 4639 4806 260 4973 5140 5307 5474 5641 5808 5974 6141 6308 6474 261 6641 6807 6973 7139 >306 ';472 7638 7804 7970 8135 262 8301 8467 8633 8798 8964 9129 9295 9460 9625 9791 263 9956 .121 .286 .451 .616 .781 .945 1110 1275 1439 264 421604 1788 1933 2097 2261 2426 2590 2754 2918 3082 265 3246 3410 3574 3737 3901 4065 4228 4392 4555 4718 266 4882 5045 5208 5371 5534 5697 5860 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 3290 3450 3610 3770 3930 4090 4249 4^109 272 4569 4729 4888 5048 5207 5367 5526 5685 5844 6004 273 6163 6322 6481 6640 6800 6957 7116 7275 7433 7592 274 7751 7909 8067 8226 8384 8542 8701 8859 9017 9175 275 9333 9491 9648 9806 9964 .122 .279 .437 .594 .762 276 440909 1066 1224 1381 1538 1695 1852 2009 2166 1'323 277 2480 2637 2793 2950 3106 3263 3419 3576 3732 3889 278 4045 4201 4357 4613 4669 4825 4981 5137 5293 6449 279 6604 5760 5915 6071 6226 6382 6537 6692 6848 7003 280 7158 7313 7468 7623 7778 7933 8088 8242 8397 8552 281 8706 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 4540 4692 285 4845 4997 5150 5302 5454 5606 5758 5910 6062 6214 286 6366 6518 6670 6821 6973 7125 ;276 7428 7579 7731 287 7882 8033 8184 8336 8487 8638 8789 8940 9091 9242 288 9392 9543 9694 9845 9995 .146 .296 .417 .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 3893 4042 4191 4340 4490 4639 4788 4936 5085 5234 292 5383 5532 5680 5829 5977 6126 6274 6423 6571 6719 293 6868 7016 7164 7312 7460 7608 7756 7904 8052 8200 294 8347 8495 8643 8790 8938 9085 9233 9380 9627 9676 295 9822 9969 .116 .263 .410 .557 .704 .851 .998 1145 296 471292 1438 1585 1732 1878 2025 2171 2318 i464 2610 297 2756 2903 3049 3195 3341 3487 3633 3779 3925 4071 298 4216 4362 4508 4653 4799 4944 5090 5235 5381 5526 299 5671 5816 5962 6107 6252 6397 6542 6687 6832 6976 OF NUMBERS. 7 N. I 2 3 4 6 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 0869 1012 1156 1299 303 1443 1586 1729 1872 2016 2169 2302 2445 2588 2731 304 2874 3016 3159 3302 3445 3687 3730 3872 4015 4157 305 4300 4442 4585 4727 4869 5011 5153 5295 5437 5579 306 5721 6863 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 9255 9396 9537 9667 9818 309 9959 ..99 .239 .380 .520 .661 .801 .941 1081 1222 310 491362 1502 1642 1782 1922 2062 2201 2341 2481 2621 311 2760 2900 3040 3179 3319 3458 3697 3737 3876 4015 312 4155 4294 4433 4672 4711 4860 4989 5128 6267 5406 313 6544 5683 5822 6960 6099 6238 63-76 6515 6653 6791 314 6930 7068 7206 7344 7483 7621 7759 7897 8035 8173 315 8311 8448 8586 8724 8862 8999 9137 9275 9412 9550 316 9687 9824 9962 ..99 .236 .374 .511 .648 .785 .922 317 501059 1196 1333 1470 1607 1744 1880 2017 2154 2291 318 2427 2564 2700 2837 2973 3109 3246 3382 3518 3655 319 3791 3927 4063 4199 4335 4471 4607 4743 1878 5014 320 5150 5286 5421 5567 5693 5828 5964 6099 6234 6370 321 6605 6640 6776 6911 7046 7181 7316 7451 7586 7721 322 7866 7991 8126 8260 8395 8530 8664 8799 8934 9008 323 9203 9337 9471 9606 9740 9874 ...9 .143 .277 .411 324 510545 0679 0813 0947 1081 1215 1349 1482 1616 1760 325 1883 2017 2151 2284 2418 2551 2684 2818 2951 3084 326 3218 3361 3484 3617 3760 3883 401o 4149 4282 4414 327 4548 4681 4813 4946 5079 5211 5344 6476 5609 5741 328 5874 6006 6139 6271 6403 6535 6668 6800 6932 70G4 329 7196 7328 7460 7592 7724 7856 7987 8119 8251 8382 330 8514 8646 8777 8909 9040 9171 9303 9434 9566 9697 331 9828 9959 ..90 .221 .363 .484 .615 .745 .876 1007 332 521138 1269 1400 1530 1661 1792 1922 2063 2183 2314 333 2444 2675 2706 2835 2966 3096 3226 3356 3480 3616 334 3746 3876 4006 4136 4266 4396 4526 4656 4785 4915 335 5045 5174 5304 5434 5563 5693 5822 5961 6081 6210 336 6339 6469 6598 6727 6856 6985 7114 7243 7372 7501 337 7630 7759 7888 8016 8145 8274 8402 8631 8660 8788 338 8917 9046 9174 9302 9430 9659 9687 9816 9943 ..72 339 530200 0328 0456 0684 0712 0840 0968 1090 1223 1351 340 1479 1607 1734 1862 1960 2117 2245 2372 2500 2627 341 2764 2882 3009 3136 3264 3391 3618 3645 3772 3899 342 4026 4163 4280 4407 4534 4661 4787 4914 5041 5167 343 5294 6421 5547 5674 6800 6927 6058 6180 6306 6432 344 6668 6686 6811 6937 7060 7189 7315 7441 7567 7693 345 7819 7945 8071 8197 8322 8448 8574 8699 8825 8951 346 9076 9202 9327 9452 9578 9703 9829 9954 ..79 .204 347 540329 0465 0680 0705 0830 0956 1080 1206 1330 1464 348 1679 1704 1829 1953 2078 2203 2327 2452 2576 2701 349 2825 2960 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 5678 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 .196 355 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 4862 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 9303 9428 9548 9667 9787 363 9907 .26 .146 .265 .385 .504 .624 .743 .863 .982 364 561101 li21 1340 1469 1578 1698 1817 1936 2055 2173 365 2293 2412 2531 2650 2769 2887 3006 3125 3244 3362 366 3481 3600 3718 3837 3955 4074 4192 4311 4429 4648 367 4666 4784 4903 5021 5139 5257 5376 5494 5612 6730 368 5848 5966 6084 6202 6320 6437 6555 6673 6791 6909 369 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 570543 0660 0776 0893 1010 1126 1243 1359 1476 1592 373 1709 1825 1942 2058 2174 2291 2407 2522 1 2^39 2756 374 2872 2988 3104 3220 3336 3452 3568 360J4 3800 £915 375 4031 4147 4263 4379 4494 4610 4726 4841 4957 6072 376 5188 5303 5419 5534 5650 5765 5880 6996 6111 6226 377 6341 6457 6572 6687 6802 6917 7032 7147 7262 7377 378 7492 7607 7722 7836 7951 8066 8181 8296 8410 8626 379 8639 8754 8868 8983 9097 9212 9326 9441 9666 9669 380 9784 9898 ..12 .126 .241 .355 .469 .583 .697 .811 381 580925 1039 1153 1267 1381 1495 1608 1722 1836 1960 382 2063 2177 2291 2404 2518 2631 2745 J858 2972 3086 383 3199 3312 3426 3539 3652 3765 3879 i992 4105 4218 384 4331 4444 4557 4670 4783 4896 5009 1)122 5236 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 317C 392 3286 3397 3508 3618 3729 3840 3950 4061 4171 4282 393 4393 4503 4614 4724 4834 4945 5055 5165 5276 6386 394 5496 5606 5717 5827 5937 6047 6157 6267 6377 6487 395 6597 6707 6817 6927 7037 7146 7256 7366 7476 7586 396 7695 7805 7914 8024 8134 8243 8353 8462 8672 8681 397 8791 8900 9009 9119 9228 i>337 9446 b556 9666 9774 398 9883 9992 .101 .210 .319 .428 .537 .646 .755 .864 399 600973 1082 1191 1299 1408 1517 1625 1734 1843 1951 OF NUMBERS. 9 N. 1 2 3 4 5 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 5089 5197 403 5305 5413 5521 5628 5736 6844 5951 6059 6166 6274 404 6381 6489 6596 6704 6811 6919 7026 7133 7241 7348 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 .554 408 610660 0767 0873 0979 1086 1192 1298 1405 1511 1617 409 1723 1829 1936 2042 2148 2264 2360 2466 2572 2678 410 2784 2890 2996 3102 3207 3313 3419 3525 3630 3736 411 3842 3947 4053 4159 4264 4370 4476 4581 4686 4792 412 4897 5003 5108 5213 5319 5424 6529 5634 6740 5845 413 5950 6055 6160 6265 6370 6476 6581 6686 6790 6896 414 7000 7105 7210 7315 7420 7525 7629 7734 7839 7943 415 8048 8153 8257 8362 8466 8571 8676 8780 8884 8989 416 9293 9198 9302 9406 9511 9615 9719 9824 9928 ..32 417 620136 0140 0344 0448 0662 0666 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 4691 4695 4798 4901 5004 6107 5210 422 5312 5415 5518 5621 5724 6827 5929 6032 6135 6238 423 6340 6443 6546 6648 6751 6853 6956 7068 7161 7263 424 7366 7468 7571 7673 7775 7878 7980 8082 8185 8287 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 1961 2062 2153 2255 2366 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 6785 5886 5986 6087 6187 0287 6388 433 6488 6588 6688 6789 6889 6989 7089 7189 7290 7390 434 7490 7590 7690 7790 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 1376 438 1474 1673 1672 1771 1871 1970 2069 2168 2267 2366 439 2465 2563 2662 2761 2860 2959 3058 3156 3256 3364 440 3453 3551 3650 3749 3847 3946 4044 4143 4242 4340 441 4439 4537 4636 4734 4832 4931 5029 5127 5226 5324 442 5422 5521 5619 5717 5815 5913 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 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 1669 1666 1762 1859 1956 2063 2160 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 4G58 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 455 8011 8107 8202 8298 8393 8488 8584 8679 8774 8870 456 8965 9060 9155 9250 9346 9441 9536 9631 9726 9821 457 9916 ..11 .106 .201 .296 .391 .486 .581 .676 .771 458 660865 0960 1065 1150 1245 1339 1434 1529 1623 1718 459 1813 1907 2002 2096 2191 2286 2380 2476 2669 2663 460 2758 2852 2947 3041 3135 3230 3324 3418 3612 3607 461 3701 3795 3889 3983 4078 4172 4266 4360 4464 4548 462 4642 4736 4830 4924 5018 5112 6206 629S 6393 6487 463 5581 5675 6769 5862 5966 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 9876 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 2660 2652 2744 2836 2929 471 3021 3113 3205 3297 3390 3482 3574 3666 3768 3850 472 3942 4034 4126 4218 4310 4402 4494 4686 4677 4769 473 4861 4953 6045 5137 6228 6320 5412 6503 5595 6687 474 5778 5870 5962 6063 6145 6236 6328 6419 6511 6602 475 6694 6785 6876 6968 7059 7151 7242 7333 7424 7516 476 7607 7698 7789 7881 7972 8063 8154 8246 8336 8427 477 8518 8609 8700 8791 8882 8972 9064 9165 9246 9337 478 9428 9519 9610 9700 9791 9882 9973 ..63 .154 .245 479 680336 0426 0517 0607 0698 0789 0879 0970 1060 1161 480 1241 1332 1422 1513 1603 1693 1784 1874 1964 2055 481 2146 2235 2326 2416 2506 2596 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 5026 6114 5204 5294 6383 6473 5563 5652 485 5742 5831 5921 6010 6100 6189 6279 6368 6468 6547 486 6636 6726 6816 6904 6994 7083 7172 7261 7361 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 9576 9664 9753 9841 9930 ..19 .107 490 690196 0285 0373 0362 0550 0639 0728 0816 0905 0993 491 1081 1170 1268 1347 1435 1624 1612 1700 1789 1877 492 1965 2053 2142 2230 2318 2406 2494 2583 2671 2759 493 2847 2935 3023 3111 3199 3287 3375 3463 3551 3639 494 3727 3816 3903 3991 4078 4166 4264 4342 4430 4517 495 4605 4693 4781 4868 4956 5044 6131 6210 5307 5394 496 5482 5569 5657 5744 5832 6919 6007 6094 6182 6269 497 6356 5444 6631 6618 6706 6793 6880 6968 7055 7142 498 7229 7317 7404 7491 7578 7665 7752 7839 7926 8014 499 8101 8188 8276 8362 8449 8536 8622 8709 8796 8883 OF NUMBERS. 11 N. I 2 3 4 5 6 7 8 9 500 698970 9057 9144 9231 9317 9404 9491 9678 9664 9751 501 9838 9924 ..11 ..98 .184 .271 .358 .444 .531 .617 602 700704 0790 0877 0963 1050 1136 1222 1309 1395 1482 503 1668 1654 1741 1827 1913 1999 2086 2172 2258 2344 504 2431 2517 2603 2t89 2775 2861 2947 3033 3119 3205 506 3291 3377 3463 3549 3635 3721 3807 3895 3979 4065 508 4161 4236 4322 4408 4494 4579 4665 4751 4837 4922 507 6008 5094 5179 5265 5350 5436 5522 6607 5693 5778 508 6864 6949 6035 6120 6206 6291 6376 6462 6547 6632 509 6718 6803 6888 6974 7059 7144 7229 7316 7400 7485 510 7570 7655 7740 7826 7910 7996 8081 8166 8251 8336 511 8421 8606 8591 8676 8761 8846 8931 9015 9100 9185 512 9270 9365 9440 9624 9609 9694 9779 9863 9948 ..33 513 710117 0202 0287 0371 0466 0540 0625 0710 0794 0879 514 0963 1048 1132 1217 1301 1386 1470 1654 1639 1723 515 1807 1892 1976 2060 2144 2229 2313 2397 2481 2666 516 2660 2734 2818 2902 2986 3070 3154 3238 3326 3407 517 3491 3675 3659 3742 3826 3910 3994 4078 4162 4246 518 4330 4414 4497 4681 4665 4749 4833 4916 5000 5084 519 5167 6261 5335 5418 5502 5586 6669 5753 5836 6920 520 6003 6087 6170 6254 6337 6421 6504 6588 6671 6754 521 6838 6921 7004 7088 7171 7254 7338 7421 7504 7587 522 7671 7754 7837 7920 8003 8086 8169 8253 8336 8419 523 8602 8585 8668 8751 8834 8917 9000 9083 9166 9248 524 9331 9414 9497 9580 9663 9745 9828 9911 9994 ..77 525 720169 0242 0325 0407 0490 0573 0656 0738 0821 0903 526 0986 1068 1151 1233 1316 1398 1481 1563 1646 1728 527 1811 1893 .976 2058 2140 2222 2305 2387 2469 2652 528 2634 2716 2798 2881 2963 3045 3127 3209 3291 3374 529 3466 3638 3620 3702 3784 3866 3948 4030 4112 4194 530 4276 4358 4440 4622 4604 4685 4767 4849 4931 5013 631 5095 5176 6258 5340 5422 5503 5585 5667 5748 5830 532 5912 5993 6075 6156 6238 6320 6401 6483 6564 6646 533 6727 6809 6890 6972 7053 7134 7216 7297 7379 7460 634 7641 7623 7704 7785 7866 7948 8029 8110 8191 8273 535 8354 8436 8516 8597 8678 8759 8841 8922 9003 9084 636 9165 9246 9327 9403 9489 9570 9651 9732 9813 9893 637 9974 ..66 .136 .217 .298 .378 .459 .440 .621 .702 638 730782 0863 0944 1024 1105 1186 1266 1347 1428 1608 539 1689 1669 1760 1830 1911 1991 2072 2162 2233 2313 540 2394 2474 2655 2635 2715 2796 2876 2956 3037 3117 541 3197 3278 3358 3438 3518 3598 3679 3759 3839 3919 642 3999 4079 4160 4240 4320 4400 4480 4560 4640 4720 643 4800 4^80 4960 5040 6120 5200 5279 5359 5439 5519 644 5599 5679 5759 5838 5918 5998 6078 6167 6237 6317 545 6397 6476 6566 6636 6715 6795 6874 6954 7034 7113 646 7193 7272 7352 7431 7511 7590 7670 7749 7829 7908 647 7987 8067 8146 8225 8305 8384 8463 8543 8622 8701 548 8781 8860 8939 9018 9097 9177 9256 9335 9414 9493 549 9672 9661 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 561 1152 1230 1309 1388 1467 1546 1624 1703 1782 1860 552 1939 2018 2096 2175 2254 2332 2411 2489 2568 2646 553 2725 2804 2882 2961 3039 3118 3196 3275 3353 3431 554 3510 3558 3667 3745 3823 3902 3980 4058 4136 4215 555 4293 4371 4449 4528 4606 4684 4762 4840 4919 4997 556 5075 5153 5231 5309 5387 5465 5543 5621 5699 5777 557 5855 5933 6011 6089 6167 6245 6323 6401 6479 6556 558 6634 6712 6790 6868 6945 7023 7101 7179 7256 7334 559 7412 7489 7567 7645 7722 7800 7878 7955 8033 8110 560 8188 8266 8343 8421 8498 8576 8653 8731 8808 8885 561 8963 9040 9118 9195 9272 9350 9427 9504 9582 9659 562 9736 9814 9891 9968 ..45 .123 .200 .277 .354 .431 563 750508 0586 0663 0740 0817 0894 0971 1048 1125 1202 564 1279 1356 1433 1510 1587 1664 1741 1818 1895 1972 565 2048 2125 2202 2279 2356 2433 2509 2586 2663 2740 566 2816 2893 2970 3047 3123 3200 3277 3353 3430 3506 567 3582 3660 3736 3813 3889 3966 4042 4119 4195 4272 568 4348 4425 4501 4578 4654 4730 4807 4883 4960 5036 569 5112 5189 5265 5341 5417 5494 5570 5646 5722 5799 570 5875 5951 6027 6103 61F0 6256 6332 6408 6484 6560 571 6636 6712 6788 6864 6940 7016 7092 7168 7244 7320 572 7396 7472 7548 7624 7700 7775 7851 7927 8003 8079 573 8155 8230 8306 8382 8458 8533 8609 8685 8761 8836 574 8912 8988 9068 9139 9214 9290 9366 9441 9517 9592 575 9668 9743 9819 9894 9970 ..45 .121 .196 .272 .347 576 760422 0498 0573 0649 0724 0799 0875 0950 1025 1101 577 1176 1251 1326 1402 1477 1552 1627 1702 1778 1853 578 1928 2003 2078 2153 2228 2303 2378 2453 2529 2604 579 2679 2754 2829 2904 2978 3053 3128 2203 3278 3353 580 3428 3503 3578 3653 3727 3802 3877 3952 4027 4101 581 4176 4251 4326 4400 4475 4550 4624 4699 4774 4848 582 4923 4998 5072 5147 5221 5296 5370 5446 5520 6594 583 5669 5743 5818 5892 5966 6041 6115 6190 0264 6338 584 6413 6487 6562 6636 6710 6785 6859 6933 7007 7082 585 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 591 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 595 4517 4590 4663 4736 4809 4882 4955 5028 5100 6173 596 5246 5319 5392 5465 5538 5610 5683 5756 5829 5902 597 5974 6047 6120 6193 6265 6338 6411 6483 6566 6629 598 6701 6774 6846 6919 6992 7064 7137 7209 7282 7354 599 7427 7499 7572 7644 7717 7789 7862 7934 8006 8079 OF NUMBERS. 13 N. 1 2 3 4 5 6 7 8 9 600 778151 8224 8296 8368 8441 8513 8585 8658 8730 8802 601 8874 8947 9019 9091 9163, 9236 9308 9380 9452 9624 602 9596 6669 9741 9813 9885 9967 ..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 605 1755 1827 1899 1971 2042 2114 2186 2258 2329 2401 606 2473 2644 2616 2688 2769 2831 2902 2974 3046 3117 607 3189 3260 3332 3403 3475 3646 3618 3689 3761 3832 608 3904 3975 4046 4118 4189 4261 4332 4403 4475 4646 609 4617 4689 4760 4831 4902 4974 5045 5116 5187 5259 610 5330 5401 5472 5543 5615 5686 5757 5828 5899 5970 611 6041 6112 6183 6254 6325 6396 6467 6638 6609 6680 612 6751 6822 6893 6964 7035 7106 7177 7248 7319 7390 613 7460 7631 7602 7673 7744 7815 7885 7956 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 .216 617 790285 0356 0426 0496 0667 0637 0707 0778 0848 0918 618 0988 1069 1129 1199 1269 1340 1410 1480 1560 1620 619 1691 1761 1831 1901 1971 2041 2111 2181 2252 2322 620 2392 2462 2532 2602 2672 2742 2812 2882 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 4906 4976 5045 6115 624 5185 5254 5324 5393 5463 5532 5602 5672 5741 5811 625 5880 5949 6019 6088 6158 6227 6297 6366 6436 6505 626 6574 6644 6713 6782 6862 6921 6990 7060 7129 7198 627 7268 7337 7406 7475 7646 7614 7683 7752 7821 7890 628 7960 8029 8098 8167 8236 8305 8374 8443 8613 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 0648 632 0717 0786 0854 0923 0992 1061 1129 1198 1266 1335 633 1404 1472 1641 1609 1678 1747 1816 1884 1952 2021 634 2089 2158 2226 2295 2363 2432 2500 2568 2637 2705 635 2774 2842 2910 2979 3047 3116 3184 3252 3321 3389 636 3467 3626 3594 3662 3730 3798 3867 3935 4003 4071 637 4139 4208 4276 4354 4412 4480 4548 4616 4685 4753 638 4821 4889 4967 5026 6093 5161 6229 5297 5366 5433 639 5601 5669 5637 5706 5773 5841 6908 5976 6044 6112 640 6180 6248 6316 6384 6451 6519 6587 6655 6723 6790 641 6858 6926 6994 7061 7129 7167 7264 7332 7400 7467 642 7535 7603 7670 7738 7806 7873 7941 8008 8076 8143 643 8211 8279 8346 8414 8481 8649 8616 8684 8751 8818 644 8886 8953 9021 9088 9166 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 1576 1642 1709 1776 1843 1910 1977 2044 2111 2178 649 2246 2312 2379 2445 2512 2679 2646 2713 2780 2847 14 LOGARITHMS N. I 2980 2 3 4 5 6 7 8 9 650 812913 3047 3114 3181 3247 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 6113 5179 5246 5312 5378 5446 5611 654 5578 5644 5711 5777 6843 5910 5976 6042 6109 6175 655 6241 6308 6374 6440 6506 6573 6639 6705 6771 6838 666 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 8666 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 0630 0595 0661 0727 0792 662 0858 0924 0989 1065 1120 1186 1261 1317 1382 1448 663 1514 1579 1645 1710 1775 1841 1906 1972 2037 2103 664 2168 2233 2299 2364 2430 2496 2660 2626 2691 2756 665 2822 2887 2952 3018 3083 3148 3213 3279 3344 3409 666 3474 3539 3606 3670 3735 3800 3866 3930 3996 4061 667 4126 4191 4256 4321 4386 4461 4516 4681 4646 4711 668 4776 4841 4906 4971 5036 5101 5166 6231 5296 5361 669 5426 5491 5566 5621 6686 5761 5815 5880 5945 6010 670 6075 6140 6204 6269 6334 6399 6464 6528 6593 6658 671 6723 6787 6862 6917 6981 7046 7111 7175 7240 7305 672 7369 7434 7499 7563 7628 7692 7767 7821 7886 7951 673 8015 8080 8144 8209 8273 8338 8402 8467 8631 8595 674 8660 8724 8V89 8853 8918 8982 9046 9111 9176 9239 675 9304 9368 9432 9497 9561 9625 9690 9764 9818 9882 676 9947 ..11 ..75 .139 .204 .268 .332 .396 .460 .625 677 830589 0663 0717 0781 0845 0909 0973 1037 1102 1166 678 1230 1294 1358 1422 1486 1660 1614 1678 1742 1806 679 1870 1934 1998 2062 2126 2189 2263 2317 2381 2445 680 2509 2573 2637 2700 2764 2828 2892 2956 3020 3083 681 3147 3211 3275 3338 3402 3466 3530 3593 3667 3721 682 3784 3848 3912 3976 4039 4103 4166 4230 4294 4357 683 4421 4484 1 4648 4611 4676 4739 4802 4866 4929 4993 684 5056 5120 5183 5247 5310 6373 5437 5500 5664 5627 685 5691 6754 ! 5817 5881 5944 6007 6071 6134 6197 6261 686 6324 6387 6461 6514 6677 6641 6704 6767 6830 6894 687 6957 7020 7083 7146 7210 7273 7336 7399 7462 7525 688 7688 7652 7716 7778 7841 7904 7967 8030 8093 8166 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 9865 9918 9981 ..43 692 840106 0169 0232 0294 0367 0420 0482 0546 0608 0671 693 0733 0796 0859 0921 0984 1046 1109 1172 1234 1297 694 1359 1422 1485 1647 1610 1672 1735 1797 1860 1922 695 1985 2047 2110 2172 2235 2297 2360 2422 2484 2547 696 2609 2672 2734 2796 2869 2921 2983 3046 3108 3170 697 3233 3295 3357 3420 3482 3644 3606 3669 3731 3793 698 3865 3918 3980 4042 4104 4166 4229 4291 4353 4415 e. 4477 4539 4601 4664 1 4726 4788 4860 4912 4974 5036 OF NUMBERS. 15 N. 1 2 3 4 5 6 7 8 9 700 846098 6160 6222 5284 5346 5408 6470 5532 5694 5656 701 5718 5780 6842 5904 5966 6028 6090 6161 6213 6275 702 6337 6399 6461 6523 6586 6646 6708 6770 6832 6894 703 6955 7017 7079 7141 7202 7264 7326 7388 7449 7511 704 7573 7634 7676 7768 7819 7831 7943 8004 8066 8128 705 8189 8251 8312 8374 8435 8497 8569 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 708 850033 0095 0156 0217 0279 0340 0401 0462 0524 0586 709 0646 0707 0769 0830 0891 0952 1014 1075 1136 1197 710 1258 1320 1381 1442 1603 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 3029 713 3090 3150 3211 3272 3333 3394 3455 3616 3577 3637 714 3698 3759 3820 3881 3941 4002 4063 4124 4185 4245 715 4306 4367 4428 4488 4649 4610 4670 4731 4792 4852 716 4913 4974 6034 5095 5156 6216 6277 6337 5398 5459 717 5519 5580 5640 5701 5761 5822 6882 5943 6003 6064 718 6124 6185 6246 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 7755 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 9439 9499 9559 9619 9679 724 9739 9799 9859 9918 9978 ..38 ..98 .168 .218 .278 '725 860338 0398 0458 0618 0578 0637 0697 0757 0817 0877 726 0937 0996 1066 1116 1176 1236 1295 1355 1416 1475 727 1634 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 3086 3144 3204 3263 730 3323 3382 3442 3501 3561 3620 3680 3739 3799 3858 731 3917 3977 4036 4096 4166 4214 4274 4333 4392 4452 732 4611 4570 4630 4689 4148 4808 4867 4926 4986 5045 733 6104 5163 5222 6282 6341 5400 5459 6619 5678 5637 734 6696 5755 6814 6874 5933 5992 6051 6110 6169 6228 735 6287 6346 6405 6466 6524 6583 6642 6701 6760 6819 736 6878 6937 6996 7055 7114 7173 7232 7291 7350 7409 737 7467 7526 7685 7644 7703 7762 7821 7880 7939 7998 738 8056 8115 8174 8233 8292 8360 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 9936 9994 ..53 .111 .170 .228 .287 .345 742 870404 0462 0521 0679 0638 0696 0756 0813 0872 0930 743 0989 1047 1106 1164 1223 1281 1339 1398 1456 15i5 744 1573 1631 1690 1748 1806 1866 1923 1981 2040 2098 745 2156 2215 2273 2331 2389 2448 2506 2564 2622 2681 746 2739 2797 2855 2913 2972 3030 3088 3146 3204 3262 747 3321 3379 3437 3495 3653 3611 3669 3727 3786 3844 748 3902 3960 4018 4076 4134 4192 4260 4308 4360 4424 749 4482 4640 4698 4666 4714 4772 4830 4888 4945 5003 16 LOGARITHMS N. 1 2 3 4 6 6 7 8 9 750 8750G1 5119 5177 5235 5293 5351 5409 5466 5524 5582 761 5640 5698 5756 6813 5871 5929 6987 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 755 7947 8004 8062 8119 8177 8234 8292 8349 8407 8464 756 8522 8579 8637 8694 8762 8809 8886 8924 8931 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 0756 760 0814 0871 0928 0985 1042 1099 1156 1213 1271 1328 761 1385 1442 1499 1656 1613 1670 1727 1784 1841 1898 762 1955 2012 2069 2126 2183 2240 2297 2354 2411 2468 763 2525 2581 2638 2696 2762 2809 2866 2923 2980 3037 764 3093 3150 3207 3264 3321 3377 3434 3491 3548 3606 765 3661 3718 3775 3832 3888 3945 4002 4059 4115 4172 766 4229 4285 4342 4399 4455 4512 4569 4626 4682 4739 767 4795 4852 4909 4965 6022 5078 5136 6192 5248 5305 768 6361 5418 5474 5631 6587 6644 6700 5757 5813 5870 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 7606 7561 772 7617 7674 7730 7786 7842 7898 7956 8011 8067 8123 773 8179 8236 8292 8348 8404 8460 8616 8573 8629 8666 774 8741 8797 8853 8909 8965 9021 9077 9134 9190 9246 775 9302 9358 9414 9470 9626 9582 9638 9694 9760 9806 776 9862 9918 0974 ..30 ..86 .141 .197 .253 .309 .365 777 890421 0477 0533 0589 0646 0700 0756 0812 0868 0924 778 0980 1035 1091 1147 1203 1259 1314 1370 1426 1482 779 1537 1693 1649 1705 1760 1816 1872 1928 1983 2039 780 2095 2150 2206 2262 2317 2373 2429 2484 2540 2595 781 2651 2707 2762 3818 2873 2929 2985 3040 3096 3161 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 4638 4593 4648 4704 4759 4814 785 4870 4926 4980 5036 5091 5146 6201 5267 5312 6367 786 5423 5478 5533 5688 5644 5699 5754 6809 5864 6920 787 5976 6030 6085 6140 6196 6261 6306 6361 6416 6471 788 6526 6581 6636 6692 6747 6802 6867 6912 6967 7022 789 7077 7132 7187 7242 7297 7362 7407 7462 7517 7672 790 7627 7683 7737 7792 7847 7902 7967 8012 8067 8122 791 8176 8231 8286 8341 8396 8451 8506 8661 8615 8670 792 8725 8780 8836 8890 8944 8999 9054 9109 9164 9218 793 9273 9328 9383 9437 9492 9647 9602 9656 9711 9766 794 9821 9875 9930 9986 ..39 ..94 .149 .203 .258 .312 795 900367 0422 0476 0631 0586 0640 0695 0749 0804 0869 796 0913 0968 1022 1077 1131 1186 1240 1295 1349 1404 797 1458 1513 1567 1622 1676 1736 1785 1840 lfc94 1948 798 2003 2057 2112 2166 2221 2275 2329 2384 2438 2492 799 2547 2601 2656 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 3968 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 805 5796 5850 5904 5958 6012 6066 6119 6173 6227 6281 806 6335 6389 6443 6497 6551 6604 6658 6712 6766 6820 807 6874 6927 6981 7035 7089 7143 7196 7250 7304 7358 808 7411 7465 7619 7573 7626 7680 7734 7787 7841 7896 809 7949 8002 8056 8110 8163 8217 8270 8324 8378 8431 810 8485 8539 8592 8646 8699 8763 8807 8860 8914 8967 811 9021 9074 9128 9181 9236 9289 9342 9396 9449 9603 812 9556 9610 9663 9716 9770 9823 9877 9930 9984 ..37 813 910091 0144 ^197 0251 0304 0358 0411 0464 0518 0571 814 0624 0878 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 2063 2115 2169 817 2222 2275 2323 2381 2435 2488 2541 2594 2645 2700 818 2753 2806 2859 2913 2966 3019 3072 3125 3178 3231 819 3284 3337 3390 3443 3496 3649 3602 3656 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 5083 5136 5189 5241 5594 6347 823 5400 5453 5505 5558 5611 5664 5716 6769 5822 5875 824 5927 6980 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 7606 7558 7611 7663 7716 7768 7820 7873 7925 7978 828 8030 8083 8185 8188 8240 8293 8345 8397 8450 8602 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 0641 0693 1114 833 0645 0897 0749 0801 0863 0906 0958 1010 1062 834 1166 1218 1270 1322 1374 1426 1478 1530 1682 1634 835 1686 1738 1790 1842 1894 1946 1998 2060 2102 2154 836 2206 2258 2310 2362 2414 2466 2618 2570 2622 2674 837 2725 2777 2829 2881 2933 2985 3037 3089 3140 3192 838 3244 3296 3348 3399 3451 3603 3556 3607 3668 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 6054 6106 5157 6209 5261 842 5312 6364 5415 6467 6518 6570 5621 5673 6725 5776 843 5828 5874 6931 5982 6034 6086 6137 6188 6240 6291 844 6342 6394 6445 6497 6548 6600 6651 6702 6754 6805 845 6857 6908 6959 7011 7062 7114 7165 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 9061 9112 9163 9216 9266 9317 9368 18 LOGARITHMS N. 929419 1 1 3 3 4 5 6 7 8 9 850 9473 9521 9672 9623 9674 9725 9776 9827 9879 851 9930 9981 ..32 ..83 .134 .185 .236 .287 .338 .389 852 930440 0491 0542 0592 0643 0694 0745 0796 0847 0898 853 0949 1000 1051 1102 1163 1204 1254 1305 1356 1407 854 1458 1609 1560 1610 1661 1712 1763 1814 1865 1915 855 1966 2017 2068 2118 2169 2220 2271 2322 2372 2423 856 2474 2624 2676 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 4196 4246 4269 4347 4397 4448 860 4498 4549 4599 4650 4700 4751 4801 4852 4902 4953 861 5003 6054 5104 5154 5205 6255 6306 535t? 6406 5467 862 6507 5558 6608 5658 6709 5769 5809 68G0 5910 5960 863 6011 6061 6111 6162 6212 6262 6313 6363 6413 6463 864 6514 6664 6614 6665 6715 6765 6816 6865 6916 6966 865 7016 7066 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 8620 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 871 940018 0068 0118 0168 0218 0267 0317 0367 0417 0467 872 0616 0566 0616 0666 0716 0765 0815 0866 0916 0964 873 1014 1064 1114 1163 1213 1263 1313 1362 1412 1462 874 1611 1561 1611 1660 1710 1760 1809 1869 1909 1958 875 2008 2058 2107 2157 2207 2256 2306 2355 2405 2455 876 2504 2664 2603 2653 2702 2762 2801 2851 2901 2950 877 3000 3049 3099 3148 3198 3247 3297 3346 3396 3445 878 3495 3644 3693 3643 3692 3742 3791 3841 3890 3939 879 3989 4038 4088 4137 4186 4236 4285 4336 4384 4433 880 4483 4632 4581 4631 4680 4729 4779 4828 4877 4927 881 4976 5025 5074 5124 6173 6222 5272 5321 6370 5419 882 6469 5618 5667 6616 5665 6715 5764 6813 6862 5912 883 6961 6010 6059 6108 6157 6207 6256 6305 6354 6403 884 6462 6501 6661 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 887 7924 7973 8022 8070 8119 8168 8217 8266 8315 8365 888 8413 8462 8511 8560 8609 8667 8706 8755 8804 8a53 889 8902 8961 8999 9048 9097 9146 9195 9244 9:^y2 9341 890 9390 9439 9488 9636 9585 9634 9683 9731 9780 9829 891 9878 9926 9976 ..24 ..73 .121 .170 .219 .267 .316 892 950365 0414 0462 0511 0560 0608 0657 0706 0754 0803 893 0851 0900 0949 0997 1046 1096 1143 1192 1240 1289 894 1338 1386 1435 1483 1532 1580 1629 1677 1726 17.5 895 1823 1872 1920 1969 2017 2066 2114 2163 2211 2260 896 2308 2356 2405 2453 2602 2550 2599 2647 6696 2744 897 2792 2841 2889 2938 2986 3034 3083 3131 3180 3228 898 3276 3325 3373 3421 3470 3518 3566 3615 3663 3711 899 3760 3808 3856 3905 3963 4001 4049 4098 4146 4194 OF NUMBERS. 19 N. 1 2 3 4 « 6 7 8 9 900 954243 4291 4339 4387 4435 4484 4532 4580 4628 4677 901 4725 4773 4821 4869 4918 4966 5014 5062 5110 5158 902 5207 5255 5303 5351 5399 5447 5495 5543 5592 5640 903 5688 5736 5784 5832 5880 5928 5976 6024 6072 6120 904 6168 6216 6265 6313 6361 6409 6457 6505 6553 6601 905 6649 6697 6745 6793 6840 6888 6936 6984 7032 7080 906 7128 7176 7224 7272 7320 7368 7416 7464 7612 7659 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 9471 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 U36 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 2606 2653 2701 2748 2795 918 2843 2890 2937 2985 3032 3079 3126 3174 3221 3268 919 3316 3363 3410 3457 3604 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 5108 6155 923 5202 5249 5296 5343 5390 5437 5484 5531 5578 5626 924 5672 5719 5766 5813 5860 6907 5964 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 1925 1971 2018 2064 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 3969 4005 942 4051 4097 4143 4189 4235 4281 4327 4374 4i20 4466 943 4512 4558 4604 4650 4696 4742 4788 4834 4880 4926 944 4972 5018 5064 5110 5156 5202 5248 5294 5340 5386 945 5432 5478 5524 5570 5616 5662 5707 5753 6799 5845 946 5891 5937 5983 6029 6075 6121 6167 6212 6258 6304 947 6350 6396 6442 6488 6533 6579 6925 6671 6717 6763 948 6808 6854 6900 6946 6992 7037 7083 7129 7176 7220 949 7266 7312 7358 j 7403 7449 7495 7541 7586 7632 7678 19 20 LOGARITHMS N. 1 2 3 4 5 6 7 8 ■ 9 950 977724 7769 7815 7861 7906 7952 7998 8043 8089 8135 951 8181 8226 8272 8317 8363 8409 8454 8500 8546 8591 952 8637 8683 8728 8774 8819 8865 8911 895(3 9002 9047 953 9093 9138 9184 9230 9275 9321 9360 9412 9457 9508 954 9548 9594 9639 9685 9730 9776 9821 9867 9912 9958 955 980003 0049 0094 0140 0185 0231 0276 0322 0367 0412 956 0458 0503 0549 0594 0640 0685 0730 0776 0821 0867 957 0912 0957 1003 1048 1093 1139 1184 1229 1275 1320 958 1366 1411 1466 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 962 3175 3220 3265 3310 3356 3401 3446 3491 3536 3581 963 3626 3671 3716 3762 3807 3862 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 6202 6247 5292 5337 6382 967 B426 6471 6516 5661 6606 6651 5699 6741 5786 5830 968 6875 5920 5965 6010 6055 6100 6144 6189 6234 6279 969 6324 6369 6413 6468 6503 6548 6593 6637 6682 6727 970 6772 6817 6861 6906 6961 6996 7040 7035 7130 7176 971 7219 7264 7309 7353 7398 7443 7488 7532 7577 7622 972 7666 7711 7766 7800 7846 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 9301 9405 976 9450 9494 9539 9583 9628 9672 9717 9761 9803 9850 977 9895 9939 9983 ..28 ..72 .117 .161 .208 .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 1369 1403 1448 1492 1536 1580 1626 981 1669 1713 1768 1802 1846 1890 19.'?5 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 3668 3613 3657 3701 3745 3789 3833 986 3877 3921 3966 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 6066 5108 5152 989 5196 5240 6284 6328 5372 6416 5460 6504 5647 5691 990 5635 5679 5723 5767 5811 5864 6898 5942 5986 ^30 991 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 7617 7661 7605 7648 7692 7736 7779 995 7823 7867 7910 7954 7998 8041 8086 8129 8172 8216 996 8259 8303 8347 8390 8434 8477 8521 8564 8608 8652 997 8695 8739 8792 8826 8869 8&13 8956 9000 9043 9087 998 9131 9174 9218 9261 9306 9348 9392 9435 9479 9522 999 9565 9609 9652 9696 9739 9783 9826 9870 9913 9957 TABLE ir. Log. Sines and Tangents. (0°) Natural Sines. 21 / Sine. D.IO" Cosine. D.IO" Tang. D.IO" Coiang. N.sine. N. COS. 0.000000 10.000000 0.000000 Infinite. 00000 100000 60 1 6.463726 000000 6.463726 18.536274 00029 100000 59 2 764756 000000 764756 235244 00058 100000 58 3 940847 000000 940847 059153 00087 100000 57 4 7.065786 000000 7.065786 12.934214 00116 100000 56 5 162696 000000 162696 837304 00145 100000 55 6 241877 9.999999 241878 758122 00175 100000 54 7 308824 999999 308825 691175 00204 lOOODO 53 8 366816 999999 366817 633183 00233 100000 52 9 417968 999999 417970 582030 00262 100000 51 10 463725 999998 463727 536273 00291 100000 50 11 7.505118 9.999998 7.505120 12.494880 003201 99999 49 12 542908 999997 542909 457091 00349 99999 48 13 577668 999997 577672 422328 00378 99999 47 14 609853 999996 609857 390143 00407 99999 46 15 639816 999996 639820 360180 00436 99999 46 16 667845 999995 667849 332151 00465 99999 44 17 694173 999995 694179 305821 00495 99999 43 18 718997 999994 719003 280997 00524 99999 42 19 742477 999993 742484 257516 00553 99998 41 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 089106 00814 99997 32 29 928119 999985 926134 073866 00844 99996 31 30 940842 999983 940858 059142 00873 99996 30 31 7.955082 2298 2227 2161 2098 2039 1983 9.999982 0.2 0.2 7.955100 2298 2227 2161 2098 2039 1983 1930 1880 1833 1787 1744 1703 1664 1627 1591 1557 1524 1493 1463 1434 1406 1379 1353 1328 1304 1281 1259 1238 1217 12.044900 00902 99996 29 32 968870 999981 968889 031111 00931 99996 28 33 ' 982233 999980 982253 017747 00960 99995 27 34 995198 999979 0*2 995219 004781 00989 99995 26 35 8.007787 999977 0-2 0-2 8.007809 11.992191 01018 99995 26 36 020021 999976 020045 979955 01047 99995 24 37 031919 999975 0-2 031945 968055 01076 99994 23 38 043501 1930 1880 1832 1787 1744 1703 1664 1626 1591 1557 1524 1492 1462 1433 1405 1379 1353 1328 1304 1281 1259 1237 1216 999973 0-2 0-2 0-2 0-2 0-2 0*2 0'2 0*3 0'3 o;3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.4 0.4 0.4 0.4 043527 956473 01105 99994 22 39 054781 999972 054809 945191 01134 99994 21 40 085776 999971 065806 934194 01164 99993 20 41 8.076500 9.999969 8.076531 11.923469 01193 99993 19 42 086965 999968 086997 913003 01222 99993 18 43 097183 999966 097217 902783 01251 99992 17 44 107167 999964 107202 892797 01280 99992 16 45 116926 999963 116963 883037 01309 99991 15 46 126471 999961 126610 873490 01338 99991 14 47 135810 999959 135851 864149 01367 99991 13 48 144953 999958 144996 855004 01396 99990 12 49 153907 999956 153952 846048 01425 99990 11 50 162681 999954 162727 837273 01454 99989 10 51 8.171280 9.999952 8.171328 11.828672 01483 99989 9 52 179713 999950 179763 820237 01513 99989 8 53 187985 999948 188036 811964 01542 99988 7 54 196102 999946 196156 803844 01571 99988 6 55 204070 999944 204126 795874 01600 99987 6 56 2X1895 999942 211953 788047 780359 01629 99987 4 57 219581 999940 219641 01658 99986 3 58 227134 999938 227195 772805 i 01687 99986 2 59 234557 999936 234621 765379 ! 01716 99985 1 60 241855 999934 241921 758079 ; 01745 99985 / Cosine. Sine. Co til lie:. 'I'anff. ' N. COS. N. sine- 89 Degrees. | 22 Log. Sines and Tangents. (1°) Natural Sines, TABLE II. Sine. 8.241855 1 9.dQf«5i 1 249033 256094 3 263042 4 "'^"'^- 5 6 7 8 9 10 118 12 13 14 15 16 17 18 19 20 21 8 22 23 24 25 26 27 1196 1177 1158 1140 1122 1105 1088 1072 1056 - -: 1041 o^^ 1027 ''^" 1012 276614 283243 289773 296207 302546 308794 .314954 321027 327016 332924 338763 344504 350181 985 971 959 946 355783 322 361315 qi/j 366777 .372171 377499 ^;; 382762 Q^i 387962 ^L 393101 g^ 398179 Q^ 403199 ^i 408161 gio 413068 SAq 417919 S^n 8.422717 ^XV I 427462 'l^ 432156 ^«f 436800 7fi^ 441394 l^ 446941 'Tq 460440 742 454893 «oK 459301 70? 40 463666 LiL 41 8.467986 if^ 42 472263 XA^ 43 476498 '^ 44 480693 ^^ 46 484848 ^^^ 46 488963! ^^^ 47 4930401 ^'^ 48 4970781 XA« 49 601080 X^; 60 606045 1 ^J 51 8.608974 ^^J 62 512867 ^J^ 516726 ^^^ 520651 ^^' 624343 ?.t>i 628102 ^^^ 631828 ^;^ 635523 ^Ji 539186 ^^^ 642819 ^^^ 29 30 31 32 33 34 35 36 37 38 39 40 D.IO" Cosine. 53 54 55 56 57 68 59 9.999934 999932 999929 999927 999925 999922 999920 999918 999915 999913 999910 9.999907 999905 999902 999899 999897 999894 999891 999888 I Cosir D.IO' 9.999879 999876 999873 999870 999867 999864 999861 999858 999854 999861 9.999848 999844 999841 999838 999834 999831 999827 999823 999820 999816 9.999812 999809 999805 999801 999797 999793 999790 999786 999782 999778 9.99977.f 999769 999765 999761 999757 999753 999748 999744 999740 999735 0-4 0.4 0.5 0.5 0.5 0.5 06 0.5 0.5 0.6 0.6 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0-6 0.6 0.6 Sine. 0.6 0.6 0-6 0.6 0.6 0-6 0-6 0.6 0.6 0.7 0-7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 8.241921 249102 256165 263116 269956 276691 283323 289856 296292 302634 308884 8.315046 321122 327114 333025 333856 344610 350289 355895 361430 366895 8.372292 377622 Tang. 388092 393234 398315 403338 408304 413213 418068 8.422869 427618 432315 436962 441560 446110 450613 455070 459481 463849 8.468172 472454 476693 480892 485060 489170 493250 497293 501298 505267 509200 513098 5169ol 620790 524586 528349 532080 535779 539447 543084 Cnlaii^r- D.IO' 1197 1177 1158 1140 1122 1105 1089 1073 1057 1042 1027 1013 999 985 972 959 946 934 922 911 899 888 879 867 857 847 837 828 818 809 800 791 783 774 766 758 750 743 735 728 720 713 707 700 693 686 680 674 668 661 656 650 644 638 633 627 622 616 611 606 Cotang. 11.758079 750898 743835 736885 730044 723309 716677 710144 703708 697366 691118 11-684954 678878 672886 666975 661144, 055390 649711 644105 638570 633105 11.627708 622378 617111 611908 606766 ■ 601685 596662 691696 586787 581932 11.577131 572382 667685 563038 558440 553890 549387 644930 540519 536161 11 .531828 527546 523307 619108 614950 510830 506750 502707 498702 494733 11.490800 486902 483039 479210 475414 471651 467920 464221 460653 456916 N. sine. N. cos. 01742 01774 01803 01832 01862 01891 01920 i 01949 01978 02007 02U31 02065 02094 0212s miou 02181 02211 02240 02269 02298 02327 02356 02385 02414 02443 024' 02501 02530 02560 99985 99984 99934 99983 99983 99982 99982 99^81 99980 99980 ^9979 )y979 99978 99977 J:.977 99976 99976 i>9975 Jy974 99974 99973 yi>972 99972 99971 99970 2 y9969 99969 99968 99967 02589 99966 'i"^ 02618 02647 02076 02705 02734 102763 02792 02821 02850 02879 02908 02938 02967 0299G 03026 03054 03083 03112 03141 03170 03199 03228 0q257 I 03286 103316 103645 I I 0337 ! 103403 1:03432 j( 03461 I ! 03490 I ' N. COS. ^9966 99965 99964 99963 99963 99962 99961 99960 99959 99959 99958 99957 99956 99955 99954 99953 99952 99952 99951 99950 99949 99948 99947 99946 99945 99944 4,99943 99942 99941 99940 y9939 88 Degreep. TABLE II. Log. Sines and Tangents. (-P) Natural Sines. 23 iS.ne. D. 10" Cosine. D. 10"| Tang. D. 10" Colang. j jN. sine. N. cos 8.542819 54G422 549995 553539 557054 560540 663999 567431 570836 574214 677566 8.580892 684193 587469 590721 593948 597152 600332 603489 606623 609734 8.612823 615891 618937 621962 624965 627948 630911 633854 636776 639680 8.642663 645428 648274 651102 65.3911 656702 659475 662230 664968 667689 670393 673080 675751 678405 681043 683665 686272 688863 691438 693998 8.696543 699073 701589 704090 706577 709049 711507 713952 716383 718800 Cosine. 600 595 591 586 581 576 672 667 563 559 654 550 646 642 538 634 530 526 622 619 615 511 608 504 501 497 494 490 487 484 481 477 474 471 468 466 462 459 466 453 451 448 445 442 440 437 434 432 429 427 424 422 419 417 414 412 410 407 405 403 ,999735 999731 999726 999722 999717 999713 999708 999704 999G99 999694 999689 .999685 999680 999675 999670 999665 999660 999665 999650 999645 999640 .999635 999629 999324 999619 999614 999608 999603 999597 999592 999586 .999581 999575 999570 999564 999558 999553 999547 999641 999535 999629 .999624 999518 999512 999506 999500 999493 999481 999475 999469 .999463 999456 999450 999443 999437 999431 999424 999418 999411 999404 Si I 0.7 0.7 0.7 0-8 0-8 0-8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.9 0.9 0.9 0.9 0.9 09 0.9 0.9 0.9 0.9 0-9 0.9 0.9 0.9 0.9 1.0 1-0 1.0 1.0 1-0 1.0 1-0 1.0 1-0 1-0 1.0 1.0 1.0 1.0 1.0 1.0 1 1 1 .643084 546691 650268 553817 557336 560828 564291 567727 571137 574520 577877 .581208 584614 587795 691051 594283 597492 600677 603839 606978 610094 .613189 616262 619313 622343 625352 628340 631308 634256 637184 640093 .642982 646863 648704 651537 654362 657149 659928 662689 665433 668160 .670870 673663 676239 678900 681544 684172 6 6784 689381 691963 694529 .697081 699617 702139 704246 707140 709618 702083 714634 716972 719396 Cotang. 602 596 591 587 682 677 573 668 664 659 555 551 547 543 539 535 531 527 523 519 516 512 508 605 501 498 495 491 488 485 482 478 475 472 469 466 463 460 457 454 453 449 446 443 442 438 435 433 430 428 425 423 420 418 415 413 411 408 406 404 11.456916! I 03490 453309 1 1 03519 4497321103548 446183; 0357 442664; 439172: 435709 1 1 432273 1' 428863 I 425480 ; 422123 I 11.418792, 415486 412205 408949 405717 402508 399323 396161 393022 389906 11.386811 383738 380687 03606 03635 03664 03693 03723 03752 03781 03810 03839 03868 03897 03926 03955 03984 04013 04042 04071 04100 03129 04159 377657 104188 374648! 04217 371660:104246 04275 04304 04333 04362 04391 04420 04449 04478 04507 04536 04 04594 04623 04653 04682 04711 04740 04769 04798 04827 368692'! 365744 ' 362816 359907 ,357018 354147 351296 348463 346648 342851 340072 337311 334567 331840 11.329130 326437 323761 321100 318456 316828 313216 1 104856 310619! 104885 308037 1104914 305471 302919 300383 297861 295354 292860 290382 287917 285465 283028 280604 04943 04972 05001 05030 05059 0508b 051 1: U5146 05175 05205 05234 Tanff. ! N. COS. N.sine. 99939 99938 99937 99936 99935 99934 99933 99932 99931 99930 99929 99927 99926 99925 99924 99923 99922 99921 99919 99918 99917 99916 99915 99913 99912 99911 99910 99909 99907 99906 99905 99904 99902 99901 99900 99898 99897 05-99896 99894 ^9893 99892 99890 99889 99888 99886 99885 99883 99882 99881 99879 99878 99876 99875 99873 998 ?2 99870 99869 9^867 99866 99864 99S6o 60 59 58 57 66 56 64 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 87 Degrees. 24 Log, Sines and Tangents. (3°; Natural Sines. TABLE II. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Sine. 8.718800 721204 723595 725972 728337 730688 733027 735354 737667 739969 742259 8.744536 746802 749055 751297 753528 755747 757955 760151 762337 764511 8.766675 768828 770970 773101 775223 777333 779434 781524 783605 785675 8.787736 789787 791828 793859 795881 797894 799897 801892 803876 805852 8.807819 809777 811726 813667 815599 817522 819436 821343 823240 825130 8.827011 828884 830749 832607 834456 836297 838130 839956 841774 843585 Cosine. D. 10" Cosine. 401 398 396 394 392 390 388 386 384 382 380 378 376 374 372 370 364 362 361 359 357 355 353 352 350 348 347 345 343 342 340 339 337 335 334 332 331 329 328 326 325 323 322 320 319 318 316 315 313 312 311 309 308 307 306 304 303 302 1.999404 999398 999391 999384 999378 999371 999364 999357 999350 999343 999336 1.999329 999322 999315 999308 999301 999294 999286 999279 999272 999265 1.999257 999250 999242 999235 999227 999220 999212 999205 999197 999189 .999181 999174 999166 999158 999150 999142 999134 999126 999118 999110 .999102 999094 999086 999077 999069 999061 999053 999044 999036 999027 .999019 999010 999002 998993 998984 998976 998967 998958 998950 998941 Sine. D. 10' 1. 1. 1. 1. 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.5 1.5 1.5 Tang. D. 10' 8.719396 721806 724204 726588 728959 731317 733663 735996 738317 740626 742922 8.745207 747479 749740 751989 754227 756453 758668 760872 763065 765246 8.767417 769578 771727 773866 775995 778114 780222 782320 784408 786486 8.788554 790613 792662 794701 796731 798752 800763 802765 804858 806742 8.808717 810683 812641 814589 816629 818461 820384 822298 824205 826103 8.827992 829874 831748 833613 835471 837321 839163 840998 842825 844644 Cotanjr. 402 399 397 395 393 391 389 387 385 383 381 379 377 375 373 371 369 367 365 364 362 360 358 356 355 353 351 350 348 346 345 343 341 340 338 337 335 334 332 331 329 328 326 325 323 322 320 319 318 316 315 314 312 311 310 308 307 306 304 303 05234 I 05263 ] 05292 05321 I 05350 i 05379 05408 05437 I 05466 ; 05495 05524 05553 05582 05611 05640 05669 05698 05727 05756 05785 05814 05844 05873 05902 05931 Cotang. | fN. sine 11.280604 278194 275796 273412 271041 268683 266337 264004 261683 259374 257078 11 .254793 252521 250260 248011 245773 243547 241332 239128 236935 234754 11.232583 230422 228273 226134 224005 221886 219778 217680 215592 213514 11.211446 209387 207338 205299 203269 201248 199237 197235 195242 193258 11.191283 189317 187359 185411 183471 181539 179616 177702 175795 173897 11.172008 170126 168252 166387 164529 162679 160837 159002 157175 155356 99863 99861 99860 99858 9985 99856 99854 99852 99851 99849 99847 99846 99844 99842 99841 99839 99838 99836 99834 99833 99831 99829 9982 99826 99824 0596099822 05989 06018 06047 I 06076 06105 06134 06163 1106192 ! 106221 06250 06279 06308 06337 06366 06395 06424 06453 06482 06511 06540 01)569 06598 06627 06656 , 06685 I 06714 :! 06743 ;! 06773 ;i 06802 ;' 06831 I I 06860 1 06889 108918 106947 ': 06976 Tang. N. COS. 99821 99819 9981 99815 99813 99812 99810 99808 99806 99804 99803 99801 99799 99797 99795 99793 99792 99790 99788 99786 99784 99782 99780 99778 99776 99774 99772 99770 99768 99766 99764 99762 99760 99758 99756 N. COS. N.sine, 86 Degrees. TABLE II. Log. Sines and Tangents. (4°) Natural Sines. 25 Sine. D. 10" Cosine. Cosine. .998941 998932 998923 998914 998905 998887 998878 998869 998860 998861 .998841 998832 998823 998813 998804 998795 998785 998776 998766 998757 .998747 998738 998728 998718 998708 998679 998669 998669 .998649 998639 998629 998619 998609 998599 998589 998678 998568 998558 .998548 998537 998527 998516 998506 998495 998486 998474 998464 998463 .998442 998431 998421 998410 998399 r98388 998377 998366 998355 998344 Sine. D. 10" Tang. .844644 846455 848260 850057 851846 853628 855403 867171 858932 860686 862433 .864173 865906 867632 869361 871064 872770 874469 876162 877849 879529 .881202 882869 884530 886185 887833 889476 891112 892742 894366 895984 ;. 897596 899203 900803 902398 903987 905570 907147 908719 910285 911846 ;. 913401 914951 916495 918034 919568 921096 922619 924136 925649 927156 !. 928658 930155 931647 933134 934616 936093 937565 939032 940494 941952 Cotang. D. 10" 302 301 299 298 297 296 295 293 292 291 290 289 288 287 285 284 283 282 281 280 279 278 277 276 275 274 273 272 271 270 269 268 267 266 265 264 263 262 261 260 259 258 257 256 256 255 254 253 252 251 250 249 24y 248 247 246 245 244 244 243 Cotang. I IN. sine. N. cos. 11 11.155356 153545 151740 149943 148154 146372 144597 142829 141068 139314 137567 135827 134094 132368 130649 128936 127230 125531 123838 122151 120471 11.118798 117131 115470 113815 112167 110524 108888 107258 105634 104016 11.102404 100797 099197 097602 096013 094430 092853 091281 089715 088154 11.086599 085049 083505 081966 080432 078904 077381 075864 074351 072844 071342 069845 068353 066866 065384 063907 062435 060968 059506 058048 06976 07005 07034 07063 07092 07121 07150 07179 07208 07237 07266 07295 07324 07353 07382 07411 07440 07469 07498 07627 07556 07585 07643 07672 07701 07730 I j 07759 I! 077 li 07817 !j 07846 II 07875 ■ I 07904 I 07933 ^07962 ! I 07991 i 108020 108049 i 08078 I 08107 99756 99754 99752 99760 99748 99746 99744 99742 99740 99738 99736 99734 99731 99729 99727 99725 99723 99721 99719 99716 99714 99712 0761499710 99708 99705 99703 99701 99699 99696 99694 99692 99689 99687 99685 99683 99680 99678 99676 99673 y9671 08136 99668 11 108165 108194 108223 j I 08252 1 1 08281 i] 08310 1 108339 '! 08368 I j 08397 i 08426 I j 08455 i 1 08484 108513 : 08542 -085 I i 08600 '108629 : 108658 ijose : 08716 N. COS. N.sine. 99666 99664 99661 99659 99657 99654 99652 99649 99647 99644 99642 99639 99637 99635 99632 99630 9y627 99625 99622 99619 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 46 44 43 42 41 40 39 38 37 36 35 34 33 33 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 85 Degrees. 26 Log. Sines and Tangents. (5°) Natural Sines. TABLE 11, 2 3 4 B 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Sine. 8.940296 941738 943174 944608 946034 947456 948874 950287 951696 953100 954499 8.955894 957284 958670 960052 961429 962801 964170 965534 966893 968249 8.969600 970947 972289 973628 974962 976293 977619 978941 980259 981573 8.982883 984189 985491 986789 988083 989374 990660 991943 993222 994497 8.995768 997036 998299 999560 [9.000816 002089 003318 004563 005805 007044 9.008278 009510 010737 011962 013182 014400 015613 016824 018031 019235 D. 10 240 239 239 238 237 236 235 235 234 233 232 232 231 230 229 229 228 227 227 226 225 224 224 223 222 222 221 220 220 219 218 218 217 216 216 215 214 214 213 212 212 211 211 210 209 209 208 208 207 206 206 205 205 204 203 203 202 202 201 201 Cosine. 9.998344 998333 998322 998311 998300 998289 998277 998266 998255 998243 998232 998220 998209 998197 998186 998174 998163 998151 998139 998128 998116 9.998104 998092 998080 998068 998056 998044 998032 998020 997996 9.997984 997972 997959 997947 997935 997922 997910 997897 997885 997872 9.997860 997847 997835 997822 997809 997797 997784 997771 997758 997745 997732 997719 997706 997693 997680 997667 997654 997641 997628 997614 Cosine. D. 10" 1.9 1.9 1.9 1.9 1.9 1.9 1.9 1.9 1.9 1.9 1.9 1.9 1.9 1.9 1.9 1.9 1.9 1.9 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 Tang. 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.2 2.2 2.2 2.2 2.2 2.2 Sine. 8.941952 943404 944852 946295 947734 949168 950597 952021 953441 954856 956267 8.957674 959075 960473 961866 963255 964639 966019 967394 968766 970133 8.971496 972855 974209 975560 976906 978248 979586 980921 982251 983577 8.984899 986217 987532 988842 990149 991461 992750 994045 995337 996624 8.997908 999188 000465 001738 003007 004272 005534 006792 008047 009298 010546 011790 013031 014268 015502 016732 017959 019183 020403 021620 D. 10"l Cotang. {iN. sine. N. cos Cotang. 242 241 240 240 239 238 237 237 236 235 234 234 233 232 231 231 230 229 229 228 227 226 226 225 224 224 223 222 222 221 220 220 219 218 218 217 216 216 215 215 214 213 213 212 211 211 210 210 209 208 208 207 207 206 206 205 204 204 203 203 11.058048! '08716 056596 i I 08745 0551481108774 0537051108803 052266! 1 08831 050832 1 08860 049403 047979! 108918 046559! 108947 045144 'i 08976 043733 11.042326 040925 039527 038134 036745 035361 033981 032606 031234 029867 11.028504 027145 025791 '■ 09005 ■ 09034 ' 09063 1 09092 109121 109150 i 09179 1 09208 109237 ! 09266 1 09295199567 I 09324 99564 ' 09353199562 : 09382!99559 024440 j I 09411 199556 023094 :l09440i99553 021752 1 109469199551 0204141 1 09498199548 0190791 1 09527199545 99619 99617 99614 99612 99609 9960 99604 99602 99599 99596 99594 99591 99588 99586 99583 99580 99578 99575 99572 99570 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 46 44 43 42 41 40 39 38 37 36 36 34 33 32 017749 016423 .015101 013783 012468 011158 009851 09556 1995421 31 09585199540 09614J99537 09642 199534 09671,99531 09700199528 09729199526 008549 ii06758|99523 007250 l| 09787 99520 005955 I j 09816199517 004663 11 09845199514 003376 1 09874199511 .002092 000812 10.999535 998262 996993 995728 09903 ;99508 09932 J99o0t 09961199503 09990199500 10019:99497 10048|99494 994466!! 10077199491 9932081110106 99488 991953! 1 10135|99485 990702 i: 1016499482 10192,99479 1022199476 10250 99473 10279 99470 10308 199467 983268 I i 1033/199464 983041 1 1 10366,99461 980817;! 10395199458 979597 1 1 10424199155 978380 ij 10453 J99452 Tang. I ! N. cos.|N.pine. 10.989454 988210 686969 985732 984498 84 Degrees. TABLE II. Log. Sines and Tangents. Natural Sines. C!otang. | N. sine. N. cos. Sine. D. 10" 9.019235 020436 021632 022825 024016 025203 026386 027567 028744 029918 031089 9.032257 033421 034582 035741 030896 038048 039197 040342 041485 042625 9.043762 044895 046026 047154 048279 049400 050519 051635 052749 053859 9.054966 056071 057172 058271 059367 060460 061551 062639 063724 064806 9.066886 066962 068036 069107 070176 071242 072306 073366 074424 075480 9.076633 077683 078631 079676 080719 081759 082797 083832 084864 085894 Cosine. 200 199 199 198 198 197 197 196 196 195 195 194 194 193 192 192 191 191 190 190 189 189 180 188 187 187 186 186 185 185 184 184 184 183 183 182 182 181 181 180 180 179 179 179 178 178 177 177 176 176 176 176 176 174 174 173 173 172 172 172 Cosine. D. 10" .997614! 997601 i 997588 j 997574 ! 997561 I 997547 j 997534 997520 ! 997507 I 997493 I 997480 i .997466! 997452 997439 ! 997425 i 997411 997397 997383 997369 997355 997341 .997327 997313 997299 997285 997271 997257 997242 997228 997214 997199 .997186 997170 997156 997141 997127 997112 997098 997083 997068 997053 .997039 997024 997009 996994 996979 996964 996949 996934 996919 996904 .996889 996874 996858 996843 996828 996812 996797 996782 996766 996751 Sine. 2.2 2.2 2.2 2.2 2.2 2.2 2.3 2.3 2.3 2 2 2 2 2. 2 2.3 2-3 2-3 2-3 2-3 2.3 2-4 2-4 2.4 2.4 2-4 2.4 2.4 2.4 2 2 2 2. 2 2 2.4 2.4 2.4 2.5 2.5 2-5 2-6 2-5 2.6 2.6 2.6 2-5 2.5 2.5 2.5 2.5 2.6 2.6 2.6 2.6 2.5 2.6 2 6 2.6 2.6 Tang. iD. 10" 9.021620 022834 024044 025251 026455 027655 028852 030046 031237 032425 033609 9.034791 035969 037144 038316 039485 040651 041813 042973 044130 045284 9.046434 047582 048727 049869 061008 052144 053277 054407 055636 066659 9.057781 058900 060016 061130 062240 063348 064453 065666 066655 067762 9.068846 069038 071027 072113 073197 074278 076356 076432 077505 078576 9.079644 080710 081773 082833 083891 084947 086000 087050 088098 089144 Cotang. > Degrees. 202 202 201 201 200 199 199 198 198 197 197 196 196 195 195 194 194 193 193 192 192 191 191 190 190 189 189 188 188 187 187 186 186 185 185 185 184 184 183 183 182 182 181 181 181 180 180 179 179 178 178 178 177 177 176 176 176 175 175 174 10.9783801 977166 ; 975956 ' 974749 i 973545 972345 971148 969954 ! 968763 [ 967676 i 966391 [ 10.965209! 964031 962856 i 961684 1 960515 I 959349 I 958187 I 957027 ! 955870 1 964716 I 10.953566 952418 951273 950131 948992 947856 946723 945693 944465 943341 10.942219 941100 939984 i 938870 1 937760 i 936652 I 935547 I 934444 I 933346 I 932248 ! 10.931164 930062 I 928973 927887 926803 925722 924644 923568 922495 921424 10.920356 919290 918227 917167 916109 915063 914000 912960 911902 910856 I Ta^g^ 99452 99449 99446 99443 99440 99437 99434 99431 9y428 99424 99421 99418 y9415 99412 99409 99406 99402 99399 99396 99393 99390 99386 99383 99380 99377 99374 99370 99367 99364 99360 99357 99354 99351 99347 99344 99341 99337 99334 99331 99327 99324 99320 99317 99314 99310 99307 99303 99300 99297 99293 99290 99286 99283 99279 99276 99272 99269 99205 99262 99268 i>9255 N. COS. N.sine, 0463 0482 0511 0540 0569 0.39'- 0626 0655 0684 0713 0742 0771 0800 0829 0858 0887 0916 0945 0973 002 031 060 089 118 147 176 205 234 263 291 320 349 378 407 436 465 494 523 652 580 609 638 667 696 725 754 783 812 840 869 898 927 956 985 2014 2043 2071 2100 2129 2158 2187 28 Log. Sines and Tangents. (7°) Natural Sines. TABLE n. D. I0"j Cosine. Sine. 9.085894 086922 087947 088970 089990 091008 092024 093037 094047 095056 096062 9.097065 0980S6 099065 100062 101056 102048 103037 104025 105010 105992 9.106973 107951 108927 109901 110873 111842 112809 113774 114737 115698 9.116656 117613 118567 119519 120469 -121417 122362 123306 124248 126187 9.126125 127060 127993 128925 129864 130781 131706 132630 133551 134470 51 9.135387 136303 137216 138128 139037 139944 140850 141754 142655 143555 Cosine. 171 171 170 170 170 169 169 168 168 168 167 167 166 166 166 165 165 164 164 164 163 163 163 162 162 162 161 161 160 160 .996751 996735 996720 996704 996673 996657 996641 996625 996610 996694 9.996578 996562 996546 996530 996514 996482 996466 996449 996433 9.996417 996400 996384 996368 996351 996336 996318 996302 996285 ,f.(. 996269 i^n 9.996252 996236 996219 996202 996186 996168 996161 996134 996117 996100 996083 996066 996049 996032 996015 995998 169 159 159 168 168 168 157 167 157 156 156 156 155 155 164 154 154 163 163 153 152 162 152 152 161 161 151 160 150 D. lU' 995963 995946 995928 .995911 995894 996876 995869 995841 995823 995806 995788 996771 995753 Sine. 2.6 2.6 2.6 2.6 2.6 2.6 2.6 2.6 2.6 2.6 2.6 2.7 2.7 2.7 2.7 2.7 2.7 2.7 2.7 2.7 2.7 2.7 2.7 2.7 2.7 2.7 2.7 2.7 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.9 2.9 2.9 2.9 2.9 2.9 2.9 2.9 2.9 2.9 2.9 2.9 2.9 2.9 2.9 2.9 2.9 2.9 2.9 Tang. 9. .089144 090187 091228 092286 093302 094336 095367 096395 097422 098446 099468 .100487 101504 102519 103532 104642 105650 106556 107559 108660 109659 .110556 111561 112543 113533 114521 115507 116491 117472 118452 119429 .120404 121377 122348 123317 124284 126249 126211 127172 128130 129087 130041 130994 131944 132893 133839 134784 136726 136667 137605 138642 139476 140409 141340 142269 143196 144121 145044 145966 146886 147803 Cotang. D. JO ^ 174 173 173 173 172 172 171 171 171 170 170 169 169 169 168 168 168 167 167 166 166 166 165 165 165 164 164 164 163 163 162 162 162 161 161 161 160 160 160 159 159 159 168 158 158 157 157 157 156 156 156 165 165 155 154 164 164 153 153 153 Uotaug. N. sin(/ 10.910856 909813 908772 907734 906698 905664 904633 903605 902578 901554 900532 10.899513 898496 897481 896468 895458 894450 893444 892441 891440 890441 10.889444 888449 887467 886467 885479 884493 883509 882528 881648 880571 10.879596 878623 877652 876683 875716 874751 873789 872828 871870 870913' 10.869959; 869006 868056 867107 866161 865216 864274 I 863333 862395 861458 10.860524 859591 858660 857731 856804 855879 ; 864956 854034 I 853116 852197 ;: 1218-/ i 12216 l! 12245 it 12274 ; 112302 1 112331 ; 112360 j 112389 '1 12418 ii 12447 j 1 12476 i| 12504 1112533 !l 12562 1112591 Ji 12620 i| 12649 112678 12706 12735 12764 12793 12822 12851 12880 1 12908 12937 12966 1299 13024 13053 13081 13110 13139 13168 1319 13226 ! 13254 ; 13283 13312 ! 13341 13370 1 13399 i 13427 j 1345G ' 13485 113514 i 13543 13572 13600 13629 13658 13687 1371G 13744 13773 ; 13802 i 13831 1 13860 ! 13889 113917 N. COK. 99256 99251 99248 99244 99240 99237 9233 99230 99226 99222 99219 99215 99211 99208 99204 99200 99197 99193 99189 99186 99182 99178 99176 99171 99167 99163 99160 99166 99152 99148 99144 99141 99137 99133 99129 99125 99122 99118 99114 99110 99106 )9102 99098 99094 99091 99087 99083 99079 99075 99071 99067 ^9063 :>y059 139055 99051 :;9047 J9043 99039 99035 99031 99027 Tang. liN. C0.1. .V.sine. 82 Degrees. TABLE II. Log. Sines and Tangents. (8°) Natural Sines. 29 Sine. 9.143655 144453 145349 146243 147136 148026 148915 149802 150886 151569 152451 9.153330 154208 156083 155957 156830 167700 158569 159436 160301 161164 9.162026 162885 163743 164600 165454 166307 167159 168008 168856 169702 9.170547 171389 172230 173070 173908 174744 175578 176411 177242 178072 9.178900 179726 180551 181374 182196 183016 183834 184651 186466 186280 9.187092 187903 188712 189619 190326 191130 191933 192734 193634 194332 D. 10" 150 149 149 149 148 148 148 147 147 147 147 146 146 146 145 146 146 144 144 144 144 143 143 143 142 142 142 142 141 141 141 140 140 140 140 139 139 139 139 138 138 138 137 137 137 137 136 136 136 136 136 136 135 135 134 134 134 134 133 133 Cosine. Cosine. .995753 996736 996717 995699 995681 995664 996646 996628 995610 995591 996573 .995556 995537 996619 995601 995482 995464 995446 995427 996409 996390 .996372 996353 995334 996316 996297 995278 995260 995241 995222 995203 .995184 995165 995146 995127 996108 995089 995070 996061 995032 995013 .994993 994974 994955 994935 994916 994896 994877 994857 994838 994818 .994798 994779 994759 994739 994719 994700 994680 994660 994640 I 994620 : Sinel I D. lU" 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.1 3.1 3.1 3.1 3.1 3.1 3.1 3.1 3.1 3.1 3.1 3.1 3.1 3.1 3.2 3.2 3.2 3.2 3.2 3.2 3.2 3.2 3.2 3.2 3.2 3.2 3.2 3.2 3.2 3,2 3.2 3.3 3.3 3.3 3.3 3.3 3.3 3.3 3.3 3.3 3.3 Tang. ). 147803 148718 149632 150644 151454 152363 163269 154174 156077 155978 156877 ). 157776 168671 159566 160467 161347 162236 163123 164008 164892 166774 >. 166664 167532 168409 169284 170167 171029 171899 172767 173634 174499 M 76362 176224 177084 177942 178799 179656 180508 181360 182211 183069 1.183907 184752 186597 186439 187280 188120 188958 189794 190629 191462 1.192294 193124 193953 194780 195606 196430 197253 198074 198894 1 99713 Co tang. D, 10' 153 152 162 152 151 151 151 150 150 160 160 149 149 149 148 148 148 148 147 147 147 146 146 146 145 146 146 145 144 144 144 144 143 143 143 142 142 142 142 141 141 141 141 140 140 140 140 139 139 139 139 138 138 138 138 137 137 137 137 136 Cotang. j N. sine 10.852197 851282 850368 849456 848546 847637 846731 846826 844923 844022 843123 10.842226 841329 840436 839643 838653 837764 836877 835992 835108 834226 10.833346 832468 831691 830716 829843 828971 828101 827233 826366 825501 10.824638 823776 822916 822058 821201 820345 819492 818640 817789 816941 10.816093 815248 814403 813561 812720 811880 811042 810206 809371 808638 10.807706 806876 806047 805220 804394 803570 802747 801926 801106 800287 3917 3946 3975 4004 4033 4061 4090 4119 4148 4177 4205 42S4 4263 4292 4320 4349 4378 4407 4436 4464 4493 4522 4561 4580 4608 4637 4666 4695 4723 4752 4781 4810 4838 4867 4896 4926 4954 4982 5011 5040 5069 5097 5126 5155 5184 521^ 5241 5270 5299 532/ 5356 5585 5414 544'^ 5471 5500 6629 5557 5586 5616 564Jj Tang. N. cos. N.sine 60 69 68 67 66 55 64 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 36 34 33 32 31 SO 29 28 27 98884 r26 98880 98876 98871 98867 98863 98858 98854 98849 98845 98841 988a6 98832 9882-; 98823 98818 98814 98809 98805 98800 98'; 96 98791 98787 98782 98778 98773 98769 99027 99023 99019 99015 99011 99006 99002 98998 98994 98990 98986 98982 98978 98973 98969 98965 98961 98957 98953 98948 98944 98940 98936 98931 98927 98923 98919 98914 98910 98906 98902 98897 98893 81 Degrees. 30 Log. Sines and Tangents. (9°) Natural Sines. TABLE n. Sine. 9.194332 195129 195925 196719 197511 198302 199091 199879 200666 201451 202234 9.203017 203797 204577 205354 206131 206906 207679 208452 209222 209992 9.210760 211526 212291 213055 213818 214579 215338 216097 216854 217609 9.218363 219116 219868 220618 221367 222115 222861 223608 224349 225092 9.225833 226573 227311 228048 228784 229518 230252 230984 231714 232444 ^.233172 233899 234625 235349 236073 236795 237515 238235 238953 239670 Cosine. D. 10' 133 133 132 132 132 132 131 131 131 131 130 130 130 130 129 129 129 129 128 128 128 128 127 127 127 127 127 126 126 126 126 125 125 125 125 125 124 124 124 124 123 123 123 123 123 122 122 122 122 122 121 121 121 121 120 120 120 120 120 119 Cosine. .994620 994600 994580 994560 994540 994519 994499 994479 994459 994438 994418 .994397 994377 994367 994336 994316 994295 994274 994254 994233 994212 .994191 994171 994150 994129 994108 994087 994066 994045 994024 994003 .993981 993960 993939 993918 993896 993875 993854 993832 993811 993789 .993768 993746 993725 993703 993681 993660 993638 993616 993594 993672 .993550 994528 993506 993484 993462 933440 993418 993396 993374 993361 Sine. D. 10' 3.3 3.3 3.3 3.4 3.4 3.4 3.4 3.4 3.4 3.4 3.4 3.4 3.4 3.4 3.4 3.4 3.4 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.6 3.5 3.6 3.6 3.5 3.5 3.6 3.6 3.6 3.6 3.6 3.6 3.6 3.6 3.6 3.6 3.6 3.7 3.7 3.7 3.7 3.7 3.7 3.7 3.7 3.7 3.7 Tans .199713 200529 201345 202159 202971 203782 204592 205400 206207 207013 207817 .208619 209420 210220 211018 211815 212611 213405 214198 214989 215780 .216568 217356 218142 218926 219710 220492 221272 222052 222830 223606 .224382 225156 225929 226700 227471 228239 229007 229773 230539 231302 .232066 232826 233586 234345 235103 235859 236614 237368 238120 238872 .239622 240371 241118 241865 242610 243354 244097 244839 245579 246319 136 136 136 135 135 135 135 134 134 134 134 133 133 133 133 133 132 132 132 132 131 131 131 131 130 130 130 130 130 129 129 129 129 129 128 128 128 128 127 127 127 127 127 126 126 126 126 126 125 125 125 125 125 124 124 124 124 124 123 123 Cotang. Cotang. I IN. sine. N. cos 10.800287 799471 798655 797841 797029 790218 795408 794600 793793 792987 792183 10.791381 790580 789780 788982 788185 787389 786595 785802 735011 784220 10.783432 782644 781858 781074 780290 779508 778728 777948 777170 776394 10.775618 774844 774071 773300 772529 771761 770993 770227 769461 768698 10.767935 767174 766414 765655 764897 764141 763386 762632 761880 761128 10.760378 759629 758882 758135 757390 756646 755903 755161 754421 753681 15643 16672 16701 16730 15758 15787 15816 15845 15873 15902 15931 15959 15988 16017 16046 16074 16103 16132 16160 16189 16218 16246 16275 16304 16333 16361 16390 16419 16447 16476 16505 16533 16562 16691 16620 98769 98764 98760 98755 98751 98746 98741 98737 98732 98728 98723 98718 98714 98709 98704 98700 98695 98690 98686 98681 98676 98671 98667 98662 98657 98652 98648 98643 98638 98629 98624 98619 98614 98609 116648 98604 !| 16677 98600 J! 16706 98596 116734 98590 116763 .98585 1116792 98580 ii 16820 98575 i 1 16849198570 l'l6878|98565 i 1690Gi9S561 ; 116935,98556 16964 98561 i;16992!98646 17021 !98o41 |17050'98536 ;17078!9S531 :17107;9S526 I 17136,98521 Ii 17164198516 i 17193198511 i! 17222 98506 ii 17250 98.501 17279 17308 17336 17365 Tang. N. cos. N 98496 98491 98486 98481 80 Degrees. TABLE II. Log. Sines and Tangents. (10°) Natural Sines. 31 Sine. 9.239670 240386 241101 241814 242626 243237 243947 244666 245363 246069 246775 9.247478 248181 248883 249583 250282 250980 251677 252373 253067 263761 254453 255144 255834 256523 257211 257898 258583 259268 259951 260633 261314 261994 262673 263351 264027 264703 265377 266051 266723 267395 9.268065 268734 269402 270069 270735 271400 272064 272726 273388 274049 274708 275367 276024 276681 277337 277991 278644 279297 279948 280599 Cosine. D. 10" 119 119 119 119 118 118 118 118 118 117 117 117 117 117 116 116 116 116 116 116 115 115 115 115 115 114 114 114 114 114 113 113 113 113 113 113 112 112 112 112 112 11 11 11 11 11 11 110 110 110 110 110 110 109 109 109 109 109 109 108 Cosine. ,993351 993329 993307 993285 993262 993240 993217 993195 993172 993149 993127 .993104 993081 993059 993036 993013 992990 992967 992944 992921 992898 .992875 992852 992829 992806 992783 992769 992736 992713 992690 992666 .992643 992619 992596 992672 992649 992525 992501 992478 992464 992430 .992406 992382 992369 992335 992311 992287 992263 992239 992214 992190 .992166 992142 992117 992093 992069 992044 992020 991996 991971 991947 Sine. D. 10" 3.7 3.7 3.7 3.7 3.7 3.7 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.9 3.9 3.9 3.9 3.9 3.9 3.9 3.9 3.9 3.9 3.9 3.9 3.9 3.9 3.9 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.1 4.1 4.1 4.1 4.1 4.1 4.1 Tang. .246319 347057 247794 248530 249264 249998 250730 251461 252191 252920 253648 .254374 255100 255824 256547 267269 267990 258710 259429 260146 260863 .261678 262292 263005 263717 264428 266138 265847 266555 267261 267967 1.268671 269376 270077 270779 271479 272178 272876 273573 274269 274964 '.275658 276351 277043 277734 278424 279113 279801 280488 281174 281868 1.282542 283225 283907 284588 285268 285947 286624 287301 287977 288662 Cotang. D. 10" 123 123 123 122 122 122 122 122 121 121 121 121 121 120 120 120 120 120 120 119 119 119 119 119 118 118 118 118 118 118 117 117 117 117 117 116 116 116 116 116 116 115 115 115 115 115 115 114 114 114 114 114 114 113 113 113 113 113 113 112 Cotang. 10.763681 762943 752206 761470 750736 750002 749270 748539 747809 747080 746352 10.745626 744900 744176 743463 742731 742010 741290 740571 739854 739137 10.738422 737708 736995 736283 736572 734862 734163 733445 732739 732033 10.731329 730626 729923 729221 728521 727822 727124 726427 726731 726036 10.724342 723649 722967 722266 721576 720887 720199 719512 718826 718142 10.717468 716776 716093 715412 714732 714053 713376 712699 712023 711348 N.sine. N. cos Tang. 17365 17393 17422 17451 17479 17508 17537 17665 17694 17623 17661 17680 17708 17737 17766 17794 17823 17852 98481 98476 98471 98466 98461 98455 98450 98445 98440 98435 98430 98425 98420 98414 98409 98404 98399 98394 17880 98389 17909 17937 17966 17995 18023 18052 18081 18109 18138 18166 18195 18224 18252 18281 18309 18338 18367 18395 18424 18452 18481 18509 98383 98378 98373 98368 98362 98357 98352 98347 98341 98336 98331 98325 98320 98315 98310 98304 98299 98294 9828S 98283 98277 98272 18538198267 1866798261 1869598256 18624 98250 18652 18681 18710 18738 18767 98245 98240 98234 98229 98223 18796 98218 18824 98212 1885298207 1888198201 1891098196 18938 98190 18967I9S185 1899598179 19024198174 19052198168 1908198163 N. COS. N.Fine. 79 Degrees. 32 Log. Sines and Tangents. (11°) Natural Sines. TABLE II. Sine. DTTi)^ 280599 281248 281897 282544 283190 283836 284480 285124 285766 286408 287048 287687 288326 288964 289600 290236 290870 291504 292137 292768 293399 9.294029 294658 295286 295913 296539 297164 297788 298412 299034 299655 9.300276 300895 301514 302132 302748 303364 303979 304593 305207 305819 9.306430 307041 307650 308259 308867 309474 310080 310685 311289 311893 312495 313097 313698 314297 314897 315495 316092 316689 317284 317879 Cosine. 108 108 108 108 108 107 107 107 107 107 107 106 106 106 106 106 106 105 105 105 105 105 105 104 104 104 104 104 104 104 103 103 103 103 103 103 102 102 102 102 102 102 102 101 101 101 101 101 101 100 100 100 100 100 100 100 100 99 99 99 Cosine. ID. lu .991947 991922 991897 991873 991848 991823 991799 991774 991749 991724 991699 L 991674 991649 991624 991599 991574 991549 991524 991498 991473 991448 '.99L422 991397 991372 991346 991321 991295 991270 991244 991218 991193 .991167 991141 991115 991090 991064 991038 991012 990986 990960 990934 990908 990882 990855 990829 990803 990777 990750 990724 990697 990671 1.990644 990618 990591 990566 990538 990511 990485 990458 990431 990404 Sine. 4.1 4.1 4.1 4.1 4.1 4.1 4.1 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.3 4.3 4.3 4.3 4.3 4.3 4.3 4.3 4.3 4.3 4.3 4.3 4.3 4.3 4.3 4.3 4.3 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.5 4.5 4.5 4.5 Tiiug. ID. iv 1.288652 289326 289999 290671 291342 292013 292682 293360 294017 294684 295349 ► .296013 296677 297339 298001 298662 299322 299980 300638 301295 301951 1.302607 303261 303914 304667 305218 305869 306519 307168 307815 308463 • .309109 309754 310398 311042 311685 312327 312967 313608 314247 314885 1.316623 316159 316795 317430 318064 318697 319329 319961 320592 321222 1.321851 322479 323106 323733 324358 324983 325607 326231 326853 327476 Cotang. 112 112 112 112 112 111 111 111 111 111 111 111 110 110 110 110 110 110 109 109 109 109 109 109 109 108 108 108 108 108 108 107 107 107 107 107 107 107 106 106 106 106 106 106 106 105 105 105 106 105 105 105 104 104 104 104 104 104 104 104 lug. jN. sine. N. cos 10.711348 710674 710001 709329 708658 707987 707318 706650 705983 705316 704651 10.703987 703323 702661 701999 701338 700678 700020 699362 698705 698049 10.697393 696739 696086 696433 694782 694131 693481 692832 692185 691637 10-690891 690246 689602 688958 688315 687673 687033 686392 686753 686115 10-684477 683841 683205 682670 681936 681303 680671 680039 679408 678778 10.678149 677521 676894 676267 675642 675017 674393 673769 673147 672526 11908198163 ! 19109 98157 119138 i 19167 19195 19224 1 19252 19281 ' 19309 19338 19366 19395 98101 19423 19452 19481 19509 19538 19566 19595 19623 19652 19680 19709 19737 19766 197^4 19823 19861 19880 1 19908 19937 19965 ! 19994 1 20022 •20061 '20079 20108 20136 20166 20193 20260 20279 20307 20336 20364 20393 20421 {20450 120478 1 20507 120535 1 20563 20592 1 20620 120649 i 20677 1 20706 120734 i 2076a 120791 Tang. 98152 98146 98140 98135 98129 98124 98118 98112 98107 98096 98090 98084 98079 98073 98067 98061 98056 98050 98044 98039 98033 98027 98021 98016 98010 98004 97998 97992 97987 97981 97975 97969 97963 97958 97952 97946 97940 20222(97934 97928 97922 97916 97910 97905 97899 97893 97887 97881 97875 97869 9/863 97857 97851 97846 97839 97833 97827 97821 97815 N. cos. N.pine 78 Degrees. TABLE II. Log. Sines and Tangents. (12°) Natural Sines. 33 1 2 3 4 5 6 7 8 9 10 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 ; 44 ! 45 I 46 ii 47 48 i 49 50 I 51 62 53 54 55 56 57 58 59 60 Sine. 9.317879 318473 3190i)ti 319658 320249 320840 321430 322019 322607 323194 323780 9.324366 324950 325534 326117 326700 327281 327862 328442 329021 329599 9.330176 330753 331329 331903 332478 333051 333624 334195 334766 335337 9.335906 336475 337043 337610 338176 338742 339306 339871 340434 340996 9.341558 342119 342679 343239 343797 344355 344912 345469 346024 346579 19.347134 347687 348240 348792 349343 349893 350443 330992 351540 352088 D. 10" Cosine. 99.0 98.8 98.7 98.6 98.4 98.3 98.2 98.0 97.9 97.7 97.6 97.5 97.3 97.2 97.0 96.9 96.8 96.6 96.5 96.4 96.2 96.1 96.0 95.8 95.7 95.6 95.4 95.3 95.2 95.0 94.9 94.8 94.6 94.5 94.4 94.3 94.1 94.0 93.9 93.7 93.6 93.5 93.4 93.2 93. i 93.0 92.9 92.7 92.6 92.5 92.4 92.2 92.1 92.0 91.9 91.7 91.6 91.5 91.4 91.3 Cosine. >. 990404 990378 990351 990324 990297 990270 990243 990215 990188 990161 990134 1.990107 990079 990052 990025 989997 989970 989942 989915 989887 989860 1.989832 989804 989777 989749 989721 989693 989665 989637 989609 989582 .989553 989525 989497 989469 989441 989413 989384 989356 989328 989300 .989271 989243 989214 989186 989157 989128 989100 989071 989042 989014 9.988985 988956 988927 988898 988869 988840 988811 988782 988753 988724 Sine. D. 10" 4.5 4.5 4.5 4 4 4 4 4, 4 4 4, 4, 4.6 4.6 4.6 4.6 4.6 4.6 4.6 4.6 4.6 4.6 4.6 4.6 4,7 4.7 4.7 4.7 4.7 4.7 4.7 4.7 4.7 4.7 4.7 4.7 4.7 4.7 4.7 4.7 4.7 4.7 4.7 4.7 4.7 4.7 4.8 4.8 4.8 4.8 4.8 4.8 4.8 4.8 4.8 4.8 4.8 4.9 4.9 4.9 Tan g. 9.327474 3280^5 328715 329334 329953 330570 33118 331803 332418 333033 333646 9.334259 334871 335482 336093 336702 337311 337919 338527 339133 339739 9.340344 340948 341552 342155 342757 343358 343968 344558 345157 345755 9.346353 346949 347546 348141 348735 349329 349922 350514 351106 351697 9.352287 352876 353465 354053 354640 355227 355813 356398 356982 357566 9.358149 358731 359313 359893 360474 361053 301632 362210 362787 363364 Cotang. D. lU" 103 103 103 103 103 103 103 102 102 102 102 102 102 102 102 101 101 101 101 101 101 101 101 100 100 100 100 100 100 100 100 99.4 99.3 99.2 99.1 99.0 98. 98.7 98.6 98.5 98.3 98.2 98.1 98.0 97.9 97.7 97.6 97.5 97.4 97.3 97.1 97.0 96.9 90.8 96.7 96.6 96.6 96.3 96.2 96.1 Gotaag. 10 10 10 10 10 10 .672526 671905 671285 670866 670047 669430 668813 668197 667582 666967 666354 .665741 665129 664518 663907 663298 662689 662081 661473 660867 660261 .659666 659052 658448 657845 657243 656642 656042 655442 654843 654245 .653647 653051 652456 651859 651263 650671 650078 649486 648894 648303 .647713 647124 646535 646947 645360 644773 644187 648602 643018 642434 .641851 641269 640687 640107 639626 638947 638368 637790 637213 636636 N. 8!Ue. x\. COK. 20791 97815 20820 97809 20848 97803 20877 97797 20905 97791 20933 97784 20962 97778 20990 97772 21019 97766 21047 97760 21076 97764 21104 97745 21132 97742 21161 97735 21189 97729 21218 97723 ! 21246 97717 21275 97711 21303 97705 2133197698 21360 97692 21388 97686 21417:97680 21445 97673 2147497667 21502 97661 21530 97655 21569 97648 21587 97642 2161697636 2164497630 21672:97623 21701 '97617 21729 97611 21758 '97604 |i2178a!97598 ! 21814:97592 121843 97585 ;i2187l!97579 1121899:97573 ;|21928'97566 ! 1 21956:97660 !|21985;97653 !:22013!97547 1^2204197541 1 22070,97534 !i2209b!97528 |i 22126:97521 j; 22155 97515 i 22183 97608 !' 22212 '97502 i 22240197496 [: 22268 197489 i' 2229 7 197483 1 122326 197476 '122353 97470 ! 122382 97463 122410 i 22438 ! 22467 I 22495 Tang. N. cos. N.sine 97457 97450 97444 97437 77 Degrees. 34 Log. Sines and Tangents. (13°) Natural Sines. TABLE IL Sine. 352088 352635 353181 353726 354271 354815 355358 355901 356443 356984 357524 358064 358603 359141 359678 360215 360752 361287 361822 362356 362889 9.363422 363954 364485 365016 365546 366075 366604 367131 367669 368185 9.368711 369236 369761 370286 370808 371330 371852 372373 372894 373414 9.373933 374452 374970 376487 376003 376519 377036 377549 378063 378677 9.379089 379601 380113 380624 381134 381643 382162 382661 383168 383676 C)osine. D. 10'' Cosine. D. 10" Tang. D. 10" Cotang. | N.sine IN. cos, 91.1 91.0 90.9 90.8 90.7 90.5 90.4 90.3 90.2 90.1 89.9 89.8 89.7 89.6 89.5 89.3 89.2 89.1 89.0 88.9 88.8 88.7 88.5 88.4 88.3 88.2 88.1 88.0 87.9 87.7 87.6 87.6 87.4 87.3 87.2 87.1 87.0 86.9 86.7 86.6 86.6 86.4 86.3 86.2 86.1 86.0 86.9 85.8 86.7 85.6 85.4 85.3 85.2 85.1 86.0 84.9 84.8 84.7 84.6 84.5 .988724 988695 988666 988607 988578 988548 988519 988489 988460 988430 .988401 988371 988342 988312 988282 988352 988223 988193 988163 988133 .988103 988073 988043 988013 987983 987953 987922 987892 987862 987832 .987801 987771 987740 987710 987679 987649 987618 987588 987557 987526 .987496 987466 987434 987403 987372 987341 987310 987279 987248 987217 .987186 987155 987124 987092 987061 987030 986967 986936 986904 Sme. 4.9 4.9 4.9 4.9 4.9 4.9 9 9 9 9 9 9 4.9 4.9 5.0 5.0 5.0 5.0 5.0 5.0 6.0 5.0 6.0 5.0 6.0 6.0 5.0 5.0 6.0 6.0 5.1 5.1 5.1 5.1 5.1 5.1 5.1 5.1 5.1 5.1 5.1 5.1 6.1 5.1 5.2 5.2 6.2 5.2 6.2 5.2 5.2 5.2 5.2 5.2 5.2 5.2 6.2 5.2 5.2 6.2 9.363364 363940 364515 365090 365664 366237 366810 367382 367953 368524 369094 9.369663 370232 370799 371367 371933 372499 373064 373629 374193 374756 9.376319 375881 376442 377003 377663 378122 378681 379239 379797 380364 9.380910 381466 382020 382676 383129 384234 384786 385337 9.386438 386987 387636 388631 389178 389724 390270 390815 391360 9.391903 392447 393531 394073 394614 395154 395694 396233 396771 Cotang. Degrees. 10.636636 636060 635485 634910 634336 633763 633190 632618 632047 ; 631476 j 630906' 10.630337 629768 I 629201 i 628633 628067' 627501 626936 626371 625807 625244 10.624681 6241 19 623558 622997 622437 621878 621319 620761 620203 619646 10.619090 618534 617980 617425 616871 616318 615766 615214 614663 614112 10.613562 613013 612464 611916 611369 610822 610276 609730 609185 60SG40 10.608097 607553 607011 606469 605927 605386 604846 604306 603767 603229 22495 97437 22523 97430 22552 97424 2258097417 22608 97411 22637 97404 22665 97398 22693 97391 22722 97384 22750197378 22778 22807 22836 97371 97365 97358 22863(97351 22892197345 Tang. 22920 22948 22977 23005 23033 23062 23090 23118 28146 23175 23203 23231 23260 23288 23316 23345 23373 23401 23429 23458 23486 23514 23542 23571 23627 23656 23684 23712 23740 23769 23797 97127 23826 97120 23853 23882 23910 23938 23966 23995 24023 24051 24079 24108 24136 24164 2419-2 97338 97331 97325 97318 97311 97304 97298 97291 97ii84 97278 97271 97264 97257 97251 97244 97237 97230 97223 97217 97210 97203 97' 96 97189 97182 23599 97176 97169 97162 97156 97148 97141 97134 97113 97106 97100 97093 )iom 97079 97072 97065 97068 97051 97044 7037 97030 >". COS. N.sine, TABLE II. Log. Sines and Tangents. (14°) Natural Sines. 35 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 65 56 57 58 69 60 Sine. 1.383675 384182 384687 385192 385697 386201 386704 387207 387709 388210 388711 .389211 389711 390210 390708 391206 391703 392199 392695 393191 393685 .394179 394673 395166 395658 396150 396641 397132 397621 398111 398600 .399088 399575 400062 400549 401035 401520 402005 402489 402972 403455 .403938 404420 404901 406382 405862 406341 406820 407299 407777 408254 .408731 409207 409682 410157 410632 411106 411579 412052 412524 412996 D. 10" Cosine. 84.4 84.3 84.2 84.1 8i.0 83.9 83.8 83.7 83.6 83.5 83.4 83.3 83.2 83.1 83.0 82.8 82.7 82.6 82.5 82.4 82.3 82.2 82.1 82.0 81.9 81.8 81.7 81.7 81.6 81.5 81.4 81.3 81.2 81.1 81.0 80.9 80.8 80.7 80.6 80.5 80.4 80.3 80,2 80.1 80.0 79.9 79.8 79.7 7y.6 79.5 79.4 79.4 79.3 79.2 79.1 79. U 78.9 78. b 78.7 78.6 Cosine. 1.986904 986873 986841 986809 986778 986746 986714 986683 986651 986619 986587 1.986555 986523 986491 986459 986427 986395 986363 986331 986299 986266 .986234 986202 986169 986137 986104 986072 986039 986007 985974 985942 .985909 985876 985843 985811 985778 985745 985712 985679 985646 986613 .985580 985547 985514 985-180 985447 985414 985380 985347 985314 985280 .985247 985213 985180 985146 985113 985079 985045 985011 984978 984944 D. 10" Sine. 5.2 5.3 6.3 5.3 6.3 6.3 5.3 5.3 5.3 6.3 5.3 5.3 6.3 5.3 6.3 6.3 5.3 5.4 6.4 6.4 5.4 5.4 6.4 6.4 5.4 5.4 5.4 5.4 5.4 5.4 5.4 5.5 5.5 6.5 5.5 5.5 5.5 5.6 5.6 5.5 5.5 6.6 6.5 5.5 6.5 6.6 6.6 6.6 6.6 5.6 5.6 5.6 5.6 6.6 6.6 6.6 5.6 6,6 5.6 5.6 Tang. 9.396771 397309 397846 398383 398919 399456 399990 400624 401058 401691 402124 9.402656 403187 403718 404249 404778 405308 405836 406364 406892 407419 407945 408471 408997 409521 410045 410669 411092 411616 412137 412668 9.413179 413699 414219 414738 416267 416775 416293 416810 417326 417842 418358 418873 419387 419901 420415 420927 421440 421952 422463 422974 9.423484 423993 424503 425011 425519 426027 426634 427041 427547 428052 D. 10' Co tang. Cotans !N. sine. N. cos 10.603229 602691 602154 601617 601081 600545 600010 599476 598942 698409 697876 10.697344 696813 696282 595751 596222 694692 694164 593636 693108 692581 10.592055 591529 591003 690479 689956 589431 588908 588385 687863 587342 10.686821 586301 585781 585262 584743 j 684225 1 583707 I 583190 j 58267411 582168 I j 10.5816421 1 581127 || 580613 I i 68009911 579685 I 679073 I 6785601 578048 i 677637 i 677026 I 10.676516 576007 676497 574989 674481 573973! I 573466 I 672969;! 572463 1 1 571948! I 24192 24220 24249 24277 24305 24333 24362 24390 24418 24446 24474 24503 24531 24559 24587 24615 24644 24672 24700 24728 24766 24784 24813 24841 24869 24897 24925 24954 24982 25010 25038 26066 26094 26122 26151 25179 2520 25236 25263 25291 25320 25348 25376 26404 25432 25460 25488 26516 25545 25573 25601 25629 26667 26685 25713 26741 25766 25798 25826 26864 26882 97030 97023 97015 97008 97001 96994 96987 96980 96973 96966 96969 96952 96945 96937 96930 96923 96916 96909 96902 96894 96887 96880 96873 96866 96858 96851 96844 96837 96829 96822 , 96816 96807 96800 96793 96786 96778 96771 96764 96756 96749 96742 96734 96727 96719 96712 96705 96697 96690 96682 96675 96067 96660 96663 96645 96638 96630 96623 96616 96608 96600 96593 Tang. N. COS. N.pine. ' 75 Degrees. 20 36 Log. Sines and Tangents. (15°) Natural Sines. TABLE II. Sine. 412996 413467 413938 414408 414878 415347 416815 416283 416751 417217 417684 418150 418615 419079 419544 420007 420470 420933 421395 421857 422318 9.422778 423238 4236y7 424156 424615 425073 425530 425987 426443 426899 9.427354 427809 428263 428717 429170 429623 430075 430527 430978 431429 9.431879 432329 432778 433226 433676 434122 434569 435016 435462 435908 9.436353 436798 437242 437686 438129 438572 439014 439456 439897 440338 Cosine. D. 10' 70 78 78.0 77.9 77.8 77.7 77.6 77.5 77.4 77.3 77.3 77.2 77.1 77.0 76.9 76.8 76.7 76.7 76.6 76.5 76.4 76.3 76.2 76.1 76.0 76.0 75.9 75.8 75.7 75.6 75.5 75 75 75 75 75 75 74.9 74.9 74.8 74.7 74 6 74.5 74.4' 74.4 74.3 74.2 74.1 74.0 74.0 73.9 73.8 73.7 73.6 73.6 73.6 Cosine. .984944 984910 984876 984842 984808 984774 984740 984706 984672 984637 984603 .984569 984535 984500 984466 984432 984397 984363 984328 984294 984259 .984224 984190 984156 984120 984085 984050 984015 983981 983946 983911 .983875 983840 983805 983770 983736 983700 983664 983629 983594 983558 .983523 983487 983452 983416 983381 983345 983309 983273 983238 9S3202 .983166 983130 983094 983058 983022 982986 982960 982914 982878 982842 Sine. D. Wi Tang. 5.7 5.7 5.7 5.7 5.7 5.7 5.7 5.7 5.7 6.7 5.7 6.7 5.7 5.7 5.7 6.8 5.8 6.8 5.8 5.8 5.8 5.8 5.8 5.8 5.8 5.8 6.8 5.8 5.8 5.8 5.8 5.8 6.9 6.9 6.9 5.9 5.9 6.9 5.9 5.9 6.9 5.9 6.9 6.9 5.9 6.9 6.9 5.9 6.0 6.0 6.0 6.0 6.0 6.0 6.0 6.0 6.0 6.0 6.0 6.0 9.428052 428557 429062 42956U 430070 430573 431075 431577 432079 432680 433080 9.433580 434080 434579 435078 435576 436073 436570 437067 437563 438059 9.438554 439048 439543 440036 440529 441022 441514 442006 442497 442988 9.443479 443968 444458 444947 445435 445923 446411 446898 447384 447870 9.448356 448841 449326 449810 450294 460777 451260 461743 452225 462706 9.463187 453668 464148 454628 455107 455586 456064 450542 457019 457496 Cotang. D. 10" Cotaug. N. sine, 10.571948 571443 570938 57U434 569930 51)94-27 668925 568423 567921 567420 666920 10.566420 2688 2591 U 2593 2596 25994 2602-i 26050 26079 26107 26135 26163 26191 5659-20 26219 665421 i 26247 564922: 1 26275 564424 I! 26303 5C3927;; 26331 6G3430 I 26359 96463 562933! 26387 662437! 126415 561941 1126443 26471 26500 10.561446 660952 560457 659964 1 1 26556 5594711! 26584 558978 I 558486! 126640 557994! {26668 567503; 26696 5570121:26724 10.656521 !! 26752 556032 jj 267 SO 555542 : 1 2680b 555053! 26836 554566 1 126864 654077! I 26892 553689! 126920 653102 i{ 2694b 552616; 1 2697b 552130:127004 10.551644; 27032 651159; 2; 000 550674:: 27088 550190!' 27110 549706 ii27l4-i 549223 548740 548257 547775 647294 10,546813 546332 545852 545372 544893 644414 543936 543458 642981 642504 27172 27200 2722b 2725(> 127284 127312 27340 ! 2736b 127396 I 27424 j 27462 127480 27508 27 — I 27564 96593 96585 96578 96570 96562 96565 96547 96540 96632 96624 96617 96509 96502 96494 96486 96479 96471 96456 96448 96440 96433 96425 26528 96417 96410 96402 26612 96394 96386 96379 96371 96363 06355 96347 96340 ;36332 96324 96316 96308 96301 96293 96285 96277 96269 962b 1 96253 96246 96238 96230 96222 96214 96206 96198 96190 96182 96174 96166 96168 96160 96142 636 96134 96126 Tang. N. cos.lN.sine, 74 Degrees. or TH I.A £AUhO^^>^ TABLE II. Log. Sines and Tangents. (16°) Natural Sines. 37 Sine. D. 10" Cosine. D. 10" Tang, D. 10" Cotang. j N. sine. N. cos. .440338 440778 441218 441658 442096 442535 442973 443410 443847 444284 444720 ,445155 445590 446025 446459 446893 447326 447759 448191 448623 449054 .449485 449915 450345 450775 451204 451632 462060 452488 452915 453342 ,453768 454194 454619 455044 455469 455893 456316 456739 457162 457584 .458006 458427 458848 459208 459688 460108 460527 460946 461364 461782 .462199 462616 463032 463448 463864 464279 464694 466108 465522 465935 73.4 73.3 73.2 73-1 73.1 73.0 72.9 72.8 72.7 72.7 72.6 72.6 72.4 72.3 72.3 72.2 72.1 72.0 72.0 71.9 71.8 71.7 71.6 71.6 71.5 71.4 71.3 71.3 71.2 71.1 71.0 71.0 70.9 70.8 70.7 70.7 70.6 70.5 70.4 70.4 70.3 70.2 70.1 70.1 70.0 69.9 69.8 69.8 69.7 69.6 69.5 69.5 69.4 69.3 69.3 69.2 69.1 69.0 69.0 68.9 .982842 982805 982769 982733 982696 982660 982624 982587 982551 982614 982477 .982441 982404 982367 982331 982294 982257 982220 982183 982146 982109 .982072 982035 981998 981961 981924 981886 981849 981812 981774 981737 .981699 981662 081625 981587 981549 981512 981474 981436 981399 981361 .981323 981285 981247 981209 981171 981133 981095 981057 981019 980981 ',980942 980904 980866 980827 980789 980750 980712 980673 980636 980596 Sine. 6.0 6.0 6.1 6.1 6.1 6.1 6.1 6.1 6.1 6.1 6.1 6.1 6.1 6.1 6.1 6.1 6.1 6.2 6.2 6.2 6.2 6.2 6.2 6.2 6.2 6.2 6.2 6.2 6.2 6.2 6.2 6.3 6.3 6.3 6.3 6.3 6.3 6.3 6.3 6.3 6.3 6.3 6.3 6.3 6.3 6.3 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 9.457496 457973 458449 468925 459400 469875 460349 460823 461297 461770 462242 9.462714 463186 463658 464129 464699 466069 466639 466008 466476 466946 9 467413 '467880 468347 468814 469280 469746 470211 470676 471141 471606 9 472068 '472532 472995 473457 473919 474381 474842 476303 475763 476223 9 476683 477142 477601 478059 478517 478975 479432 479889 480345 ■ 480801 9 481257 481712 482167 482621 483076 483529 483982 484436 484887 485339 Cotang. 10 10 10 10 10 10 542604 542027 541651 541076 540600 640125 539651 539177 638703 538230 637768 .537286 636814 636342 635871 536401 534931 634461 533992 633524 633056 .532687 632120 531653 531186 530720 630254 529789 629324 528859 628395 ,627932 527468 527005 526643 526081 525619 525158 524697 524237 523777 ,523317 622868 522399 521941 521483 621026 620668 620111 519655 519199 ,518743 618288 517833 517379 516925 516471 516018 515565 516113 614661 27664 27692 27620 27648 27676 27704 27731 27769 27787 27816 27843 27871 27899 27927 27956 27983 28011 28039 28067 28095 28123 28150 28178 28206 28234 28262 28290 28318 28346 28374 28402 28429 28457 28485 28513 28641 28569 28597 28626 28652 28680 28708 28736 28764 28792 2882U 28847 j 28875 28903 28931 2895*.. 2898'*/ 29015 29042 290/0 29098 29126 29154 29182 29201' 29247 Tang. li N. cos. N.sine. 96126 96118 96110 96102 96094 96086 96078 96070 96062 96064 96046 96037 96029 96021 96013 96006 95997 95989 95981 95972 95964 95956 96948 95940 96931 95923 95916 95907 95898 95890 95882 96874 95865 96867 95849 96841 95832 95824 95816 y5807 96799 96791 95782 95774 95766 95767 96749 95740 95732 95724 95715 95707 95698 95690 95681 95673 95664 95056 95647 95639 95630 73 Degrees. 38 Log. Sines and Tangents. (17°) Natural Sines. TABLE IL Sine. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 40 47 48 49 50 51 52 53 54 55 56 57 58 59 60 0(9.465935 466348 466761 467173 467585 467996 468407 468817 469227 469637 470046 9.470455 470863 471271 471679 472086 472492 472898 473304 473710 474115 9.474619 474923 475327 475730 476133 476536 476938 477340 477741 478142 9.478542 478942 479342 479741 480140 480539 480937 481334 481731 482128 9.482525 482921 483316 483712 484107 484501 484895 485289 485682 480075 9.486467 486860 487251 487643 488034 488424 488814 489204 489593 489982 D. 10" Cosine. 68.8 68.8 68.7 68.6 68.5 68.5 68.4 68.3 68.3 68.2 68.1 68.0 68.0 67.9 67.8 67.8 67.7 67.6 67.6 67.5 67.4 67.4 67.3 67.2 67.2 67.1 67.0 66.9 66.9 66.8 66.7 66.7 66.6 66.5 66.5 66.4 66.3 66.3 66.2 66.1 66.1 66.0 65.9 65.9 65.8 65.7 65.7 66.6 65.5 65.5 65.4 66.3 65.3 65.2 65.1 65.1 66.0 65.0 64.9 64.8 Cosine. 9.980596 980558 980519 980480 980442 980403 980364 980325 980286 980247 980208 .980169 980130 980091 980052 980012 979973 979934 979895 979855 979816 97977a 979737 979697 979668 979618 979579 979539 979499 979459 979420 9.979380 979340 979300 979260 979220 979180 979140 979100 979059 979019 9.978979 978939 978898 978858 978817 978777 978736 978696 978655 978615 9,978574 978533 978493 978462 978411 978370 978329 978288 978247 978206 D. 10'^ Sine. 6.4 6.4 6.5 6.5 6.5 6.5 6.5 6.5 6.5 6.5 6.5 6.5 6.5 6.5 6.5 6.5 Tang. ID. 10 6.6 6.6 6.6 6.6 6.6 6.6 6.6 6.6 6.6 6.6 6.6 6.6 6.7 6.7 6.7 6.7 6.7 6.7 6.7 6.7 6.7 6.7 6.7 6.7 6.7 6.7 6.7 6.8 6.8 6.8 6.8 6.8 6.8 6.8 6.8 6.8 6.8 6.8 6.8 9.485339 485791 486242 486693 487143 487593 488043 488492 488941 489390 489838 9.490286 490733 491180 491627 492073 492519 492965 493410 493854 494299 9.494743 495186 495630 496073 496515 496957 497399 497841 468282 498722 9.499163 499603 500042 500481 500920 501359 501797 502235 502672 503109 503546 503982 504418 504854 505289 505724 506159 5U6593 507027 607460 507893 508326 508759 509191 509622 510054 510486 610916 511346 511776 Co tang. 3 75.2 76.1 75.1 75.0 74.9 74.9 74.8 74.7 74.7 74.6 74.6 74.5 74.4 74.4 74.3 74.3 74.2 74.1 74.0 74.0 74.0 73.9 73.8 73.7 73.7 73.6 73.6 73.5 73.4 73.4 73.3 73.3 73.2 73.1 73.1 73.0 73.0 72.9 72.8 72.8 72.7 72.7 72.6 72.5 72.5 72.4 72.4 72.3 72.2 72.2 72.1 72.1 72.0 71.9 71.9 71.8 71.8 71.7 71.6 Cotang. IN. sine 10.514661 514209 513758 513307 512867 612407 511967 511508 511059 510610 610162 10.509714 509267 508820 508373 607927 507481 507035 506590 i 506146 505701 10.505257 504814 604370 503927 603485 503043 502601 502159 601718 501278 10.500837 500397 499968 499619 499080 498041 498203 497765 497328 496891 10.496454 496018 495682 496146 494711 494276 493841 1 493407 492973 I 492540 1 10.492107; 491674 491241 490809 490378 489946 489515 489084 488654 488224 1:29237 I i 29265 1^29293 29321 29348 29376 29404 29432 95630 95622 95613 95606 95596 95588 95579 95671 29460196562 !2948 129615 ! 29543 129571 I 29599 29620 ! 29654 j 29682 129710 i 29737 j 29765 1 29793 I 29821 95654 '95545 95536 95528 95519 95611 95502 95493 95485 95476 95467 95459 95450 2984995441 29876 95433 29904 95424 Tang. 29932 29960 2998 130015 I 30043 I 30071 1 30098 30126 30154 30182 30209 30237 30265 30292 30320 30348 30376 30403 30431 30459 30486 30514 30542 30570 3059 ( 30625 30053 30080 30708 30703 30791 30819 30840 30874 30902 j N. COS. N.pine, 96416 95407 95398 95389 95380 95372 95303 95354 95345 95337 95328 95319 95310 95301 95293 95284 95276 95266 95257 95248 95240 95231 95222 95213 95204 95195 95186 95177 95168 30730 95159 95160 95142 95133 95124 96115 95100 60 69 68 67 66 65 54 63 62 51 60 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 26 24 23 22 21 20 19 18 17 16 15 14 13 12 U 10 9 8 7 6 6 4 3 2 1 7? Degrees. TABLE II. Log. Sines and Tangents. (18°) Natural Sines. 39 Sine. D. 10' 9.489982 490371 490759 491147 491535 491922 492308 492695 493081 493466 493861 9.494236 494621 495005 495388 495772 496154 496537 496919 497301 497682 498064 498444 498826 499204 499584 499963 500342 500721 501099 501476 9.501854 502231 502607 502984 503360 503735 504110 504485 504860 505234 9.505608 505981 506354 506727 507099 507471 507843 508214 508586 508956 9.509326 509696 510065 510434 510803 511172 511540 511907 512275 512642 Cosine. 64.8 64.8 64.7 64.6 64.6 64.6 64.4 64.4 64.3 64.2 64.2 64.1 64.1 64.0 63.9 63.9 63.8 63.7 63.7 63.6 63.6 63.5 63.4 63.4 63.3 63.2 63.2 63.1 63.1 63.0 62.9 62.9 62.8 62.8 62.7 62.6 62.6 62.5 62.5 62.4 62.3 62.3 62.2 62.2 62.1 62.0 62.0 61.9 61.9 61.8 61.8 61.7 61.6 61.6 61.5 61.5 61.4 61.3 61.3 61.2 9. Cosine. D. 10' 9. 978206 978165 978124 978083 978042 978001 977959 977918 977877 977835 977794 977752 977711 977669 977628 977686 977544 977503 977461 977419 977377 977335 977293 977251 977209 977167 977126 977083 977041 976999 976957 976914 976872 976830 976787 976745 976702 976660 976617 976574 976532 976489 976446 976404 976361 976318 976275 976232 976189 976146 976103 ,976060 976017 976974 975930 975887 976844 975800 975767 975714 976670 Sine. 6.8 6.8 6.8 6.9 6.9 6.9 6.9 6.9 6.9 6.9 6.9 6.9 6.9 6.9 6.9 6.9 7.0 7.0 7.0 7.0 7.0 7.0 7.0 7.0 7.0 7.0 7.0 7.0 7.0 7.0 7.0 7.0 7.1 7.1 7.1 7.1 7.1 7.1 7.1 7.1 7.1 7.1 7.1 7.1 7.1 7.1 7.1 7.2 7.2 7.2 7.2 7.2 7.2 7.2 7.2 7.2 7.2 7.2 7.2 7.2 Tang. D. 10" 611776 512206 612636 613064 513493 513921 514349 514777 616204 515631 616067 616484 616910 517335 617761 518185 518610 519034 519458 519882 520306 9.520728 621161 621573 521995 522417 522838 523259 523680 624100 524620 9.524939 525369 525778 526197 626615 627033 527451 527868 528285 528702 9.629119 629636 629960 530366 530781 531196 631611 532025 632439 632853 633266 533679 534092 534504 534916 636328 636739 636160 536561 636972 Cotang. 71.6 71.6 71.5 71.4 71.4 71.3 71.3 71.2 71.2 71.1 71.0 71.0 70.9 70.9 70.8 70.8 70.7 70.6 70.6 70.5 70.6 70.4 70.3 70.3 70.3 70.2 70.2 70.1 70.1 70.0 69.9 69.9 69.8 69.8 69.7 69.7 69.6 69.6 69.6 69.5 69.4 69.3 69.3 69.3 69.2 69.1 69.1 69.0 69,0 68.9 68.9 68.8 68.8 68.7 68.7 68.6 68.6 68.6 68.6 68.4 Cotang. I N. sine. N. cos 10 10 10 10 10 10 .488224 i 487794 ! 487366 I 486936 486607 486079 486651 485223 484796 484369 483943 .483516 483090 482665 482239 481815 481390 480966 480542 480118 479695 .479272 478849 478427 478005 477583 477162 476741 476320 475900 476480 .475061 474641 474222 473803 473386 472967 472649 472132 471716 471298 .470881 470466 470050 469634 469219 468804 468389 467975 467661 467147 .466734 466321 466908 466496 465084 464672 464261 463850 463439 463028 Tang. 30902 30929 30957 30985 31012 31040 31068 31095 31123 31151 31178 31206 31233 31261 31289 31316 31344 31372 94952 31399 31427 31464 31482 31510 31537 31565 31593 31620 31648 31675 31703 31730 31768 31786 31813 31841 31868 31896 31923 31961 31979 32006 32034 32061 32089 32116 32144 32171 32199 32227 96106 96097 96088 95079 95070 95061 95052 96043 95033 95024 95016 95006 94997 94988 94979 94970 94961 94943 94933 94924 94916 94906 94897 94888 94878 94869 94860 94851 94842 94832 94823 94814 94805 94795 94786 94777 94768 94768 94749 94740 94730 94721 94712 94702 94693 94684 94674 94666 32250 94656 32282 94646 32309 32337 32364 32392 32419 32447 32474 32502 32629 32557 94637 94627 94618 94609 94599 94590 94580 94571 94561 94652 N. cos. N-sine. 71 Degrees. 40 Log. Sines and Tangents. (19°) Natural Sines. TABLE IL 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 Sine. 9.512642 513009 513375 513741 514107 514472 514837 515202 515566 515930 516294 9.516657 517020 517382 517745 518107 518468 518829 519190 519551 519911 9.520271 520631 520990 521349 521707 522066 522424 522781 523138 523495 9.523852 524208 524664 524920 525275 625680 525984 626339 626693 527046 9.527400 627753 628106 628468 528810 529161 629613 529864 530215 530565 530915 531265 531614 531963 532312 532661 533009 533357 533704 534052 D. lu' 61.2 61.1 61.1 61.0 60.9 60.9 60.8 60.8 60.7 60.7 60.6 60.5 60.5 60.4 60.4 60.3 60.3 60.2 60.1 60.1 60.0 60.0 59.9 59.9 59.8 59.8 59.7 69.6 59.6 59.5 59.6 4 59.4 59.3 59.3 69.2 69.1 59.1 59.0 59.0 58.9 58.9 58.8 58.8 58.7 58 7 58.6 58.6 58.5 Losme. Cosine. 58.4 68-3 68.2 58-2 58.1 58.1 58.0 58.0 67.9 975670 975627 975583 975539 97.5496 975452 975408 975365 975321 975277 975233 9.975189 975145 975101 975057 975013 974969 974925 974880 974836 974792 9.974748 974703 974659 974614 974570 974525 974481 974436 974391 974347 ,974302 974267 974212 974167 974122 974077 974032 973987 973942 973897 9.973852 973807 973761 973716 973671 973625 973580 973535 973489 973444 19.973398 973352 973307 973261 973215 973169 973124 973078 973032 972986 D. lu' Sine. 7.3 7.3 7.3 7.3 7.3 7.3 7.3 7.3 7.3 7.3 7.3 7.3 7.3 7.3 7.3 7.3 7.4 7.4 7.4 7.4 7.4 7.4 7.4 7.4 7.4 7.4 7.4 7.4 7.4 7.4 7.5 7.5 7.5 7.6 7.6 7.6 7.6 7.6 7.5 7.6 7.6 7.6 7.6 7.6 7.6 7.6 7.6 7.6 7.6 7.6 7.6 7.6 7.6 7.6 7.6 7.6 7.6 7.6 7.6 7.7 i' ang. (l>^ 322480 I 42972 322164 321829 321504 10.321179 320854 320529 3202U5 319880 43182 319556 319232 318908 ij 43261 3185841; 43287 318260 43313 10.3179371:43340 3176131143366 3172901 43392 316967; 43418 316644! 43445 3163211 43471 315999 1 1 43497 315676 143523 315354': 43549 315032: 43575 10.3147lu: 43602 3143881: 43628 314066 : 43654 313745 313423 313102 312781 312460 312139 311818 42999 43025 43051 43077 43104 43130 43156 43209 43235 43680 43706 4373J^ 43759 43785 43811 43837 Tanf 90631 90613 90606 90594 90582 90569 90557 90545 90532 90520 90507 90495 90483 90470 90458 90446 90433 90421 90408 90396 90383 90371 90358 90346 90334 90321 9030y 90296 90284 90271 90259 90246 90233 90221 90208 90196 90183 90171 90158 90146 90133 90120 90108 90095 90082 90070 90057 90045 90032 90019 90007 89994 89981 89968 89956 89943 89930 89918 89906 89892 89879 N. CO?. X.t'iDe. 64 Degrees. TAELE II. Log. Sines and Tangents. (26°) Natural Sines. 47 Sine. .641842 642101 642360 642618 642877 643135 643393 643650 643908 644165 644423 .644680 644936 645193 645450 645706 645962 646218 646474 646729 646984 .647240 647494 647749 648004 648258 648512 648766 649020 649274 649527 .649781 650034 650287 650539 650792 651044 651297 651549 651800 652052 .652304 652555 652806 653057 653308 653558 653808 654059 654309 654558 .654808 655058 655307 655556 655805 656054 656302 656551 65G799 657047 D. 10' 43, 43. 43, 43, 43, 43, 43, 42. 42. 42, 42, 42, 42, 42, 42. 42, 42, 42, 42. 42, 42, 42. 42, 42. 42. 42, 42. 42, 42. 42. 42. 42. 42. 42, 42. 42. 42. 42. 42. 41. 41. 41. 41, 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41, 41. 41. 41. 41. Cosine. Cosine. 9.953660 953599 953537 953475 953413 953352 953290 953228 953166 953104 953042 9.952980 952918 952855 952793 952731 952669 952606 952544 952481 952419 9.952356 952294 952231 952168 952106 952043 951980 951917 951854 951791 951728 951665 951602 951539 951476 951412 951349 951286 951222 951159 951096 951032 950968 950905 960841 950778 950714 950650 950586 950522 9.950458 950394 950330 950366 950202 950138 950074 950010 949945 949881 Sine. D. 10'' 10.3 10.3 10.3 10.3 10.3 10.3 10.3 10.3 10.3 10.3 10.3 10.4 10.4 10.4 10.4 10.4 10.4 10,4 10.4 10.4 10.4 10.4 10.4 10.4 10.5 10.5 10.5 10.5 10,5 10.5 10.5 10.5 10.5 10.5 10.5 10.5 10.5 10; 6 10.6 10.6 10.6 10.6 10.6 10.6 10.6 10.6 10.6 10.6 10.6 10.6 9. Tang. 688182 688502 688823 689143 689433 689783 690103 690423 690742 691062 691381 1.691700 692019 692338 692656 692975 693293 693612 693930 694248 694566 1.694883 695201 695518 695836 696153 696470 696787 697103 697420 697736 1.698053 698369 698685 699001 699316 699632 699947 700263 700578 700893 1.701208 701523 701837 702152 702466 702780 703095 703409 703723 704036 (.701350 704GG3 704977 705290 705603 705916 706228 706541 706854 7071G6 Cotang. [D. lO'l 53.4 53.4 53.4 53.3 53.3 53.3 53.3 53.3 53,2 53.2 53.2 53.1 53.1 53,1 53.1 53.1 53.0 53.0 53.0 53.0 52.9 52.9 52.9 52.9 52.9 52.8 52.8 52.8 52.8 52.7 52.7 52.7 52.7 62,6 52.6 52.6 62.6 52.6 9 K 52.5 62.5 52.4 62.4 62.4 52.4 62.4 52.3 62.3 52.3 52.3 52.2 52.2 52.2 52.2 52.2 52.1 52.1 52.1 52.1 52.1 Cotang. ; N. sine. N 10.311818 311498 311177 310857 310537 310217 309897 309577 309258 308938 308619 10.308300 307981 307662 307344 307025 306707 306388 30S070 305752 305434 10.305117! 304799 3044S2 304164 1 303847 i 303530 1 303213 302897 302680 302264 10-301947 301631 301315 300999 300GS4 300368 300053 299737 299422 299107 10-298792 298477 298163 297848 297534 297220 296906 2S6691 296277 295964 10.295650 295337 295023 294710 294^97 294084 293772 293469 29^146 292834 Tang. 43837 43863 43889 43916 43942 43968 43994 44020 44046 44072 44098 44124 44161 89879 89867 89854 89841 89828 89816 89803 89790 89777 89764 89752 89739 89726 44177 89713 44203 44229 44265 44281 44307 44333 44359 44386 44411 44437 44464 44490 44616 44542 44568 44694 44620 44646 44672 44698 44724 44750 44776 44802 4482« 44854 89700 89687 89674 89662 89649 89636 89623 89610 89597 89584 89571 89558 89646 89532 89519 89480 89467 89454 89441 89428 89416 89402 89389 89376 4488089363 44906 44932 44958 44984 45010 45036 46062 45088 46140 45166 45192 46218 45243 45269 46296 45321 45C47 45373 45G99 89503 31 89493 ! 30 89360 89337 89324 89311 89298 89285 89272 89269 45114 89246 89232 89219 89206 89193 89180 89167 89153 89140 89127 89114 89101 2v. COS. N.sine. 63 Degrees 48 Log. Sines and Tangents. (27°) Natural Sines. TABLE n. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Sine. D. 10' 9.657047 657295 657642 657790 658037 658284 658531 658778 659025 659271 659517 9.659763 660009 660255 660501 660746 660991 661236 661481 661726 661970 9.662214 662459 662703 662946 663190 663433 663677 663920 664163 664406 9.664648 664891 665133 665375 665617 665859 666100 666342 666583 666824 9.667065 667305 667546 667786 668027 668267 668506 668746 668986 669225 9.669464 669703 669942 670181 670419 670658 67089G 671134 671372 671609 CJosine. 41.3 41.3 41.2 41.2 41.2 41.2 41.1 41.1 41.1 41.0 41.0 41.0 40.9 40.9 40.9 40.9 40.8 40.8 40.8 40.7 40.7 40.7 40.7 40.6 40.6 40.6 40.5 40.5 40.5 40.5 40.4 40.4 40.4 40.3 40.3 40.3 40.2 40.2 40.2 40.2 40.1 40.1 40.1 40.1 40.0 40 40.0 39.9 39.9 39.9 39.9 39.8 39.8 39.8 39.7 39.7 39.7 39.7 39.6 39.6 Cosine. .949881 949816 949752 949688 949623 949558 949494 949429 949364 949300 949235 .949170 949105 949040 948975 948910 948845 948780 948715 948650 948684 .948519 948454 9. 948323 948257 948192 948126 948060 947995 947929 947863 947797 947731 947665 947600 947533 947467 947401 947335 947269 947203 947 \36 947070 947004 946937 946871 946804 946738 946671 946604 946538 946471 946404 946337 946270 946203 946136 946069 946002 945936 Sine. D. 10' 10.7 10.7 10.7 10.8 10.8 10.8 10.8 10.8 10.8 10.8 10.8 10.8 10.8 10.8 10.8 10.8 10.8 10.9 10.9 10.9 10.9 10.9 10.9 10.9 10.9 10.9 10.9 10.9 10.9 11.0 11.0 11.0 11.0 11.0 11.0 11.0 11.0 11.0 11.0 11.0 11.0 11.0 11.1 11.1 11.1 11.1 11.1 11.1 11.1 11.1 11.1 11.1 11.1 11.1 11.1 11.2 11.2 11.2 11.2 11.2 Tang. 9.707166 707478 707790 708102 708414 708726 709037 709349 709660 709971 710282 9.710593 710904 711215 711525 711836 712146 712456 712766 713076 713386 ,713696 714005 714314 714624 714933 715242 715551 715860 716168 716477 9.716785 717093 717401,-. o 717709 f;-^ 718017 718325 718633 718940 719248 9} •;! 719556 I ^J -2 9.719862 :^}-^ 720169 °f-f 720476 °J-} 720783 i^j-; 721089;°} -J 721396 j^;-; 721702 °;-;. 722009 ;°J-^ 722316^}-" 722621 I ^}-X .722927 ! 5 -^ 723232 ^/.-^ 723538 °X'^ 723844 724149 724-i64 D. lO'l Cotang. ! N. sine. N. cos. 52.0 52.0 52.0 52.0 51.9 51.9 51.9 51.9 51.9 51.8 51.8 51.8 51.8 51.8 61.7 51.7 51.7 51.7 51.6 51.6 51.6 61.6 51.6 51.5 51.5 51.5 61.5 51.4 51.4 61.4 51.4 51.4 51.3 61.3 61.3 51.3 61.2 60.9 50.9 50.9 724759 1^.-1 725056 °^-^ 725369 725674 50.8 50.8 CJctang. 10.292834 292622 292210 291898 291586 291274 290963 290651 290340 290029 289718 10.289407 289096 288786 288475 288164 287854 287544 287234 286924 286614 10.286304 285996 285686 285376 285067 284758 284449 284140 283832 283623 10.283215 282907 282699 282291 281983 281676 281367 281060 280762 280445 10.280138 279831 279524 279217 278911 278604 278Ji98 277991 277685 277379 10.277073 276768 276462 276166 275851 275546 275241 274935 274631 274326 45399 45426 ' 45451 i 46477 I 45503 1 45529 j 46554 ! 45580 1 45606 ! 45632 45658 45684 45710 45736 46762 45787 45813 45839 45865 i 45891 j 45917 146942 145968 45994 I 46020 46046 46072 46097 46123 46149 46175 46201 46226 Ij 46252 46278 46304 46330 46365 46381 46407 46433 46458 46484 46510 46536 46561 4658'. 4G613 46639 46664 46690 46716 46742 46767 46793 I 46819 i 46844 46870 46896 46921 46947 Tang. 89101 89087 89074 89061 89048 89035 89021 89008 88995 88981 88968 88955 88942 88928 88915 88902 88888 88876 88862 88848 88835 88822 88808 88795 88782 88768 88755 88741 88728 88715 88701 88688 88674 88661 88647 88634 88620 88607 88593 88580 88566 88653 88539 88526 88512 88499 88485 88472 88468 88445 88431 88417 88404 88390 88377 88363 88349 88336 88322 88308 88295 N. COS. .N.piiic. 62 TABLE II. Log. Sines and Tangents. (28°) Natural Sines. 49 Sine. 9.671609 671847 672084 672321 672568 672795 673032 673268 673506 673741 673977 9.674213 674448 674684 674919 675155 675390 675624 675859 676094 676328 9.676562 676796 677030 677264 677498 677731 677964 678197 678430 678663 678895 679128 679360 679592 679824 680056 680288 680519 680750 680982 9.681213 681443 681674 681905 682135 682365 682595 682825 683055 683284 9.683514 683743 683972 684201 684430 684658 684887 685115 685343 685571 Cosine. D. 10" 39.6 39.5 39.5 39.5 39.5 39.4 39.4 39.4 39.4 39.3 39.3 39.3 39.2 39.2 39.2 39.2 39.1 39.1 39.1 39.1 39.0 39.0 39.0 39.0 38.9 38.9 38.9 38.8 38.8 38.8 38.8 38.7 38.7 38.7 38.7 38.6 38.6 38.6 38.6 38.5 38.5 38.5 38.4 38.4 38.4 38.4 38.3 38.3 38.3 38.3 38.2 38.2 38.2 38.2 38.1 38.1 38.1 38.0 38.0 38.0 Cosine. .945935 945868 945800 946733 945666 945598 945631 945464 946396 945328 945261 .945193 945126 946068 944990 944922 944864 944786 944718 944650 944582 .944514 944446 944377 944309 944241 944172 944104 944036 943967 .943830 943761 943693 943624 943566 943486 943417 943348 943279 943210 .943141 943072 943003 942934 942864 942795 942726 942656 942587 942517 .942448 942378 942308 942239 942169 942099 942029 941959 941889 941819 Sine. D. 10' Tang. .725674 725979 726284 726588 726892 727197 727601 727805 728109 728412 728716 .729020 729323 729626 729929 730233 730635 730838 731141 731444 731746 .732048 732351 732663 732955 733257 733558 733860 734162 734463 734764 .735066 736367 735668 736969 736269 736670 736871 737171 737471 737771 . 738071 738371 738671 738971 739271 739670 739870 740169 740'468 740767 .741066 741365 741664 741962 742261 742559 742868 743156 743454 743752 Cotang. D. 10" 50.8 60.8 50 7 60.7 60.7 50.7 50.7 50.6 50.6 60.6 50.6 50.6 50.6 60.5 50.6 60.5 50.5 50.4 50.4 50.4 60.4 60.4 50.3 50.3 50.3 60.3 60.3 50 60 60 50 50 50 50 60 50.1 60.1 50.1 60.0 50.0 50.0 50.0 60.0 49.9 49.9 49.9 49.9 49.9 49.9 49.8 49.8 49.8 49.8 49.8 49.7 49.7 49.7 49.7 49.7 49.7 Cotang. 10.274326 274021 273716 273412 273108 272803 272499 272196 271891 271588 271284 10.270980 270677 270374 270071 269767 269465 269162 268859 268666 268254 10.267952 267649 267347 267046 266743 266442 266140 266838 265537 265236 10.264934 26^1633 264332 264031 263731 263430 263129 262829 262529 262229 10.261929 261629 261329 261029 260729 260430 260130 259831 269532 259233 10.258934 258635 258336 258038 257739 257441 257142 256844 256546 256248 N. sine. N. cos 46947 46973 46999 47024 47050 47076 47101 I 47127 88296 88281 88267 88264 88240 88226 88213 88199 47153 88185 47178 47204 47229 47255 47281 47306 47332 47358 47383 47409 88048 47434 88034 47460 88020 47486 88006 47511 87993 47537 87979 47562 87965 47588 87951 47614 87937 47639 87923 , 47665 87909 I 47690 87896 147716 87882 ! 1 47741 87868 1 .47767 87854 j 1 47793 87840 : 47818 87826 147844 87812 1147869 87798 i; 47895 87784 ^(47920 87770 I 47946 87766 88172 88158 88144 88130 88117 88103 88089 88075 88062 47971 47997 48022 48048 48073 48099 48124 48150 48176 48201 48226 48252 48277 i 48303 i 48328 ! 1 48354 87632 ; 48379 87518 1 1 48405 87504 48430 87490 48456 87476 4848187462 Tan'jc. 87743 87729 87716 87701 87687 87673 87659 87645 87631 87617 87603 87589 87575 87561 87546 N. COS. N.sine. 61 Degrees. 50 Log. Sines and Tangents. (29°) Natural Sines. TABLE n. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 26 26 27 28 29 30 31 32 33 34 36 36 37 38 39 40 41 42 43 44 46 46 47 48 49 50 61 62 63 54 56 56 67 68 69 60 Sine. D. 10" 9.685571 685799 686027 686254 686482 686709 686936 687163 687389 687616 687843 9.688069 688296 688621 688747 688972 689198 689423 689648 689873 690098 690323 690548 690772 690996 691220 691444 691668 691892 692116 692339 9.692562 692785 693008 693231 693453 693676 693898 694120 694342 694564 9.694786 695007 696229 695450 696671 695892 696113 696334 696554 696776 9.696995 697216 697436 697654 697874 698094 698313 698532 698761 698970 Cosine. 38,0 37.9 37.9 37.9 37.9 37.8 37.8 37.8 37.8 37.7 37.7 37.7 37.7 37.6 37.6 37.6 37.6 37.6 37.5 37.5 37.6 37.4 37.4 37.4 37.4 37.3 37.3 37.3 37.3 37.5 37.2 37.2 37.1 37.1 37.1 37.1 37.0 37.0 37.0 37.0 36.9 36.9 36.9 36.9 36.8 36.8 36.8 36.8 36.7 36.7 36.7 36.7 36.6 36.6 36.6 36.6 36.6 36.5 36.6 36.6 Cosine. {D. 10^' ,941819 941749 941679 941609 941639 941469 941398 941328 941258 941187 941117 ,941046 940975 940905 940834 940763 940693 940622 940661 940480 940409 .940338 940267 940196 940125 940054 939982 939911 939840 939768 939697 ,939626 939664 939482 939410 939339 939267 939196 939123 939052 938980 .938908 938836 938763 938691 938619 938547 938476 938402 938330 938268 .938185 938113 938040 937967 937895 937822 937749 937676 937604 937631 Sine. 11.7 11.7 11.7 11.7 11 11 11 11 11 11.7 11.7 11.8 11.8 11.8 11.8 11.8 11.8 11.8 11.8 11.8 11.8 11.8 11.8 11.8 11.9 11.9 11.9 11.9 11.9 11.9 11.9 11.9 11.9 11.9 11.9 11.9 12.0 12,0 12.0 12.0 12.0 12.0 12.0 12.0 12.0 12.0 12.0 12.0 12.1 12.1 12.1 12.1 12.1 12.1 12.1 12.1 12.1 12.1 12.1 12.1 Tang. .743762 744060 744348 744646 744943 745240 745638 745835 746132 746429 746726 . 747023 747319 747616 747913 748209 748506 748801 749097 749393 9.7i 749986 760281 760676 750872 761167 751462 751767 762062 762347 762642 752937 763231 763626 753820 764116 754409 754703 764997 765291 755585 755878 766172 756466 756769 767062 767346 767638 757931 758224 768517 758810 759102 769396 769687 769979 760272 760564 760856 761148 761439 Cotang. D. 10" 49.6 49.6 49.6 49.6 49.6 49.6 49.6 49.5 49.5 49.5 49.6 49.4 49.4 49.4 49.4 49.4 49.3 49.3 49.3 49.3 49.3 49.3 49,2 49.2 49.2 49.2 49.2 49.2 49.1 49.1 49.1 49.1 49.1 49.1 49,0 49.0 49,0 49.0 49.0 49.0 48.9 48.9 48,9 48.9 48.9 48.9 48,8 48.8 48.8 48.8 48.8 48. 8i 48.7 1 48.7 48.7 48.7 48.7 48.7 48.6 48.6 Cotang. I ;N. sine. N. cos 10. 10 10 10 10 10 256248 255950 255652 255355 265067 264760 254462 254165 253868 253571 263274 262977 252681 252384 252087 251791 251495 261199 260903 250607 250311 250015 249719 249424 249128 248833 248538 248243 247948 247653 247358 247063 246769 246474 246180 245885 245591 246297 246003 244709 244415 244122 243828 243636 243241 242948 242655 242362 242069 241776 241483 ,241190 240898 240605 240313 240021 239728 239436 239144 2S8852 238561 '48481 148506 : 48532 '[ 48557 48583 : 48608 48634 87462 87448 87434 87420 87406 87391 87377 4865987363 48684|87349 48710j87335 4873587321 48761187306 48786187292 48811 48837 48862 48888 48913 48938 48964 87278 87264 87250 87235 87221 87207 87193 48989187178 4901487164 49040!87150 149065 ; 49090 1 49116 49141 87136 87121 87107 87093 49166 87079 49192 49217 49242 49268 49293 87064 87050 87036 87021 87007 49318 86993 4934486978 49369 86964 4939486949 4941986935 49445186921 49470!86906 ! 49495186892 ' 49.521 186878 49546 86863 i; 49571 86849 1 1 49596186834 1 1 49622186820 ij 49647 J86805 i49672!86791 |49697j86777 i 49723 86762 : 49748 186748 ''49773 86733 49798'86719 ; 4982486704 i 1 49849 86690 j 149874 86675 '^ 49899,86661 49924^86646 |i 49950 86632 i 49975186617 i 50000186603 I Tang. ||N. co«.|]\.piue. 60 Degrees. TABLE II. Log. Sines and Tangents. (30°) Natural Sines. 51 Sine. 9.698970 699189 D. 10" 699407 699626 699844 : ^,, 700062 i^^ 700280 700498 700716 700933 701151 .701368 701585 701802 702019 702236 702452 702669 702885 703101 703317 .703533 703749 703964 704179 704395 704610 704825 705040 705254 705469 .705683 705898 706112 706326 706539 706753 706967 707180 707393 707606 .707819 708032 708245 708458 708670 708882 709094 709303 709518 709730 .709941 710153 710364 710575 710786 710967 711208 711419 711629 711839 136 Cosine. Cosine. 9.937631 937458 937385 937312 937238 937165 937092 937019 936946 936872 936799 936725 936652 936578 936505 936431 936357 936284 936210 936136 936062 935988 935914 935840 935766 935692 935618 935543 935469 935395 935320 9.935246 935171 935097 935022 934948 934873 934798 934723 934649 934574 934499 934424 934349 934274 934199 934123 934048 933973 933898 933822 9.933747 933671 933596 933520 933445 933369 933293 933217 933141 933066 Sine. D. 10" 12.1 12.2 12.2 12.2 12.2 12.2 12.2 12 12 12 12 12 12 12 12 12.3 12.3 12.3 12.3 12.3 12.3 12.3 12.3 12.3 12.4 12.4 12.4 12.4 12.4 12.4 12.4 12.4 12.4 12.4 12.4 12.4 12.4 12.5 12.5 12.5 12.5 12.5 12.5 12.5 12.5 12.5 12.5 12.5 12.5 12.6 12.6 12.6 12.6 12.6 12.6 12.6 12.6 12.6 12.6 12.6 9. Tang. 9. 9. 761439 761731 762023 762314 762605 762897 763188 763479 763770 764061 764352 764643 764933 765224 765514 765805 766095 766385 766675 766965 767255 767545 767834 768124 768413 768703 768992 769281 769570 769860 770148 770437 770726 771015 771303 771592 771880 772168 772457 772745 773033 773321 773608 773896 774184 774471 774759 775046 775333 775621 775908 776195 776482 776769 777055 777342 777628 777915 778201 778487 778774 Cotang. D. 10' 48.6 48.6 48.6 48.6 48.5 48.5 48.5 48.5 48.5 48.5 48.4 48.4 48.4 48.4 48.4 48.4 48.4 48.3 48.3 48.3 48.3 48.3 48.3 48.2 48.2 48.2 48.2 48.2 48.2 48.1 48.1 48.1 48.1 48.1 48.1 48.1 48.0 48.0 48.0 48.0 48.0 48.0 47.9 47.9 47.9 47.9 47.9 47.9 47.9 47.8 47.8 47.8 47.8 47.8 47.8 47.8 47,7 47.7 47.7 47.7 Cotang. |,N. sine. N. cos 86603 10.238561! 238269 : 237977 ' 237686 ! 237394 ' 237103 I 236812 I 236521 I 236230 I 235939 I 235648 I 10.235357 235087 1 234776 I 234486 i 234195 ! 233905 233615 233325 233035 232745 10.232455 232166 231876 231587 231297 231008 230719 230430 230140 229852 10.229563 229274 228985 228697 228408 228120 227832 227543 227255 226967 10-226679 226392 226104 225816 225529 225241 224954 224667 224379 224092 10.223805 223518 223231 222945 222658 222372 222085 221799 221612 221226 Tang. 50000 50025 50050 5007{) 50101 50126 50151 50176 5a201 60227 50252 5027 50302 50327 50362 60377 60403 50428 60463 50478 50503 50528 50563 50578 50603 60628 50664 50679 50704 50729 50754 50779 50804 50829 50854 50879 50904 50929 50954 60979 51004 61029 51054 61079 61104 51129 51154 51179 51204 61229 51254 51279 61304 61329 51354 51379 51404 51429 51454 51479 51504 86573 86559 86544 86530 86515 86601 86486 80471 86457 86442 86427 86413 86398 86384 86369 86354 86340 86326 86310 86295 86281 86266 86251 86237 86222 86207 86192 86178 86163 86148 86133 86119 86104 86089 86074 86059 86045 86030 86015 86000 85985 85970 85956 85941 85926 85911 85896 85881 85866 85861 85836 85821 85806 85792 86777 85762 85747 85732 85717 N. cos. N.sine. 59 Degrees. 21 52 Log. Sines and Tangents. (31°) Natural Sines. TABLE II. Sine. |D. 10' 1 2 3 4 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 j 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 .711839 712050 712260 712469 712679 712889 713098 713308 713517 713726 713935 .714144 714352 714561 714769 714978 715186 715394 715602 715809 716017 .716224 716432 716639 716846 717053 717259 717466 717673 717879 718085 .718291 718497 718703 718909 719114 719320 719525 719730 719935 720140 .720345 720549 720754 720958 721162 721366 721670 721774 721978 722181 .722385 722588 722791 722994 723197 723400 723603 723805 724007 724210 Cosine. 35.0 33.0 35.0 34.9 34.9 34.9 34.9 34.9 34.8 34.8 34.8 34.8 34.7 34.7 34.7 34.7 34.7 34.6 34.6 34.6 34.6 34.5 34.5 34.5 34.5 34.5 34.4 34.4 34.4 34.4 34.3 34.3 34.3 34.3 34.3 34.2 34.2 34.2 34.2 34.1 34.1 34.1 34.1 34.0 34.0 34 34.0 34.0 33.9 33.9 33.9 33.9 33.9 33.8 33.8 33.8 33.8 33.7 33.7 33.7 Cosine. .933086 932990 93-2914 932833 932762 932685 932609 932533 932457 932380 932304 .932228 932151 932075 931998 931921 931845 931768 931691 931614 931537 .931460 931383 931306 931229 931152 931076 930998 930921 930843 930766 .930688 930611 930533 930456 930378 930300 930223 930145 930067 929989 1.929911 929833 929755 929677 929599 929521 929442 929364 929286 929207 1.929129 929050 928972 928893 928815 928736 928667 928678 928499 928420 Sine. D. 10" Tang. 12.6 12.7 12.7 12,7 12.7 12.7 12.7 12.7 12.7 12.7 12.7 12.7 12.7 12.8 12.8 12.8 12.8 12,8 12.8 12.8 12.8 12.8 12.8 12.8 12.9 12.9 12.9 12.9 12.9 12.9 12.9 12.9 12.9 12.9 12.9 12.9 13.0 13.0 13.0 13.0 13.0 13.0 13.0 13.0 13.0 13.0 13.0 13.0 13.1 13.1 13.1 13.1 13.1 13.1 13.1 13.1 13.1 13.1 13.1 13.1 .778774 779060 779346 779632 779918 780203 780489 780776 781060 781346 781631 .781916 782201 782486 782771 783056 783341 783626 783910 784195 784479 .784764 785048 785332 785616 785900 786184 786468 786752 787036 787319 .787603 787886 788170 788453 788736 789019 789302 789585 789868 790151 .790433 790716 790999 791281 791563 791846 792128 792410 792692 792974 i. 793256 793538 793819 794101 794383 794664 794945 795227 795508 795789 Cotang. D. 10" 47.7 47.7 47.6 47.6 47.6 47.6 47.6 47.6 47.6 47.5 47.5 47.5 47.5 47.5 47.5 47.5 47.5 47.4 47.4 47.4 47.4 47.4 47.4 47.3 47.3 47.3 47.3 47.3 47.3 47.3 47.2 47.2 47.2 47.2 47.2 47.2 47.2 47.1 47.1 47.1 47.1 47.1 47.1 47.1 47.1 47.0 47.0 47.0 47.0 47.0 47.0 47.0 46.9 46.9 46.9 46.9 46.9 46.9 46.9 46.8 Cotang. 10.221226 220940 220654 220368 220082 219797 219511 219225 218940 218654 218369 10.218084 217799 217514 217229 216944 216639 216374 216090 215805 215621 10.216236 214952 214668 214384 214100 213816 213532 213248 212964 212681 10.212397 212114 211830 211647 211264 210981 210698 210415 210132 209849 10.209667 209284 209001 208719 208437 208164 207872 207590 207308 207026 10.206744 206462 206181 205899 205617 205336 205055 204773 204492 204211 N.sine.jA. cos.i 51504185717 51529!o5702 51o54!d5687 515/9|»5672 51604185657 51628185642 51653 65627 51678 85612 51703 83397 51728 83382 51733 83367 51778185551 51803 185536 51828I85521 51852186506 51877185491 61'902j83476 51927185461 51952I85446 51977185431 52002183416 62026185401 52051 185385 52076|85370 52101 85355 152126 86340 152151185325 !52175|85310 i 52200185294 1 52225 52230 1 52276 1 52299 1 52324 1 52349 J 52374 1 52399 1 52423 62448 52473 52498 85279 85264 85249 86234 85218 85203 85188 85173 86157 85142 85127 85112 52522 85096 52547 62672 52697 52621 52646 85081 83066 85U31 83036 85U20 52671185005 \ 5269b!84989 84974 84939 84943 84928 84913 84897 84882 i52i20 1 52746 152770 1 52794 ! 62819 i 52844 ii528o9 52893184866 Tang. 1152918 1152943 1152967 1 1 52992 84851 84836 84820 84806 N.siae. 58 Degrees. TABLE n. Log. Sines and Tangents. (32°) Natural Sines. 53 Sine. 9.724210 724412 724614 72.i816 725017 725219 725420 725922 725823 726024 726225 9.726426 726626 726827 727027 727228 727428 727628 727828 728027 728227 9.728427 728626 728825 729024 729223 729422 729621 729820 730018 730216 730415 730613 730811 731009 731206 731404 731602 731799 731996 732193 9.732390 732587 732784 732980 733177 733373 733569 733765 733961 734157 9.734353 734549 734744 734939 735135 735330 736526 735719 735914 736109 Cosine. D^10"| Cosine. |D. 10' 33.7 33.7 33.6 33.6 33.6 33.6 33.6 33.6 33.6 33.6 33.5 33.4 33.4 33.4 33.4 33.4 33.3 33.3 33.3 33.3 33.3 33.2 33.2 33.2 33.2 33.1 33.1 33.1 33.1 33.0 33.0 33.0 33.0 33.0 32.9 32.9 32.9 32.9 32.9 32.8 32.8 32.8 32.8 32.8 32.7 32.7 32.7 32.7 32.7 32.6 32.6 32.6 32.6 32.6 32.6 32.6 32.6 32,6 32.4 32.4 .928420 928342 928263 928183 928104 928025 927946 927867 927787 927708 927629 .927549 927470 927390 927310 927231 927151 927071 926991 92691 1 926831 .926751 926671 926591 926511 926431 926351 926270 926190 926110 926029 .925949 925868 925788 925707 925626 925545 926466 925384 925303 925222 1.926141 925060 924979 924897 924816 924735 924654 924572 924491 924409 1.924328 924246 924164 924083 924001 923919 923837 923755 923673 923591 Sine. 13.2 13.2 13.2 13.2 13.2 13.2 13.2 13.2 13.2 13.2 13.2 13.2 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.4 13.4 13.4 13.4 13.4 13.4 13.4 13.4 13.4 13.4 13.4 13.4 13.6 13.6 13.5 13.5 13.5 13.6 13.5 13.5 13.5 13.6 13.6 13.6 13.6 13.6 13.6 13.6 13.6 13.6 13.6 13.6 13.6 13.6 13.7 13.7 Tang. D. 10"! Cotang. | N. sine. N. cos ,796789 796070 796351 796632 796913 797194 797475 797756 798036 798316 798596 .798877 799157 799437 799717 799997 800277 800567 800836 801116 801396 .801676 801966 802234 802513 802792 803072 803361 803630 803908 804187 .804466 804746 805023 805302 805580 805859 806137 806416 806693 806971 .807249 807527 807805 808083 808361 808638 808916 809193 809471 809748 .810026 810302 810580 810857 811134 811410 811687 811964 812241 812517 Cotang. 46.8 46.8 46.8 46.8 46.8 46.8 46.8 46.8 46.7 46.7 46.7 46.7 46.7 46.7 46.7 46.6 46.6 46.6 46.6 46.6 46.6 46.6 46.6 46.5 46.5 46.5 46.5 46.6 46.6 46.6 46.6 46.4 46,4 46.4 46.4 46.4 46,4 46.4 46.3 46.3 46.3 46.3 46.3 46.3 46.3 46.3 46.2 46,2 46.2 46.2 46.2 46.2 46.2 46.2 46.2 46.1 46,1 46.1 46.1 46.1 10.204211 I 203930 ! 203649 i 203368 203087 I 202806 202625 202245 201964 201684 201404 10.201123 200843 200563 200283 200003 199723 199443 199164 198884 198604 10.198326 198045 197766 197487 197208 196928 196649 196370 196092 195813 10.195534 195256 194977 194698 194420 194141 193863 193685 193307 193029 10.192761 192473 192195 191917 191639 191362 191084 190807 190529 190252 10.189975 189698 189420 189143 188866 188590 188313 188036 187759 187483 52992 63017 63041 53066 53091 53115 84805 84789 84774 84759 84743 84728 63140J84712 53164 53189 63214 53238 53263 53288 63312 53337 84697 84681 84666 84650 84635 84619 84604 84588 5336184573 5338684557 53411 63435 53460 53484 53609 53534 84542 84526 84511 84496 84480 84464 53658184448 63583 84433 63607j844l7 5363284402 53656184386 6368184370 53705184365 53730|84339 63754!84324 i 63779184308 j 53804!84292 53828;84277 I 53853184261 53877^4245 5390284230 5392684214 5395184198 53975184182 5400084167 54G24|84161 54049|84135 54073184120 I I 64097184104 j 154122:84088 l!54146j84072 il64171'840o7 1 154 195 184041 I j 54220 84025 , 54244 84009 I 54269 '83994 54293 83978 '54317:83962 54342,83946 ' 64366 83930 154391 83915 64415 83899 i 54440:83883 1 54464183867 Tang. N. COS. N. sine. 57 Degrees. 54 Log. Sines and Tangents. (33°) Natural Sines. TABLE II. 4 6 6 7 8 9 10 11 12 13 14 16 16 17 18 19 20 21 22 23 24 26 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 60 61 62 63 64 55 66 67 68 59 60 Sine. 9.736109 736303 736498 736692 736886 737080 737274 737467 737661 737856 738048 9.738241 738434 738627 738820 739013 739206 739398 739590 739783 739975 740167 740359 740550 740742 740934 741126 741316 741508 741699 741889 9.742080 742271 742462 742662 742842 743033 743223 743413 743602 743792 9.743982 744171 744361 744550 744739 744928 745117 745306 745494 745683 9.745871 746059 746248 746436 746624 746812 746999 747187 747374 747562 D. 10' Cosine. 32.4 32.4 32.4 32.3 32.3 32.3 32,3 32.3 32.2 32.2 32.2 32.2 32,2 32.1 32.1 32.1 32.1 32.1 32.0 32.0 32.0 32.0 32.0 31.9 31.9 31.9 31.9 31.9 31.8 31.8 31.8 31.8 31.8 31.7 31.7 31.7 31.7 31.7 31.6 31.6 31.6 31.6 31.6 31.6 31.6 31.5 31.5 31.6 31.4 31.4 31.4 31.4 31.4 31.3 31.3 31.3 31.3 31.3 31.2 31.2 Cosine. 9.923591 923509 923427 923345 923263 923181 923098 923016 922933 922861 922768 9.922686 922603 922520 922438 922356 922272 922189 922106 922023 921940 9.921867 921774 921691 921607 921624 921441 921357 921274 921190 921107 .921023 920939 920856 920772 920688 920604 920520 920436 920352 920268 ,920184 920099 920015 919931 919846 919762 919677 919593 919508 919424 919339 919254 919169 919085 919000 918916 918830 918745 918669 918574 D. 10" Sine. 13.7 13.7 13.7 13.7 13.7 13.7 13.7 13.7 13.7 13.7 13.8 13.8 13.8 13.8 13.8 13.8 13.8 13.8 13.8 13.8 13.8 13.9 13.9 13.9 13.9 13.9 13.9 13.9 13.9 13.9 13.9 13.9 14.0 14.0 14.0 14.0 14.0 14.0 14.0 14.0 14.0 14.0 14.0 14.0 14.1 14.1 14.1 14.1 14.1 14.1 14.1 14.1 14.1 14.1 14.1 14.1 14.2 14.2 14.2 14.2 Tang. 9.812517 812794 813070 813347 813623 813899 814175 814452 814728 815004 815279 9.816555 816831 816107 816382 816668 816933 817209 817484 817769 818035 9.818310 818686 818860 819136 819410 1 819684 819959 820234 820508 820783 9.821057 821332 821606 821880 822164 822429 822703 822977 823250 823524 9.823798 824072 824345 824619 824893 825166 825439 826713 825986 826259 826532 826805 827078 827351 827624 827897 828170 828442 828716 828987 I D. 10' 46.1 46.1 46 1 46.0 46.0 46.0 46.0 46.0 46.0 46.0 46.0 45.9 45.9 46.9 45.9 46.9 46.9 45.9 46.9 46.9 45.8 45.8 45.8 |46.8 45.8 I45.8 45.8 45.8 45.8 45.7 45.7 45.7 45.7 45.7 45.7 45.7 45.7 46.7 45.6 46.6 46.6 46.6 46.6 45.6 45.6 45.6 45.6 46.5 45.6 46.6 45.5 45.6 46.5 45.5 45.5 45.6 46.4 45.4 45.4 46.4 Cotano N. sine Cotang. 10.187482 187206 186930 186653 186377 186101 185825 185548 185272 184996 184721 10.184445 184169 183893 183618 183342 1830G7 182791 182516 182241 181966 10.181690 181415 181140 180866 180590 180316 180041 179766 179492 179217 178943 178668 178394 178120 177846 177571 177297 177023 176750 176476 10.176202 176928 176655 176381 175107 64464 54488 64513 64537 83867 83851 83835 83819 54561183804 54586183788 54610j83772 54635 83756 54659 fi3740 54683 64708 83724 83708 10. 64732 83692 54756 183676 6478183660 64805 83645 5482983629 54854183613 54878^83597 54902:83581 64927 83565 5495183549 I 54975 '83533 54999 ;835 17 55024:83501 55048:83485 ! 55072 83469 : 56097 83453 155121183437 55145183421 56169|83405 55194I83389 55218:83373 55-242 83356 55266 '83340 I 65291 83324 5531683308 55339 83292 55363 '83276 55388 83260 55412|83244 55436 '83228 65460;83212 55484183196 55509'83179 5553383163 55557:83147 174834 '5658r83131 174661 1 55605 '83 115 174287 ii 55630 83098 174014 II 55b5 183082 173741 1 1 1>6678 '83066 10. 173468 1155702 83060 173196 11 55726'83034 172922 1 ;55750'83U17 172649 : 55775 83001 172376 j 55799 82985 172103! 1 55823 82969 171830! I 55847 82963 171658 !j 65871 '82936 171285 |55&95;82920 171013 |65919[82904 X. cos.lN.sine.l Tang. 60 59 58 57 66 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 26 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 56 Degrees. TABLE II. Log. Sines and Tangents. (34°) Natural Sines. 55 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Sine. 9.747562 747749 747936 748123 748310 748497 748683 748870 749056 749243 749426 9.749615 749801 749987 750172 750358 750543 750729 750914 751099 751284 9.751469 751654 751839 752023 752208 752392 752576 752760 752944 753128 9.753312 753495 753679 753862 i 754046 i 754229 754412 I 754595 i 754778 1 754960 i 9.755143! 755326 755508 755690 755872 756054 756236 756418 756600 756782 9.756963 757144 757326 757507 757688 757869 758050 758230 758411 , J758591_j Cosine. I D. 10"| Cosine. 31.2 31.2 31.2 31.1 31.1 31.1 31.1 31.1 31.0 31.0 31.0 31.0 31.0 30.9 30.9 30.9 30.9 30.9 30.8 30.8 30.8 30.8 30.8 30.8 30.7 30.7 30.7 30.7 30.7 30.6 30.6 30.6 30.6 30.6 30.5 30.5 30.5 30 5 30.5 30.4 30.4 30.4 30.4 30.4 30.4 30.3 30.3 30.3 30.3 30.3 30.2 30.2 30.2 30.2 30.2 30.1 30.1 30.1 30.1 30.1 918574 918489 918404 918318 918233 918147 918062 917976 917891 917805 917719 9.917634 917548 917462 917376 917290 917204 917118 917032 916946 916859 916773 916687 916600 916514 916427 916341 916254 916167 916081 915994 9.915907 915820 916733 915646 915559 915472 915385 915297 915210 915123 915035 914948 914860 914773 914685 914598 914510 914422 914334 914246 9.914158 914070 913982 913894 913806 913718 913630 913541 913453 913365 Sine. D. 10" 14.2 14.2 14.2 14.2 14.2 14.2 14,2 14.3 14.3 14.3 14.3 14.3 14.3 14.3 14,3 14.3 14.3 14.4 14.4 14.4 14.4 14.4 14.4 14.4 14.4 14.4 14.4 14,4 14.5 14.5 14.5 14.5 14.5 14.5 14.5 14.5 14,5 14.5 14.5 14.5 14.6 14,6 14.6 14.6 14.6 14.6 14.6 14.6 14.6 14.6 14.7 14.7 14.7 14,7 14.7 14.7 14.7 14,7 14.7 14.7 Tang. 9.828987 829260 829532 829805 830077 830349 830621 830893 831165 831437 831709 9.831981 832253 832525 832796 833068 833339 833611 833882 834154 834425 ,834696 834967 835238 835509 835780 836051 836322 836593 836864 837134 837405 837675 837946 838216 838487 838757 839027 839297 839568 839838 9.840108 840378 840647 840917 841187 841457 841726 841996 842266 842535 842805 843074 843343 843612 843882 844151 844420 844G89 844958 846227 D. 10" Cotang. 1 N.sine. N. cos Cotang. 45.4 45.4 45.4 46.4 45.4 45.3 45.3 45.3 45.3 45.3 45.3 45.3 45.3 45.3 45.3 45.2 45.2 45.2 45.2 45.2 45.2 45.2 45.2 45.2 45.2 45.1 45,1 45.1 45.1 45.1 45,1 45.1 45.1 45.1 45.1 45.0 45.0 45.0 45.0 45.0 45.0 45.0 45.0 45.0 44.9 44.9 44.9 44.9 44.9 44.9 44.9 44.9 44.9 44.9 44.9 44.8 44.8 44.8 44.8 44.8 10.171013 170740 170468 170195 169923 169651 169379 169107 168835 168563 168291 10.168019 167747 167475 167204 166932 166661 166118 165846 165575 10.165304 165033 164762 164491 164220 163949 163678 163407 163136 162866 10.162595 162325 162054 161784 161513 161243 160973 160703 160432 160162 10.159892 159622 159353 159083 158813 158643 168274 158004 157734 157465 10.157195 166926 166667 156388 156118 165849 155580 165311 156042 154773 Tang. 55919 55943 55968 55992 56016 56040 56064 56088 56112 56136 56160 56184 56208 56232 56256 56280 56305 56329 66363 56377 56401 56425 56449 56473 56497 56621 66545 56669 56593 56617 56641 66666 56689 66713 56736 66760 66784 56808 568S2 66856 56880 66904 56928 66962 66976 57000 57024 67047 57071 67096 67119 67143 67167 67191 67216 67238 57262 67286 57310 67334 57358 82904 82887 82871 82865 82839 82822 82806 82790 82773 82757 82741 82724 82708 82692 82675 82659 82643 82626 82610 82593 82577 82661 82644 82528 82511 82496 82478 82462 82446 82429 82413 82396 82380 82363 82347 82330 82314 82297 82281 82264 82248 82231 82214 82198 82181 82165 82148 82132 82115 82098 82082 82065 82048 82032 82015 81999 81982 81965 81949 81932 81915 N. cos. N.sine 60 59 58 57 56 55 54 63 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 S 7 6 5 4 3 2 1 55 Degrees. 56 Log. Sines and Tangents. (35°) Natural Sines. TABLE II. Sine. 768591 758772 758952 759132 759312 759492 769672 759852 760031 760211 760390 760569 760748 760927 761106 761285 761464 761642 761821 761999 762177 762356 762534 762712 762889 763067 763245 763422 763600 763777 763954 764131 764308 764485 764662 764838 765015 765191 766367 765544 766720 9.765896 766072 766247 766423 766598 766774 766949 767124 767300 767475 9.767649 767824 767999 768173 768348 768522 768697 768871 769045 769219 Cosine. D. lU' 30.1 30.0 30.0 30.0 30.0 30.0 29.9 29.9 29.9 29.9 29.9 29.8 29.8 29.8 29.8 29.8 29.8 29.7 29.7 29.7 29.7 29.7 29.6 29.6 29.6 29.6 29.6 29.6 29.5 29.6 29.6 29.5 29.5 29.4 29.4 29.4 29.4 29.4 29.4 29.3 29.3 29.3 29.3 29.3 29.3 29.2 29.2 29.2 29.2 29.2 29.1 29.1 29.1 29.1 29.1 29.0 29.0 29.0 29.0 29.0 Cosine. .913365 913276 913187 913099 913010 912922 912833 912744 912665 912566 912477 .912388 912299 912210 912121 912031 911942 911853 911763 9U674 911584 .911496 911405 911315 911226 911136 911046 910956 910866 910776 910686 .910596 910506 910415 910326 910235 910144 910054 909963 909873 909782 .909691 909601 909510 909419 909328 909237 909146 909055 908964 908873 .908781 908690 9086^9 908607 908416 908324 908233 908141 908049 907958 Sine. D. 10' 14.7 14.7 14.8 14.8 14.8 14.8 14.8 14.8 14.8 14.8 14.8 14.8 14.9 14.9 14.9 14.9 14.9 14.9 14.9 14.9 14.9 14,9 14.9 15.0 16.0 16.0 15.0 15.0 15.0 15.0 16.0 Q 15.0^ 16.0 16.0 15.1 16.1 15.1 15.1 15.1 16.1 16.1 15.1 16.1 16.1 15.1 16.2 15.2 16.2 15.2 15.2 16.2 16.2 16.2 15.2 15.2 16.3 16.3 15.3 15.3 15.3 Tang. ,846227 846496 845764 846033 846302 846570 846839 847107 847376 847644 847913 .848181 848449 848717 849254 849522 849790 860058 850326 860593 860861 851129 861396 851664 851931 852199 852466 852733 863001 863268 853636 853802 854069 864336 864603 854870 865137 866404 866671 866938 856204 856471 856737 857004 867270 857637 857803 858069 858336 858602 .868868 859134 859400 859666 859932 860198 860464 860730 860995 861261 Cotang. Cotang. I N. sine. N. cos 10, 10 10 10 10 10 164773 154504 154236 163967 163698 153430 153161 152893 152624 152366 152087 161819 151551 161283 151014 150746 150478 150210 149942 149676 149407 149139 148871 148604 148336 148069 147801 147534 147267 146999 146732 146466 146198 145931 145664 145397 145130 144863 144696 144329 144062 143796 143629 143263 142996 142730 142463 142197 141931 141664 141398 141132 140866 140600 140334 140068 139802 139536 139270 139005 138739 67358 57381 57405 57429 57453 : 57477 57501 : 57524 57648 67572 57596 57619 I 57643 I 57667 i 57691 I 57715 57738 1 67762 81916 81899 81882 81866 81848 81832 81815 81798 81782 81765 81748 81731 81714 81698 81681 81664 81647 81631 57786|816]4 57810:81597 57833[81580 i 57867181563 67881 181546 57904|81530 57928:81513 57962 '81496 57976|81479 67999181462 58023!81445 58047 '81428 |58070;81412 j 58094 81396 6811881378 '58141 I 58165 I 58189 58212 58236 58260 68283 81361 81344 81327 81310 81293 81276 81259 58307 81242 58330 81226 68354 81208 68378 81191 6840181174 Tang. 58425 58449 68472 58496 68519 58543 I 58567 I 58590 58614 58637 58661 68684 58708 58731 81157 81140 81123 81106 81089 81072 81055 81038 81021 81004 80987 80970 .S0953 80^36 58765 80919 58779 809U2 N. cos. N.sine. 54 Degrees. TABLE II. Log. Sines and Tangents. (36°) Natural Sines. 57 Sine. 9.769219 769393 769666 769740 769913 770087 770260 770433 770606 770779 770952 9.771125 771298 771470 771643 771815 771987 772169 772331 772503 772675 9.772847 773018 773190 773361 773533 773704 773876 774046 774217 774388 9.774568 774729 774899 776070 775240 775410 775580 775760 775920 776090 776259 776429 776598 776768 776937 777106 777276 777444 777613 777781 777960 778119 778287 778456 778624 778792 778960 779128 779295 779463 Cosine. D. 10" 29.0 28.9 28.9 28.9 28.9 28.9 28.8 28.8 28.8 28.8 28.8 28.8 28.7 28.7 28.7 28.7 28.7 28.7 28.6 28.6 28.6 28.6 28.6 28.6 28.5 28.5 28.6 28.5 28.5 28.5 28.4 28.4 28.4 28.4 28.4 28.4 28.3 28.3 28.3 28.3 28.3 28.3 28.2 28.2 28.2 28.2 28.2 28.1 28.1 28.1 28.1 28.1 28.1 28.0 28.0 28.0 28.0 28.0 28.0 27.9 9. Cosine. .907968 907866 907774 907682 907690 907498 907406 907314 907222 907129 907037 .906945 906862 906760 906667 906576 906482 906389 906296 906204 906111 906018 906925 906832 906739 906645 906662 905459 906366 905272 906179 .905085 904992 904898 904804 904711 904617 904523 904429 904335 904241 .904147 904053 903969 903864 903770 903676 903581 903487 903392 903298 .903202 903108 903014 902919 902824 902729 902634 902539 902444 902349 Sine. D. 10" 15.3 15.3 16.3 15.3 16.3 16.3 15.3 16.4 15.4 16.4 16.4 16.4 15.4 16.4 15.4 15.4 16.4 15.5 15.5 15.6 16.5 15.5 15.5 15.6 15.6 15.5 15.5 15.5 15.6 15.6 15.6 15.6 15.6 16.6 15.6 16.6 16.6 15.6 15.7 16.7 15.7 15.7 15.7 15.7 16.7 16.7 16.7 16.7 15.7 15.8 16.8 16.8 16.8 15.8 15.8 15.8 15.8 16.8 16.9 15.9 Tang, .861261 861527 861792 862058 862323 862589 862854 863119 863385 863660 863916 .864180 864446 864710 864976 866240 866606 866770 866036 866300 866564 .866829 867094 867368 867623 867887 868152 868416 868680 868945 869209 .869473 869737 870001 870265 870529 870793 871057 871321 871586 871849 .872112 872376 872640 872903 873167 873430 873694 873967 874220 874484 .874747 876010 876273 875536 875800 876063 876326 876589 876851 877114 Cotang. D. 10" 44.3 44.3 44.2 44.2 44.2 44.2 44.2 44.2 44.2 44.2 44.2 44.2 44.2 44.2 44.1 44.1 44.1 44.1 44.1 44.1 44.1 44.1 44.1 44.1 44.1 44.1 44.0 44.0 44.0 44.0 44.0 44.0 44.0 44.0 44.0 44.0 44.0 44.0 44.0 44.0 43.9 43.9 43.9 43.9 43.9 43.9 43.9 43.9 43.9 43.9 43.9 43.9 43.9 43.8 43.8 43.8 43.8 43.8 43.8 43.8 Cotang. I N. sine. N. cos . 10.138739 138473 138208 137942 137677 137411 137146 136881 136615 136350 136085 10.135820 135656 136290 135025 134760 134495 134230 133966 133700 133436 10.133171 132906 132642 132377 132113 131848 131684 131320 131055 130791 10.130527 130263 129999 129736 129471 129207 128943 128679 128416 128151 10.127888 127624 127360 127097 126833 126570 126306 126043 126780 125516 10.125253 124990 124727 124464 124200 123937 123674 123411 123149 122886 58779 80902 58802 80885 5882680867 5884980860 68873:80833 58896 80816 68920;80799 58943;80782 68967 80765 58990,80748 59014'80730 59037:80713 69061 80696 59084 59108 69131 59154 69178 59201 59226 69248 69272 69296 59318 69342 69366 69389 59412 69436 59459 80679 80662 80644 80627 80610 80593 80576 80568 80541 80524 80507 80489 80472 80456 80438 80422 80403 69482180386 59506180368 59529180351 59552180334 5957680316 59599 80299 59622 59646 69669 59693 59716 59739 80282 80264 80247 80230 80212 80196 59763180178 69786180160 59803 80143 59832 18U126 5985(i 180108 59879|800yi 59902 80073 5992G 80056 69949 180038 699;2!80021 59995 60019 60042 60065 60089 60112 60135 60158 60182 Tang. II N. cos. N.sine. 80003 79986 79968 79951 79934 79916 79899 79881 79864 53 Degrees. 58 Log. Sines and Tangents. (37°) Natural Sines. TABLE II. D. 10" 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 20 39 31 32 33 34 35 36 37 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Sine. ,779463 779631 779798 779966 780133 780300 780467 780634 780801 780968 781134 781301 781468 781634 781800 781966 782132 782298 782464 782630 782796 782961 783127 783282 783458 783623 783788 783953 784118 784282 784447 784612 784776 784941 785105 785269 785433 785597 785761 785925 786089 786252 786416 786579 786742 786906 787069 787232 787395 787657 787720 787883 788045 788208 788370 788532 788694 788856 789018 789180 789342 Cosine. 27.9 27.9 27.9 27.9 27.9 27.8 27.8 27.8 27.8 27.8 27.8 27.7 27.7 27.7 27.7 27.7 27.7 27.6 27.6 27.6 27.6 27.6 27.6 27.5 27.5 27.5 27.5 27.5 27.5 27.4 27.4 27.4 27.4 27.4 27.4 27.3 27.3 27.3 27.3 27.3 27.3 27.2 27.2 27.2 27.2 27.2 27.2 27.1 27.1 27.1 27.1 27.1 27.1 27.1 27.0 27.0 27.0 27.0 27.0 27.0 Cosine. |D. 10" .902349 902263 902168 902063 901967 901872 901776 901681 901686 901490 901394 .901298 901202 901106 901010 900914 900818 900722 900626 900529 900433 .900337 900242 900144 900047 899961 899854 899757 899660 899664 899467 .899370 899273 899176 899078 898884 898787 898689 898592 898494 .898397 898299 898202 898104 898006 897908 897810 897712 897614 897516 .897418 897320 897222 897123 897025 896926 896828 896729 896631 896532 Sine. 15.9 15.9 15.9 15 9 15 9 15.9 15 9 15 9 15.9 15 9 16,0 16 16 16 16,0 16 16 16'0 16 16*0 16'l 16 1 16 16 16 16 16 16 16 16 16 16,2 16.2 16.2 16.2 16.2 16.2 16.2 16,2 16.2 16.3 16.3 16.3 16.3 16.3 16.3 16.3 16.3 16.3 16.3 16.3 16.4 16.4 16.4 16.4 16.4 16.4 16.4 16.4 16.4 Tang. 9. 9. .877114 877377 877640 877903 878166 878428 878691 878953 879216 879478 879741 880003 880265 880528 880790 881062 881314 881576 881839 882101 882363 882625 882887 883148 883410 883672 883934 884196 884467 884719 884980 886242 885503 885766 886026 886288 886549 886810 887072 887333 887594 887855 888116 888377 888639 888900 889160 889421 889682 889943 890204 890465 890725 890986 891247 891507 891768 892028 892289 892649 892810 D. 10' Cotang. 43.8 43.8 43.8 43.8 43.8 43.8 43.8 43.7 43.7 43.7 43.7 43.7 43.7 43.7 43.7 43.7 43.7 43,7 43.7 43.7 43.6 43,6 43.6 43.6 43.6 43.6 43.6 43.6 43.6 43.6 43.6 43.6 43.6 43.6 43.6 43.6 43.5 43.5 43.5 43.5 43.5 43.5 43.6 43.5 43.5 43.5 43,5 43.5 43.5 43.5 43.4 43.4 43.4 43.4 43.4 43.4 43.4 43.4 43.4 43.4 Cotang. [|N.sine. 10.122886 122623 122360 122097 121835 121672 121309 121047 120784 120522 120259 10.119997 119736 119472 119210 118948 118686 118424 118161 117899 117637 10.117375 117113 116852 116590 116328 116066 116804 115543 115281 115020 10.114758 114497 114236 113974 113712 113451 113190 112928 112667 112406 10.112145 111884 111623 111361 111100 110840 110679 110318 110067 109796 10.109635 109275 109014 108763 108493 108232 107972 107711 107451 107190 60182 60206 60228 60251 60274 60298 60321 60344 60367 60390 60414 60437 60460 60483 G0606 60529 60653 60576 60599 60622 60645 60668 60691 60714 60738 60761 60784 60807 60830 60853 60876 60899 60922 60946 60968 60991 61016 61038 61061 61084 61107 61130 61163 61176 61199 61222 61246 61268 61291 61314 61337 61360 61383 N. cos, 9864 79846 79829 79811 79793 79776 79758 79741 79723 79706 79688 79671 79668 79635 79618 79600 79583 79565 79547 79530 79512 79494 79477 79459 79441 79424 79406 79388 79371 79363 79336 79318 79300 79282 79264 79247 79229 79211 79193 79176 79168 79140 79122 79106 79087 79069 79061 79033 79016 78998 8980 78962 78944 61406 78926 61429 78908 61451 ; 61474 161497 1 61520 I 61643 i 61666 Tang. 78891 78873 78865 78837 78819 78801 N. COS. N.sine. 52 Degrees. TABLE n. liOg. Sines and Tangents. (38°) Natural Sines. 59 1 2 3 4 6 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 3S( 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Sine. D. 10" 9.789342 789504 789665 789827 789988 790149 790310 790471 790632 790793 790954 9.791115 791276 791436 791696 791757 791917 792077 792237 792397 792557 9.792716 792876 793035 793195 793354 793514 793673 793832 793991 794150 9.794308 7944671 794626 I 794784 ; 794942 I 795101 ' 795259 795417 795575 795733 9.795891 796049 796206 79G364 796521 796679 795836 796993 797150 797307 9.797464 797621 797777 797934 798091 798247 798403 79S560 798716 798872 26.9 26.9 26.9 26.9 26.9 26.9 26.8 26.8 26.8 26.8 26.8 26.8 26.7 9. 26.7 26.7 26.7 26.7 26.7 26.6 26.6 26.6 26.6 26.6 26.6 26.5 .6 .5 .5 .6 .6 .4 .4 .4 .4 26 26 26 26 26 26 26 26 26.4 26.4 26.4 26.3 26.3 26.3 26.3 26.3 26.3 26.3 26.2 26.2 26.2 26 26 26 26 26 26. 1 26.1 26.1 26-1 26.1 26.0 26.0 26.0 Cosine. D. 10' Cosine. 896532 896433 896335 896236 896137 896038 895939 895840 895741 895641 895542 .895443 895343 895244 895145 895045 894945 894846 894746 894646 894546 .894446 894346 894246 894146 894046 893946 893846 893745 893645 893544 .893444 893343 803243 893142 893041 892940 892839 892739 892638 892536 .892436 892334 892233 892132 892030 891929 891827 891726 891624 891623 .891421 891319 89121/ 891116 891013 890911 890809 8907U7 890605 890503 Sine. Tanf D. 10' 16.4 16.5 16.5 16.5 16.5 16.5 16.5 16.5 16.5 16.5 16.5 16.6 16.6 16.6 16.6 16.6 16.6 16.6 16.6 16.6 16.6 16.7 16.7 16.7 16.7 16.7 16.7 16.7 16.7 16.7 16.7 16.8 16.8 16.8 16.8 16.8 16.8 16.8 16.8 16.8 16.8 16.9 16.9 16.9 16.9 16.9 16.9 16.9 16.9 16.9 17.0 17.0 17.0 17.0 17.0 17.0 17.0 17.0 17.0 17.0 9. 892810 893070 893331 893591 893851 894111 894371 894632 894892 895152 895412 895672 895932 896192 896452 896712 896971 897231 897491 897761 898010 898270 898630 898789 899049 899308 899668 900086 900346 900605 .900864 901124 901383 901642 901901 902160 902419 902679 902938 903197 .903466 903714 903973 904232 904491 904760 905008 906267 905626 905784 .906043 906302 906660 906819 907077 907336 907694 907862 908111 908369 Cotang. 43.4 43.4 43.4 43.4 43.4 43.4 43.4 43.3 43.3 43.3 43.3 43.3 43.3 43.3 43.3 43.3 43.3 43.3 43.3 43.3 43.3 43.3 43.3 43.3 43.2 43 43 43 43 43 43 43 43 43 43.2 43.2 43 43 43 43 43 43 43 43.1 43.1 43.1 43.1 43.1 43.1 43.1 43.1 43.1 43.1 43.1 43.1 43.1 43.1 43.1 43.1 43.0 Cotang. I N. sine.[N. cos 10, 10. 10 10 10 10 1071901 106930 1 106669 106409 106149 106889 105629 105368 105108 104848 104688 104328 104068 103808 103548 103288 103029 102769 102509 102249 101990 101730 101470 101211 100951 100692 100432 100173 099914 099654 099395 099136 098876 098617 098358 098099 097840 097581 097321 097062 096803 09G545 096286 096027 095768 095509 096260 094992 094733 094474 094216 093957 093698 093440 093181 092923 092664 092406 092148 091889 091631 Tang. 61566 61689 61612 61635 61668 61681 61704 61726 61749 61772 61796 61818 61841 61864 61887 61909 61932 61955 61978 62001 62024 62046 62069 62092 62115 62138 62160 62183 62206 62229 62251 62274 62297 62320 62342 62366 62388 62411 62433 62466 62479 62602 62524 62547 62570 62592 62615 62638 62660 62683 6270b 62728 62/51 62774 62796 62819 62842 62864 62887 62909 62932 78801 78783 8766 78747 78729 8711 78694 78676 78668 78640 78622 78604 78586 78568 8550 78532 78514 78496 78478 78460 78442 78424 78406 78387 78369 78351 78333 78315 78297 8279 78261 8243 78225 8206 8188 78170 78152 78134 8116 8098 78079 78061 78043 78025 78007 77988 77970 77952 77934 77916 77897 77879 77861 77843 77824 77806 77788 77769 77751 77733 77715 N. COS. N.sine 51 Degrees. 60 Log. Sines and Tangents, (39°) Natural Sines. TABLE II. 1 2 3 4 5 6 7 8 9 10 Jl 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 Sine. 9.798772 799028 799184 799339 799495 799651 799806 799962 800117 800272 800427 9.800582 800737 800892 801047 801201 801356 801611 801665 801819 801973 9.802128 802282 802436 802589 802743 802897 803050 803204 -^ 803357 30 803511 D. 10" 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 61 52 53 54 56 56 57 58 59 60 9.803664 803817 803970 804123 804270 804428 804581 804734 804886 805039 9.805191 805343 805495 805647 805799 805951 806103 806254 806406 806557 9.806709 806860 807011 807163 807314 807465 807615 807766 807917 808067 Cosine. 26.0 26.0 26.0 25.9 25.9 25.9 25.9 25.9 25.9 25.8 25.8 25.8 25.8 25.8 25.8 25.8 25.7 25.7 25.7 25.7 25.7 25.7 25.6 25.6 25.6 25.6 25.6 25.6 25.6 25.5 25.5 25.5 25.5 25.5 25.5 25.4 25.4 26.4 25.4 25.4 25.4 26.4 26.3 25.3 25.3 25.3 25.3 25.3 25.3 25.2 25.2 25.2 25.2 25.2 25.2 25.2 25.1 25.1 25.1 25.1 Cosine. 1.890503 890400 890298 890195 890093 889990 889888 889785 889682 889579 889477 .889374 889271 889168 889084 888961 888766 888651 888548 888444 5.888341 888237 888134 888030 887926 887822 887718 887614 887510 887406 >. 887302 887198 887093 886989 88G885 886780 886676 886571 880466 88G362 '.886257 886152 886047 885942 885837 885732 886627 885522 885416 885311 .885205 886100 884994 884889 884783 884677 884572 884466 884360 884264 D. 10" Sine. 17.0 17.1 17.1 17.1 17.1 17.1 17.1 17.1 17.1 17.1 17.1 17.2 17.2 17.2 17.2 17.2 17.2 17.2 17.2 17.2 17.3 17.3 17.3 17.3 17.3 17.3 17.3 17.3 17.3 17.3 17.4 17.4 17.4 17.4 17.4 17.4 17.4 17.4 17.4 17.4 17.5 17.6 17.5 17.6 17.5 17.6 17.6 17.5 17.5 17.5 17.6 17.6 17.6 17.6 17,6 17.6 17.6 17.6 17.6 17.6 Tang. 9.908369 908G28 908886 90J144 909402 909660 909918 910177 910435 910693 910951 9.911209 911467 911724 911982 912240 912498 912756 913014 913271 913629 ). 913787 914044 914302 914660 914817 915075 915332 915590 915847 916104 1.911)362 916619 916877 917134 917391 917648 91/905 918163 918 120 918677 .918934 919191 919448 919705 919962 920219 920476 920733 920990 921247 .921503 921760 922017 922274 922530 922787 923044 923300 923557 923813 D. 10' Co tang. 43.0 43.0 43.0 43.0 43.0 43.0 43.0 43.0 43.0 43.0 43.0 43.0 43.0 43.0 43.0 43.0 43.0 43.0 42.9 42.9 42.9 42.9 42.9 42.9 42.9 42.9 42.9 42.9 42.9 42.9 42.9 42.9 42.9 42.9 42.9 42.9 42.9 42.9 42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.7 Cotang. 10 10 10 10.091631 091372 091114 090856 090598 090340 090082 089823 089565 089307 089049 088791 088533 088276 088018 087760 087502 087244 086986 086729 086471 .086213 085956 085698 085440 085183 084925 084668 084410 084153 033896 .083638 083381 083123 082866 082609 082352 082095 081837 081680 081323 081066 080809 080552 080295 080038 079781 J 9524 079267 079010 0/8753 .078497 078240 077983 077726 077470 077213 , 076956 : 076700 076443 076187 i N. sine. N. cos. 77715 77696 77678 77660 77641 77623 7605 77586 77568 77550 77531 77513 77494 77476 77458 77439 77421 77402 77384 77366 77347 77329 77310 77292 77273 77255 77236 77218 62932 62955- 6297 6300UI 63022 63045 63068 63090 63113 63135 63158 93180 63203 63225 63248 63271 63293 63316 63338 63361 63383 6340fj 63428 63461 63473 63496 63518 63540 63563177199 63585177181 10 10 6360.^ 63630 6365Li 63676 63698 i 63720 ji6374- 163765 1163787 I j 63810 ! I 63832 1:63854 ! 16387. j|6ab99 ||6392l: i 1 63944 |i6396{. ||63y8iJ 1164011 164033 64050 i640?S 64100 64123 ' 64146 64167 64190 64212 64234 64250 ; 64279 77162 77144 77126 77107 77088 77070 77051 77033 77014 76996 76977 76959 7()940 70921 70903 76884 76866 76847 70828 76810 76791 76772 76754 76735 76717 76698 76679 76661 76642 76623 76004 Tang. I N. cok. N.piue, 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 50 Degrees. TABLE II. Log. Sines and Tangents. (40°) Natural Sines. 61 Cotang. I N .sine. N. cos 1 2 3 4 6 6 7 8 9 10 .1 12 13 14 16 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 82 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9. Sine. D.W .808067 808218 808519 808669 808819 809119 809269 809419 809669 809718 809868 810017 810167 810316 810465 810614 810763 810912 811061 811210 811358 811507 811656 811804 811962 812100 812248 812396 812544 .812692 812840 812988 813136 81-3283 813430 813578 813726 813872 814019 .814166 814313 814460 814607 814753 814900 815046 815193 815339 815486 .816631 815778 816924 816069 816215 816361 816607 816652 816798 816943 Cosine. 25.1 25.1 25.1 26.0 26.0 25.0 26.0 26.0 25.0 24.9 24.9 24.9 24.9 24.9 24.9 24.8 24.8 24.8 24.8 24.8 24.8 24.8 24.7 24.7 24.7 24.7 24.7 24.7 24.7 24.6 24.6 24.6 24.6 24.6 24.6 24.6 24.5 24.6 24.5 24.6 24.6 24.6 24.5 24.4 24.4 24.4 24.4 24.4 24.4 24.4 24.3 24.3 24.3 24.3 24.3 24.3 24.3 24.2 24.2 24.2 Cosir D. 10" ,884254 884148 884042 883936 883829 883723 883617 883610 883404 883297 883191 ,883084 882977 882871 882764 882657 882560 882443 882336 882229 882121 ,882014 881907 881799 881692 881584 881477 881369 881261 881153 881046 ,880938 880830 880722 880613 880506 880397 880289 880180 880072 879963 .879865 879746 879637 879529 879420 879311 879202 879093 878984 878876 .878766 878656 878647 878438 878328 878219 878109 877999 877890 8 77780 Sine. 17.7 17.7 17.7 17.7 17.7 17.7 17.7 17.7 17.7 17.8 17.8 17.8 17.8 17.8 17.8 17.8 17.8 17.8 17.9 17.9 17.9 17.9 17.9 17.9 17.9 17.9 17.9 17.9 18.0 18.0 18.0 18.0 18.0 18.0 18.0 18.0 18.0 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18.2 18.2 18.2 18,2 18.2 18.2 18.2 18.2 18.2 18.3 18.3 18.3 18.3 Tang. D. 10" .923813 924070 924327 924583 924840 925352 925609 925866 926122 926378 .926634 926890 927147 927403 927669 927916 928171 928427 928683 928940 .929196 929462 929708 929964 930220 930475 930731 930987 931243 931499 •931755 932010 932266 932522 932778 933033 933289 933546 933800 934056 .934311 934567 934823 935078 935333 935589 935844 936100 936355 936610 .936866 937121 937376 937632 937887 938142 938398 938653 938908 939163 Cotang. 42.7 42.7 42.7 42.7 42.7 42.7 42.7 42.7 42.7 42.7 42.7 42.7 42.7 42.7 42.7 42.7 42.7 42.7 42.7 42.7 42.7 42.7 42.7 42.7 42.6 42.6 42.6 42.6 42.6 42.6 42.6 42.6 42.6 42.6 42.6 42.6 42.6 42.6 42.6 42.6 42,6 42.6 42,6 42.6 42,6 42,6 42.6 42.6 42,6 42.6 42,6 42,6 42.5 42.5 42,5 42,6 42,5 42,5 42,5 42.5 10.076187 076930 075673 076417 076160 074904 074648 074391 074136 073878 073622 10.073366 073110 072863 072597 072341 072086 071829 071573 071317 071060 10.070804 070548 070292 070036 069780 069625 069269 069013 068757 068501 10,068245 067990 067734 067478 067222 066967 066711 066456 066200 065944 10.066689 065433 065177 064922 064667 064411 064166 063900 063645 063390 10.063134 002879 062624 062368 062113 061868 061602 061347 061092 060837 64279 64301 64323 64346 64368 64390 64412 64435 64467 64479 64501 64524 64546 64568 64690 64612 64635 64667 64679 64701 64723 64746 64768 64790 64812 64834 64856 64878 64901 64923 64946 64967 64989 66011 65033 66066 65077 65100 66122 65144 65166 65188 65210 65232 66254 65276 65298 65320 65342 G5364 ! 1 65386 I 6540b i 1 65430 65452 65474 65496 65618 66640 65562 G5584 65006 Tang. 76604 76586 76667 76548 76530 76611 6492 76473 76455 76436 76417 76398 76380 6361 6342 76323 76304 76286 76267 76248 6229 76210 76192 76173 76154 76135 76116 76097 76078 76059 6041 6022 76003 75984 5965 5946 75927 76908 76889 75870 75851 75832 76813 76794 76775 75756 76738 76719 75700 75680 75661 76642 76623 75604 76585 75666 75647 75528 76509 76490 76471 N, COS. N.sine. 49 Degrees. 62 Log. Sines and Tangents. (41°) Natural Sines. TABLE n. 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 36 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Sine. 816943 817088 817233 817379 817524 817668 817813 817958 818103 818247 818392 9.818536 818681 818825 818969 819113 819257 819401 819545 819689 819832 9.819976 820120 820263 820406 820550 820693 820836 820979 821122 821265 9.821407 821550 821693 821835 821977 822120 822262 822404 822546 822688 9.822830 822972 823114 823255 823397 823539 823680 823821 823963 824104 9.824245 824386 824527 824668 824808 824949 825090 825230 825371 826511 D. Ky Cosine. 24.2 24.2 24.2 24.2 24.1 24.1 24.1 24.1 24.1 24.1 24 24.0 24.0 24.0 24.0 24.0 24.0 24.0 23.9 23.9 23.9 23.9 23.9 23.9 23.9 23.8 23.8 23.8 23.8 23.8 23.8 23.8 23.8 23.7 23.7 23.7 23.7 23.7 23.7 23.7 23.6 23.6 23.6 23.6 23.6 23.6 23.6 23.5 23.5 23.6 23.5 23.5 23.5 23.5 23.4 23.4 23.4 23.4 23.4 23.4 Cosine. 876678 .876568 876457 876347 876236 876125 876014 875904 875793 875682 875571 .875459 875348 876237 875126 875014 874903 874791 874680 874568 874456 .874344 874232 874121 874009 873896 873784 873672 873560 873448 873335 ,873223 873110 872998 872885 872772 872659 872547 872434 872321 872208 872095 871981 871868 871765 871641 871528 871414 871301 871187 871073 D. 10" 18.3 18.3 18.3 18.3 18.3 18.4 18.4 18.4 18.4 18.4 18.4 Sine. 18.4 18.4 18.4 18.5 18.5 18.5 18.5 18.5 18.5 18.5 18.5 18.5 18.5 18.6 18.6 18.6 18.6 18.6 18.6 18.6 18.6 18.7 18.7 18.7 18.7 18.7 18.7 18.7 18.7 18.7 18.7 18.8 18.8 18.8 18.8 18.8 18.8 18.8 18.8 18.8 18.9 18.9 18.9 18.9 18.9 18.9 18.9 18.9 18.9 Tang. 9.939163 939418 939673 939928 940183 940438 940694 940949 941204 941458 941714 9.941968 942223 942478 942733 942988 943243 943498 943752 944007 944262 5.944517 944771 945026 945281 946535 945790 946045 946299 946654 946808 J.9470t>3 947318 947572 947826 948081 948336 948590 948844 949099 949353 ). 949607 949862 950116 950370 960625 950879 951133 951388 961642 951896 ). 962150 952405 952669 962913 953167 963421 953675 953929 954183 954437 j Cotang. I D. 10' 42.5 42.5 42.5 42.5 42.5 42.5 42.5 42.5 42.5 42 42 42 42 42 42 42 42 42 42.5 42.5 42 42 42 42 42 42 42 42 42 42 42 42.4 42.4 42.4 42.4 42.4 42.4 Cotang. I N. sine 42 42 42 42 42 42 42 42.4 42.4 42.4 42.4 42.4 42.4 42.4 42.4 42.4 42.4 42.4 42.3 42.3 42.3 42.3 42.3 65759 65781 10.060837 1 165606 060582 1165628 060327 1 165650 060072 I '65672 059817!] 65094 059562 1 1 65716 069306^165738 069051 058796 058542!! 65803 068286 I 65825 10.058032 I 65847 057777! I 65869 057522!! 65891 057267 i 65913 057012 ji 65935 056757! [65956 056502 65978 056248 66000 N. cos 056993 065738 10.055483 055229 66022 66044 ! 66066 1 66088 0549741 66109 0547191 66131 054466 I! 66163 0542101166175 053955 j 166197 053701 166218 053446 1 1 66240 053192! 166262 10.052937' 66284 052682 66306 052428 66327 062174 66349 0619191 66371 0516641 66393 061410 i 66414 061156 66436 060901] 1 66458 060647] 66480 10.050393!! 66501 060138! 166523 049884 !j 66546 049630 ,66666 049375! 1 66588 049121 !!6661U 048867 I 66632 048612; 66653 048358 66675 048104; 66697 10.047850 66718 047696; 6674U 047341: 66762 0470871 66783 0468331 66806 046579 66827 046325: 66848 046071! 66870 045817 I j 66891 0465631 66913 76471 75452 76433 75414 75395 75375 75356 75337 75318 76299 76280 76261 76241 76222 5203 76184 75165 75146 75126 75107 75088 75069 76050 75030 75011 74992 74973 74953 74934 74915 74896 74876 74857 74838 74818 74799 74780 74760 74741 74722 74703 74683 74663 74644 74625 74606 74586 74567 74648 74522 74509 74489 74470 74451 74431 74412 74392 74373 74353 74334 74314 Tang. N. COS. N.sine. 48 Degrees. TABLE II. Log. Sines and Tangents. (42°) Natural Sines. 63 9.826511 825651 825791 825931 826071 826211 826361 826491 826631 826770 826910 9.827049 827189 827328 827467 827606 827746 827884 828023 828162 828301 9.828439 828578 828716 828856 Sine. 829131 829269 829407 829545 829683 9.829821 829959 830097 830234 830372 830509 830646 830784 830921 831058 9.831195 831332 831469 831606 831742 831879 832015 832152 832288 832425 9,832561 832697 832833 832969 833105 833241 833377 833512 833648 833783 D. 10" Cosine. Cosine. 23.4 23.3 23.3 23. 3 23.3 23.3 23.3 23.3 23.3 23.2 23.2 23.2 23.2 23.2 23.2 23.2 23.2 23.1 23.1 23.1 23.1 23.1 23.1 23.1 23.0 23.0 23.0 23.0 23.0 23.0 23.0 22.9 22.9 22.9 22.9 22.9 22.9 22.9 22.9 22.8 22.8 22.8 22.8 22.8 22.8 22.8 22.8 22.7 22.7 22.7 22.7 22.7 22.7 22.7 22.6 22.6 22.6 22.6 22.6 22.6 .871073 870960 870846 870732 870618 870504 870390 870276 870161 870047 869933 .869818 869704 869589 869474 869360 869245 869130 869015 868900 868785 .868670 868565 868440 868324 868209 868093 867978 867862 867747 867631 .867515 867399 867283 867167 867051 866819 866703 866586 866470 .866353 866237 866120 866004 866887 866770 866663 866636 866419 866302 .865185 865068 864950 864833 864716 864698 864481 864363 864246 864127 Sine. D. 10' 19.0 19.0 19.0 19.0 19.0 19.0 19.0 19.0 19.0 19.1 19.1 19.1 19.1 19.1 19.1 19.1 19.1 19.1 19.2 19.2 19.2 19.2 19.2 19.2 19.2 19.2 19.2 19.3 19.3 19.3 19.3 19.3 19.3 19.3 19.3 19.3 19.4 19.4 19.4 19.4 19.4 19.4 19.4 19.4 19.6 19.5 19.5 19.5 19.5 19.5 19.5 19.6 19.6 19.5 19.6 19.6 19.6 19.6 19.6 19.6 Tang. 9. 9. 9. 954437 964691 964946 966200 965464 965707 956961 956215 966469 956723 966977 967231 957485 957739 957993 958246 958600 958754 959008 969262 959516 959769 960023 960277 960531 960784 961038 961291 961546 961799 962062 962306 962560 962813 963067 963320 963574 963827 964081 964336 964588 964842 965095 965349 966602 966856 966109 966362 966616 967123 .967376 967629 967883 968136 968389 968643 968896 969149 969403 969656 Cotang. D. 10" 42.3 42.3 42.3 42.3 42.3 42.3 42.3 42.3 42.3 42.3 42.3 42.3 42.3 42.3 42.3 42.3 42.3 42.3 42.3 42.3 42.3 42.3 42.3 42.3 42.3 42.3 42.3 42.3 42.3 42.3 42.3 42.3 42.3 42.3 42.3 42.3 42.3 42.3 42.3 42.3 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42 42 42 42 42 42 42 42 42 Cotang. 10. 10 10 10 10 045663 046309 046065 044800 044546 044293 044039 043786 043631 043277 043023 042769 042516 042261 042007 041764 041600 041246 040992 040738 040484 040231 039977 039723 039469 039216 038962 038709 038455 038201 037948 037694 037440 037187 036933 036680 036426 036173 035919 035665 035412 035158 034905 034651 034398 034145 033891 10 033384 033131 032877 .032624 032371 032117 031864 031611 031357 031104 030861 030697 030344 Tang. N. sine. N. cos 66913 66935 66956 66978 66999 67021 67043 67064 67086 67107 67129 67151 67172 67194 67215 67237 67258 67280 67301 67323 67344 67366 67387 67409 67430 67462 67473 67496 67616 67638 67559 67680 67602 67628 67645 67666 67688 67709 67730 67752 67773 67795 67816 67837 67859 67880 67901 67923 67944 67965 67987 68008 68029 68051 68072 68093 68116 68136 68157 68179 68200 74314 74295 74276 74256 74237 74217 74198 74178 74159 74139 74120 74100 74080 74061 74041 74022 74002 73983 73963 73944 73924 73904 73886 73865 73846 73826 73806 73787 73767 73747 73728 73708 73688 73669 73649 73629 73610 73590 73570 73661 73631 73611 73491 73472 73462 73432 73413 73393 73373 73363 73333 73314 73294 73274 73254 73234 73215 73195 73175 73155 73135 N. cos. N.sine. 47 Degrees. 64 Log. Sines and Tangents. (43°) Natural Sines. TABLE n. Sine. Cosine. 9.864127 864010 863892 863774 863656 863538 863419 863301 863183 863064 862946 9,862827 862709 862590 862471 862363 862234 862115 861996 861877 861758 9.861638 861519 861400 861280 861161 861041 860922 860802 860682 860562 9.860442 860322 860202 860082 859962 859842 859721 859601 859480 859360 9 859239 859119 858998 858877 858756 858635 858514 858393 858272 ^. _ 858151 HJ'X 9.858029 857908 857786 857665 857543 857422 857300 857178 857056 856934 D. 10' 21.9 21.9 21.9 21.9 21.8 21.8 21.8 21.8 21.8 Sine. 19.6 19.6 19.7 19.7 19.7 19.7 19.7 19.7 19.7 19.7 19.8 19.8 19.8 19.8 19.8 19.8 19.8 19.8 19.8 19.8 19.9 19.9 19.9 19.9 19.9 19.9 19.9 19.9 19.9 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.1 20.1 20.1 20.1 20.1 20.1 20.1 20.1 20.2 20.2 20.2 20.2 20.2 20.2 20.2 20.2 20.2 20.3 20.3 20.3 20.3 20.3 20.3 Tan-; .969656 969909 970162 970416 970669 970922 971175 971429 971682 971935 972188 972441 972694 972948 973201 973454 973707 973960 974213 974466 974719 9.974973 975226 975479 975732 975985 976238 976491 976744 976997 977250 9.977503 977756 978009 978262 978515 978768 979021 979274 979527 979780 9.980033 D. 10" 980538 980791 981044 981297 981550 981803 982056 982309 .982562 982814 983067 983320 983573 983826 984079 984331 984584 984837 Cotang. 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.1 42.1 42.1 42.1 42.1 42.1 42.1 42.1 42.1 42.1 42.1 42.1 42.1 42.1 42.1 42.1 Cotang. iJN .sine. N. cos. 10.030344 030091 029838 029584 029331 029078 028825 028571 028318 028065 027812 10.027559 027306 027052 026799 026546 026293 026040 025787 025634 025281 10.026027 024774 024521 024268 024015 023762 023509 023256 023003 022750 10.022497 022244 021991 021738 021485 021232 020979 020726 020473 020220 10.019967 019714 019462 019209 018956 018703 018460 018197 017944 017691 10.0174381 017186 I 016933 I 016680 : 016427 016174 015921 i 015669 i 016416 015163 I ! 68200 168221 : 1 68242 '68264 68285 68306 68327 68349 73135 73116 73096 73076 /3056 73036 73016 72996 68370 72976 68391 68412 68434 72957 72937 72917 68455172897 68476 68497 68518 68539 68561 72877 72857 72837 72817 72797 68582172777 68603172757 68624 72737 68645 ;72717 68666 72697 68688:72677 68709'72657 6873072637 6875172617 68772172597 68793 172677 6881472657 68835 68857 68878 68899 72537 72617 72497 72477 68920 172467 68941172437 0896272417 68983172397 69004:72377 69025|72357 69040:72337 69067172317 6908872297 69109|72277 6913072267 69151 72236 69172J72216 6919372196 6921472176 69235172156 6925672136 69277172116 69298 72095 6931972076 6934072066 6936172035 69382;72015 6940371995 6942471974 69445:71954 69466171934 Tarn N. cos.lN.sine 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 10 9 8 7 6 5 4 3 2 1 TABLE n. Log. Sines and Tangents. (44°) Natural Sines. 65 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 26 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Sine. 841771 841902 842033 842163 842294 842424 842555 842686 842815 842946 843076 843206 843336 843466 843696 843726 843855 843984 844114 844243 844372 9.844502 844631 844760 845018 846147 846276 845405 845533 845662 845790 846919 846047 846176 846304 846432 846660 846688 846816 846944 847071 847199 847327 847454 847582 847709 847836 847964 848091 848218 9.848346 848472 848599 848726 848862 848979 849106 849232 849369 849486 D. 10" 21.8 21.8 21.8 21.7 21.7 21.7 21.7 21.7 21.7 21.7 21.7 21.6 21.6 21.6 21.6 21.6 21.6 21.6 21.6 21.5 21.5 21.5 21.5 21.6 21.5 21.5 21.5 21.4 21.4 21.4 21.4 21.4 21.4 21.4 21.4 21.4 21.3 21.3 21.3 21.3 21.3 21.3 21.3 21.3 21.2 21.2 21.2 21.2 21.2 21.2 21.2 21.2 21.1 21.1 21.1 21.1 21.1 21.1 21.1 21.1 Cosine. .856934 866812 866690 866668 866446 866323 866201 866078 866966 855833 856711 .855588 855465 856342 866219 865096 864973 854850 854727 854603 854480 .864356 864233 864109 863986 863862 863738 853614 853490 863366 863242 .863118 852994 862869 852745 862620 862496 862371 862247 852122 861997 ,861872 861747 851622 851497 861372 851246 861121 860996 860870 850746 860619 850493 850368 850242 850116 849990 849864 849738 849611 849486 D. 10" 20 20 20 20 20 20 20. 20.4 20.4 20.4 20.5 20.5 20.5 20.6 20.5 20.5 20.5 20.5 20.6 20.6 20.6 20.6 20.6 20.6 20.6 20.6 20.6 20.7 20.7 20.7 20.7 20.7 20.7 20.7 20.7 20.7 20.8 20.8 20.8 20.8 20.8 20.8 20.8 20.8 20.9 20.9 20.9 20.9 20.9 20.9 20.9 20.9 21.0 21.0 21.0 21.0 21.0 21.0 21.0 21.0 Tang. 9.984837 985090 986343 986696 986848 986101 986354 986607 986860 987112 987365 987618 987871 988123 988376 988629 988882 989134 989387 989640 989893 9.990146 990398 990651 990903 991156 991409 991662 991914 992167 992420 9.992672 992926 993178 993430 993683 993936 994189 994441 994694 994947 9.996199 996452 996705 995957 996210 996463 996715 996968 997221 997473 9.997726 997979 998231 998484 998737 998989 999242 999496 999748 10.000000 D.,10' 42.1 42.1 42.1 42.1 42.1 42.1 42.1 42.1 42.1 42.1 42.1 42.1 42.1 42.1 42.1 42.1 42.1 42.1 42.1 42.1 42.1 42 42 42 42 42 42 42.1 42.1 42.1 42.1 42.1 42.1 42.1 42.1 42.1 42.1 42.1 42.1 42.1 42.1 42.1 42.1 42.1 42.1 42.1 42 42 42 42 42 42 42 42.1 42.1 42.1 42.1 42.1 42.1 42.1 Co tang. 10.015163 014910 014657 014404 014162 013899 013646 013393 013140 012888 012635 iO. 012382 012129 011877 011624 011371 011118 010866 010613 010360 010107 10.009856 009602 009349 009097 008844 008591 008338 008086 007833 007580 10- 007328 007075 006822 006670 006317 006064 006811 005559 005306 006063 10-004801 004548 004295 004043 003790 003537 I 00328^ 003032 002779 002627 10.002274 002021 001769 001516 001263 001011 000758 000505 000263 000000 N. sine. N. cos 69466 69487 69608 69629 69549 69670 69691 69612 69633 69664 69675 69696 69717 69737 69758 69779 69800 69821 69842 69862 1 69883 69904 69925 69946 69966 69987 70008 70029 70049 70070 70091 70112 70132 70153 70174 70196 70215 70236 70257 70277 70298 70319 70339 70360 70381 70401 70422 I 70443 70463 70484 70505 70525 70546 70667 70587 70608 70628 70649 70670 70690 70711 71934 71914 71894 71873 71853 71833 71813 71792 71772 71752 71732 71711 71691 71671 71650 71630 71610 71590 71569 71549 71629 71608 71488 71468 71447 71427 71407 71386 71366 71346 71325 71305 71284 71264 71243 71223 71203 71182 71162 71141 71121 71100 71080 71059 71039 71019 0998 70978 70967 70937 0916 70896 70876 0865 70834 70813 70793 0772 70762 70731 70711 Cosine. Sine. Cotang. Tang. N. cos. N.pine. 4& Degrees. f OF THE X^ THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO 50 CENTS ON THE FOURTH DAY AND TO $1.00 ON THE SEVENTH DAY OVERDUE. NOV 26 1946 NOV 27 194fi t^w* ^^A m m '^m SEC'D LP unr~4 1357 ^p^^| C^'LiLD WIW %\\%^t LD 21-100m-12,'43 (8796s) .11111.11 niiliupiu^ji I iiJiii i^iuiuiiipiimiiilili A 111892 •■>