JC-NRLF 
 
 SB ZflD 135 
 
GIFT OF 
 MICHAEL REESE 
 
TABLES 
 
 OF 
 
 PARABOLIC CUKVES' 
 
 J^^N 
 
 FOR THE USE OF RAILWAY ENGINEERS 
 AND OTHERS 
 
 BY 
 
 GEORGE T. ALLEN 
 
 
 
 ASSOC. M. INST. C.E. 
 
 HonUon 
 E. & F. N. SPON, LIMITED, 125 STRAND 
 
 SPON & CHAMBERLAIN, 12 CORTLANDT STREET 
 1898 
 
 [Entered at Stationers' Hall] ft J 
 
A to 
 
PREFACE. 
 
 THE Tables of Parabolic Curves have been calcu- 
 lated by the author, and are now brought to the 
 knowledge of engineers with a view of supplying a 
 want : that is, the practical tracing of curves with a 
 gradually decreasing and increasing radius, instead 
 of the usual circular curves. By this means it is 
 hoped that the great shocks caused by change of 
 direction in such a mass as a moving train on 
 entering a circular curve may be avoided, resulting, 
 no doubt, in great saving as regards material in 
 working, as well as much greater convenience to 
 those who are travelling. 
 
 The use of the Tables is also strongly advised in 
 la} ing out irrigation and navigable canals, as tend- 
 ing largely, in the former case, to diminish erosion, 
 and in the latter to facilitate navigation, and also 
 to prevent wash. Indeed, in all engineering works 
 where a change in direction of moving masses is 
 involved, these tables are recommended. 
 
 The Author is further of opinion that Parabolic 
 Curves fit more easily into the natural surface of. 
 
 B 2 
 
mountainous or undulating country than circular 
 curves do, thus diminishing in a considerable degree 
 the amount of cutting and embankment in new 
 works. 
 
 The Tables are also useful in the design of struc- 
 tures where the contour is a parabola, the accurate 
 length being easily obtainable. 
 
 GEOEGE T. ALLEN. 
 GRAVESEND: 1898. 
 
EXPLANATION. 
 
 THE tables give, as will be seen on inspection, the 
 lengths of the curve, the tangent, the radius at 
 apex, and the half-chord for the half-angle of inter- 
 section A B D or DEC (see Fig. 1), for every ten 
 
 minutes of that angle, the height B D being unity. 
 The bisectrix B E, and the height of the curve D E, 
 are each equal to 5, or the half of unity. 
 
 In setting out the curves practically, the bi- 
 sectrix is usually fixed on the ground ; then, to 
 
obtain the actual corresponding lengths of the 
 curve, tangent, radius at apex, &c., it is necessary 
 to multiply the lengths given in the tables by twice 
 the length of the bisectrix as fixed upon. 
 
 If the length of the tangent is fixed upon and not 
 that of the bisectrix, this length, divided by the 
 quantity A B or B C in the columns, gives the total 
 height from the chord to the intersection, and the 
 true length of curve, radius at apex, or half-chord, 
 will be found by multiplying each quantity by the 
 height obtained. 
 
 As in all curve books, interpolation must be 
 resorted to for the odd minutes when the half-angle 
 is not an even ten, and the resulting numbers are 
 to be multiplied by the double bisectrix, or height 
 from chord to intersection as above. 
 
 The method of setting out the intermediary points 
 in the curve is the following : 
 
 The length of the bisectrix being determined 
 upon, the tangent is divided into so many parts as it 
 may be thought necessary to adopt of intermediary 
 points, as at F, G, H, I, K, B (see Fig. 2), in which 
 example the tangent is divided into six parts. The 
 bisectrix is then divided by the square of this 
 number, in this case by 36 ; the result is now multi- 
 plied by the squares of 1, 2, 3, &c., to 6 that is, by 
 1, 4, 9, &c,, to 36 and these are the lengths of the 
 
offsets F F,, G G,, H H,, &c., to B E, which is the bi- 
 sectrix. It is an advantage to adopt for the length 
 of the bisectrix a quantity which is the square, or a 
 multiple of the square, of the number of divisions it 
 has been decided to divide the tangent into, as this 
 
 I, 
 
 FIG. 2. 
 
 facilitates the calculations of the offsets, and is fre- 
 quently within the competency of the operator. 
 
 The offsets are in all cases parallel to the bi- 
 sectrix. 
 
 EXAMPLE I. The angle of intersection of two 
 tangents equals 139, and the bisectrix is equal to 
 5 chains ; required the length of tangents, the length 
 of the curve, and the radius at apex. 
 
 Here, the half-angle of intersection ABD (Fig. 1) 
 
8 
 
 is equal to 69 30', and the total height is equal io 
 10 chains. 
 
 Chains Links 
 
 Tangent. . . = 2-855x10 = 28 55 
 Curve . . . = 5-471x10 = 54 71 
 Radius at apex = 7-153 X 10 = 71 53 
 
 Note. Metres, yards, feet or any other standard 
 of length is used in exactly the same manner. 
 
 EXAMPLE II. The angle of intersection of two 
 tangents equals 132 40', and the bisectrix equals 
 4 ch. 90 Iks. Required the length of the curve, 
 tangent, radius at apex, and the offsets for 7 inter- 
 mediate points. 
 
 The half-angle of intersection in this case is 
 66 20', and the total height from chord to inter- 
 section 9 ch. 80 Iks. Therefore, 
 
 Chains Links 
 
 Tangent . . = 2*491x9-80 = 24 41 
 Curve . . . = 4-705x9-80 = 46 10 
 Radius at apex = 5-206 x 9-80 = 51 01 
 
 For 7 intermediate points the tangent is to be 
 divided into that number of parts, and the bisectrix 
 is to be divided by the square of this number or 49, 
 and this gives the modulus for the offsets : that is, 
 
 M = il^ = 10 links. 
 49 
 
The offsets then are 
 
 Chains Links 
 
 10 X 1 = 10 
 
 10 X 4 = 40 
 
 10 x 9 = 90 
 
 10 X 16 = 1 60 
 
 10 x 25 = 2 50 
 
 10 X 36 = 3 60 
 
 10 X 49 = 4 90 
 the last being the bisectrix. 
 
 EXAMPLE III. Kequired the length of the curve, 
 the length of the tangent, and the radius at apex 
 for the insertion of a curve between two tangents, 
 the angle of intersection being 145 8', and the 
 length of the bisectrix being 2 ch. 30 Iks. Find 
 also the offsets for 6 intermediate points. 
 
 The half-angle of intersection is 72 34'. 
 
 The tangent for 72 40' is 3 3564 
 72 30' 3-3255 
 
 Difference 0-0309 
 
 The tangent for 72 30 . 3 3255 
 Add T \ of difference . . 0-01236 
 
 Gives tangent for 72 34' 3-33786 
 
 which, multiplied by 4-60, gives 15 ch. 35 Iks., the 
 length of the tangent sought. 
 
10 
 
 The curve for 72 40' is 6-5106 
 72 30' 6-4467 
 
 Difference .... 0-0639 
 
 The curve for 72 30' is 6-4467 
 Add T %- of difference . 0-02556 
 
 Curve for 72 34' , 6-47226 
 
 K, 
 
 FIG. 3. 
 
 which, multiplied by 4*60, gives 29 ch. 77 Iks., the 
 length of the curve. 
 
 The radius at apex for 72 40' is 10-2660 
 72 30' 10-0590 
 
 Difference. 0-2070 
 
li 
 
 Badius at apex for 72 30' is 10-0590 
 Add T 4 F of difference ... 08280 
 
 Radius at apex for 72 34' . 10*14180 
 
 which, multiplied by 4* 60, gives 46 ch. 65 Iks. 
 
 The bisectrix being equal to 230 links, and the 
 intermediate points to 6, 
 
 M= = 6-388 links. 
 36 
 
 The offsets are, therefore, 
 
 Chains Links 
 
 6-388 X 1 = 6-388 
 
 6-388 X 4 = 25-552 
 
 6-388 x 9 = 57-492 
 
 6-388 x 16 = 1 2-208 
 
 6-388 x 25 = 1 59-700 
 
 6-388 x 36 = 2 29-968 
 
 or 2 ch. 30 Iks., which is the bisectrix. 
 
 The number of decimals in these results is decided 
 by the degree of accuracy required by the operator. 
 
 At times it may be found convenient, as in the 
 case of large curves, to set off from a line parallel to 
 the chord and passing through the apex. In this 
 case the half-chord is divided into a convenient 
 number of parts in the manner in which it has been 
 recommended to divide the tangent. These divisions 
 are set off along EL (Fig. 3) at K,, I,, H G,,, &o., 
 and the same ordinates are used which are calcu- 
 
12 
 
 lated for setting out from the tangent. K K,, is equal 
 to F F y . 1^ I /y i s equal to G G,, or half the curve may 
 
 FIG. 4. 
 
 be traced from the half-chord, and the other half 
 from the tangent. 
 
13 
 
 It happens occasionally in practice that it is 
 necessary to trace a curve joining two tangents 
 which are possibly parallel, or even whose angle of 
 intersection is negative, and it would appear at first 
 sight that it would be impossible to effect the de- 
 sired purpose by employing the parabola, the nature 
 of this curve requiring divergent tangents. As 
 an example of this case see Fig. 4, in which AB 
 and C D are the proposed tangents. By adopting a 
 
 FIG. 5. 
 
 point D, and secondary tangents D E and D F (such 
 lines are usually given as portions of the survey), 
 and these lines being divided respectively in the 
 points G and H, making DG and DH equal, the 
 half parabola A G is traced from the tangent AB 
 and angle EBG, and similarly the half-parabola 
 D H, and finally the whole parabola G I H is inserted 
 on the tangents G D D H within the angle G D H. 
 The radius of curvature at the apex of the curve 
 
14 
 
 (the point E in Fig. 1) is given in the column 
 under that heading, the height from chord to inter- 
 section being unity. If, however, it is necessary 
 to ascertain the radius at the tangent points, the 
 following formula applies : 
 
 = ATP 
 = AIT 
 
 The radius at any other point in the curve can be 
 obtained by calculating A B and A D on the assump- 
 tion that the point is a tangent point. 
 
 It may be stated that each half of the curve is 
 quite independent of the other, as in the case of 
 
 FIG. 6. 
 
 Fig. 5, where two different curves are tangent to 
 each other, the only condition being that the height 
 must be the same in each case. 
 
 The outer elevation and inner depression of the 
 rails in a curve can be set to parabolic ordinates by 
 making the elevation or depression at B one-eighth 
 of the total rise E F, arid at D seven-eighths of the 
 same quantity (see Fig. 6). In a long curve it is 
 
UNIVERSITY 
 15 
 
 advisable to adopt the points calculated for setting 
 out the curve as tbe ones where correction for eleva- 
 tion or depression is effected. 
 
 To set out a tangent at any point of the curve 
 (see Fig. 3), the following is the proceeding : If a 
 tangent is requii*ed passing through F,, set off on 
 AL, commencing at A, a quantity equal to the off- 
 set F F,, and this point is in the tangent ; similarly, if 
 a tangent is required at G, set off on AL a quantity 
 equal to the offset G G,, and so on, for any point. 
 The same result is obtained by setting out lengths 
 along E B which are severally equal to K, K,,, I, I, /} 
 &c. For a point falling between any of the points 
 considered above, it is necessary to know its offset 
 from the tangent, and to proceed exactly in the 
 same manner. The offsets must in all cases be 
 parallel to the bisectrix. 
 
PREFACE. 
 
 CES Tables de Courbes Paraboliques ont ete calculees 
 par 1'auteur et sont aujourd'hui portees a la connais- 
 sance des ingenieurs avec le but de remplir un 
 besoin : c'est a dire, pour fournir une methode pra- 
 tique de tracer une courbe dont le rayon decroit 
 ou croit graduellement au lieu de se servir de la 
 courbe circulaire. Par ce moyen on espere que les 
 grandes secousses que Ton eprouve par les change- 
 ments de direction d'une grande masse comme un 
 train en mouvement quand ceci entre dans une courbe 
 circulaire seront evitees; ce qui resulte en grande 
 epargne de materiel roulant et fixe, ainsi que la plus 
 grande convenance des voyageurs. 
 
 L'usage de ces tables est aussi fortement raccom- 
 mande dans le trace des canaux d'irrigation et de 
 navigation : dans le premier cas pour la diminution 
 de 1'erosion des bords, et dans le second pour faciliter 
 la navigation et aussi pour empecher les dedom- 
 magements des vagues causees par le passage des 
 bateaux. En general, quand il est question de 
 
18 
 
 changement de direction de grandes masses, ces 
 tables sont fort propres a employer. 
 
 L'auteur est aussi d'opinion, que les courbes para- 
 boliques se raccordent beaucoup plus facilement a la 
 surface naturelle du terrain dans les pays monta- 
 gneux ou ondulants que n'est le cas des courbes 
 circulaires ; ce qui diminue dans un degre appreci- 
 able la quantite de remblais et deblais necessaires 
 sur les nouveaux travaux. 
 
 Les tables sont aussi d'utilite dans les projets de 
 construction, la ou le contour est une parabole, car 
 on en obtient la longueur exacte de la courbe avec 
 grande facilite. 
 
 GEORGE T. ALLEN, 
 
 Ingtnieur. 
 GRAVESEND: 1898. 
 
EXPLICATION. 
 
 CES tables donnent la longueur de la courbe para- 
 bolique, ainsi que celle de la tangente, le rayon an 
 sommet de la courbe, et la demi-corde pour la moitie 
 de Tangle compris entre les tangentes ; c'est a dire : 
 
 FIG. 1. 
 
 pour Tangle A B D ou Tangle BBC (voir la figure 
 No. 1), pour chaque dix minutes d'ouverture; la 
 hauteur B D est en tous cas egale a Tunite. La 
 bissectrice BE et la hauteur de la courbe D E 
 equivalent 0*5 ou bien la moitie de Tunite 
 
 c 2 
 
20 
 
 Dans la pratique quand on trace la courbe, la 
 longueur de la bissectrice est ordinairement fixee 
 sur le terrain ; alors, pour obtenir la vraie longueur 
 de la courbe, la taugente, le rayon au sommet, on 
 multiplie les chiffres donnees dans les colonnes par 
 le double de la longueur de la bissectrice qui a ete 
 fixee. 
 
 Si au lieu de la bissectrice on fixe la longueur de 
 la tangente, on divise cette longueur par la quantite 
 donnee dans la colonne des tangentes A B ou B C, et 
 le numero qui resulte est la hauteur totale, ou la 
 bissectrice doublee ; et pour avoir la vraie longueur 
 de la courbe, le rayon au sommet ou la demi-corde 
 on doit multiplier le numero obtenu par le nombre 
 dans les colonnes. 
 
 On doit se servir de 1'interpolation pour les angles 
 qui ont des minutes entre les dix. 
 
 La methode de tracer les points intermediates, 
 c'est cornine suit. 
 
 Quand on a decide de la longueur de la bissectrice, 
 on divise la tangente en autant de parties egales que 
 Ton veut adopter de points intermediaires, comme on 
 voit sur la figure No. 2, F, G, H, I, K, B, dans quel 
 cas la tangente est divisee en six parties. Alors on 
 divise la bissectrice par le carre de ce nombre, c'est 
 a dire, par 36. Ce resultat est multiplie par les 
 
21 
 
 carres de 1, 2, 3, etc., jusqu'a 6, c'est a dire, par 1, 4, 
 9, etc., jusqu'a 36, et ces resultats soiit effectivement 
 les longueurs des ordonnes F F /5 G G /? H H, et cetera 
 jusqu'a B E que Ton voit est la bissectrice. On peut 
 remarquer qu'on trouve avantage de fixer pour la 
 longueur de la bissectrice une quantite qui est le 
 
 FIG. 2. 
 
 carre ou un multiple du carre du nombre dans 
 ^equel on a decide de diviser la tangente. La raison, 
 c'est qu'ainsi on facilite le calcul des ordonnes, et 
 souvent cette demarche se trouve a la competence de 
 1'operateur. Ces ordonnes sont toujours paralleles 
 a la bissectrice. 
 
 EXEMPLE I er . L'angle de Tintersection de deux 
 tangentes equivaut 139, et la bissectrice a 50 metres 
 
22 
 
 de longueur. On demande la longueur de la tan- 
 gente, celle de la courbe et le rayon au sommet (a 
 Tapex). 
 
 Dans ce cas, la moitie de Tangle de 1'intersection 
 ABD (Fig. 1) vaut 69 30', et la hauteur totale est 
 de 100 metres ; done 
 
 m. 
 
 La tangente . . = 2-8554 x 100 = 285-54 
 La courbe. . . =5-4713x100 = 547-13 
 Le rayon au sommet = 7-1536 x 100 = 715-36 
 
 qui est le resultat demande. 
 
 Nota. On peut employer de la me me fagon une 
 base de longueur quelconque, comme pieds anglais 
 ou autres. 
 
 EXEMPLE II. L'angle d'intersection de deux tan- 
 gentes vaut 132 40' et la bissectrice est de 49 metres. 
 On demande la longueur de la courbe, la tangente, 
 le rayon au sommet et les ordonnees pour 7 points 
 intermediaires. 
 
 La moitie de Tangle d'intersection dans ce cas est 
 66 20', et la hauteur totale de la corde a Tintersec- 
 tion 98 metres. Alors 
 
 La tangente . . = 2-4911 X 98 = 244-12 
 La courbe . . . = 4-7054 x 98 = 461 12 
 Le rayon au sommet = 5-2060 x 98 = 510-18 
 
. 23 
 
 Pour 7 points intermediaires on doit diviser la 
 tangente en 7 parties egales et la bissectrice doit se 
 diviser par 49, ce qui donne un module pour les 
 ordonnees ; c'est a dire : 
 
 49 
 
 M = - = 1 metre. 
 49 
 
 Les ordonnees sont alors 
 
 m. 
 
 ixi=i 
 
 1X4=4 
 1X9=9 
 1 X 16 = 16 
 1 X 25 = 25 
 1 X 36 = 36 
 1 X 49 = 49 
 
 Ce dernier on reconnait pour la bissectrice. 
 
 EXEMPLE III. On demande la longueur de la 
 courbe de la tangente et aussi le rayon de courbure 
 au soinmet pour conjoindre les deux tangentes dont 
 Tangle de 1'intersection est 145 8 f et la longueur de 
 la bissectrice est 23 metres. En plus, il faut les 
 ordonnes pour six points intermediaires. 
 
 La moitie de Tangle vient a 72 34'. 
 
 La tangente pour 72 40' est ] 
 
 / -i j. 1 1 A r 
 (voir les tables) ) 
 
 72 30' est 3-3255 
 
 Difference 0-0309 
 
24 
 
 La tangente pour 72 30' est 3 3255 
 
 Ajoutez T 4 ^ de la difference pour) 
 
 i A f 
 
 les 4 minutes ..... ( 
 
 Done la tangente pour 72 34' est 3-33786 
 
 En multipliant ce chiffre par la hauteur totale 
 46 metres on trouve la longueur de la tangente, 
 152 '541 metres. 
 
 La courbe pour 72 40' est . . 6-5106 
 72 30' est . . 6-4467 
 
 Difference 0-0639 
 
 La courbe pour 72 30' est .. 6-4467 
 Ajoutez -^ de la difference . . 0*02556 
 
 Done la courbe pour 72 34' est 6-47226 
 
 En multipliant ce nombre par 46 metres on trouve 
 la longueur de la courbe, 297-723 metres. 
 
 Le rayon au sommet pour 72 40') 
 
 est fl 
 
 72 30' est 10-0590 
 Difference 0-2070 
 
 Le rayon au sommet pour 72 30' i 
 est . . . . . f 
 Ajoutez -^ de la difference . . 0-08280 
 
 Done le rayon au sommet pour) 
 
 72 34' est .... .flO'14180 
 
25 
 
 Ce nombre multiplie par 46 metres donne 
 466*522 metres, le rayon cherche. 
 
 Pour obtenir les ordonnes intermediates, comme 
 la bissectrice equivaut 23 metres et le nombre d' or- 
 donnes 6, le module 
 
 23 
 M = = 638 metre. 
 
 DO 
 
 Les ordonnes sont alors 
 
 0-638 X 1 = 0*63 
 0-638 X 4= 2-55 
 0-638 x 9= 5-74 
 0-638 x 16 = 10-20 
 0-638 x 25 = 15-95 
 0-638 x 36 = 22-96 
 
 c'est a dire 23 metres, qui est la bissectrice. 
 
 Le nombre des points decimales depend du degre 
 d'approxirnation voulu. 
 
 Quelquefois, surtout quand il s'agit de grandes 
 courbes, on facilite les operations en faisant la tracee 
 sur une ligne parallele a la corde qui en meme temps 
 passe par le sommet de la courbe. Voir la figure 3. 
 Dans ce cas la ligne E L est divisee dans un nombre 
 de parties egales de la meme fagon que Ton a fait 
 pour la tangente, et les memes ordonnes sont calcules 
 comme si Ton voulait tracer sur la tangente, mais on 
 
26 
 
 les trace en positions inverses. De fait, K, K /y est 
 egal a F F,, et I, I u est egal a G G j5 et ainsi de suite. 
 On peut tracer la moitie de la courbe de la tangente 
 et 1'autre moitie de la ligne E L a volonte. 
 
 Dans la pratique il arrive qu'il est necessaire de 
 tracer une courbe entre deux tangentes sensiblement 
 
 Fn 
 
 I, 
 
 K, 
 
 Z^C^*- 
 
 K, 
 
 D 
 
 FIG. 3. 
 
 paralleles ou dont Tangle d'intersection est meme 
 negative, et a premiere vue on le croirait impossible, 
 parce que la nature de la parabole demande des tan- 
 gentes divergentes. Comme exemple de ce cas, voir 
 la figure No. 4 ou A B et C D sont les tangentes dont 
 il s'agit. On interpose un point D formant des 
 tangentes secondaires DE et DF (les lignes telles 
 
27 
 
 ordinairement forment partie de Farpentage). Alors 
 on divise ces lignes respectivement en G et H et de 
 
 FIG. 4. 
 
 fagon que D G = D H. La demi-parabole AG est 
 traeee de la tangente A B avec Tangle E B G, et egale- 
 
28 
 
 ment la demi-parabole D H ; et fmalement la parabola 
 GIH est tracee entre les tangentes GD, DH dans 
 Tangle G D H. 
 
 Le rayon de courbure an sominet de la courbe 
 (E, Fig. 1) est donne dans les colonnes, la hauteur 
 totale de la corde a 1'intersection etant toujours 
 Funite. Si Ton a besoin de connaitre le rayon de 
 courbure aux points de tangence on a 
 
 P = 
 
 AB 3 
 A~D* 
 
 FIG. 5. 
 
 On peut remarquer que chaque moitie de la courbe 
 est parfaitement independante de 1'autre, comnie dans 
 la figure No. 5 ou les deux courbes sont tangentes 
 Tune a 1'autre; la seule condition, c'est qu'ellesont la 
 meme hauteur. 
 
 Dans une courbe de chemin de fer, Televation du 
 rail exterieur et la depression du rail interieur peu- 
 vent s'accorder a la forme parabolique en faisant la 
 
29 
 
 hausse a B (voir Fig. 6) de la valeur d'une huitieme 
 de la hausse totale E F, et la quantite a D de sept- 
 huitiemes de la meme. Dans une grande courbe il 
 est mieux de calculer ces corrections pour les points 
 qui correspondent aux ordonnes de la tracee. 
 
 Pour tracer une tangente a un point quelconque 
 de la courbe (voir Fig. 3) on fait comme suit. Si 
 on demande une tangente qui traverse le point F, 
 on trace sur AL depuis le point A une quantite 
 egale a F F y ; le point resultant est dans la tangente 
 
 FIG. 6. 
 
 voulue. Aussi, si Ton a besoin d'une tangente a G, 
 il faut tracer sur la meme ligne la quantite G G,, et 
 ainsi de suite pour tout autre point. On obtient le 
 meme resultat en tragant sur E B des distances egales 
 a K ; K y> , I, ! pour les points K y I, et cetera. Pour un 
 point qui se trouve entre deux des points deja con- 
 sideres, il faut calculer 1'ordonne exacte et poursuivre 
 la meme construction. Les ordonnes sont en tons 
 les cas paralleles a la bissectrice. 
 
TABLES 
 
 PARABOLIC CURVES. 
 
 Angle 
 ABD 
 or 
 DBC. 
 
 Curve 
 A EC. 
 
 Tangent 
 A B or B C. 
 
 Radius at 
 Apex. 
 
 Half-Chord 
 A D or D C. 
 
 40 
 
 2-017018 
 
 1-305407 
 
 7041 
 
 -839100 
 
 IO 
 
 2-025452 
 
 1-308607 
 
 7125 
 
 844069 
 
 20 
 
 2-033939 
 
 1-311833 
 
 7209 
 
 849062 
 
 30 
 
 2-042481 
 
 1-315087 
 
 7295 
 
 854081 
 
 40 
 
 2-051078 
 
 1-318368 
 
 7381 
 
 859124 
 
 50 
 
 2-059730 
 
 1-321676 
 
 7468 
 
 864193 
 
 41 
 
 2-068437 
 
 1-325013 
 
 7557 
 
 869287 
 
 IO 
 
 2-077200 
 
 1-328378 
 
 7646 
 
 874407 
 
 20 
 
 2-086020 
 
 1-331771 
 
 7736 
 
 879553 
 
 $o 
 
 2-094897 
 
 1-335192 
 
 7827 
 
 884725 
 
 4 
 
 2-103832 
 
 1-338643 
 
 7920 
 
 889924 
 
 5 
 
 2-112825 
 
 1-342123 
 
 8013 
 
 895151 
 
32 TABLES OF PARABOLIC CURVES. 
 
 Angle 
 ABD 
 or 
 DBG. 
 
 Curve 
 A EC. 
 
 Tangent 
 A B or B C. 
 
 Radius at 
 
 Apex. 
 
 Half-Chord 
 A D or D C. 
 
 42 
 
 2-121877 
 
 1-345633 
 
 8107 
 
 900404 
 
 IO 
 
 2-130989 
 
 1-349172 
 
 8203 
 
 905685 
 
 20 
 
 2-140161 
 
 1-352742 
 
 8299 
 
 910994 
 
 3 
 
 2-149394 
 
 1-356342 
 
 8397 
 
 916331 
 
 40 
 
 2-158687 
 
 1-359972 
 
 8495 
 
 921697 
 
 50 
 
 2-168042 
 
 1-363634 
 
 8595 
 
 927091 
 
 43 
 
 2-177460 
 
 1-367327 
 
 8696 
 
 932515 
 
 10 
 
 2-186941 
 
 1-371052 
 
 8798 
 
 937968 
 
 20 
 
 2-196486 
 
 1-374809 
 
 8901 
 
 943451 
 
 30 
 
 2-206095 
 
 1-378598 
 
 9005 
 
 -948965 
 
 40 
 
 2-215769 
 
 1-382420 
 
 9111 
 
 954508 
 
 50 
 
 2-225509 
 
 1-386275 
 
 9218 
 
 960083 
 
 44 
 
 2-235315 
 
 1-390164 
 
 9326 
 
 965689 
 
 10 
 
 2-245189 
 
 1-394086 
 
 9435 
 
 971326 
 
 20 
 
 2-255130 
 
 1-398042 
 
 9545 
 
 976996 
 
 30 
 
 2-265140 
 
 1-402032 
 
 9657 
 
 982697 
 
 4 
 
 2-275219 
 
 1-406057 
 
 9770 
 
 988432 
 
 5 
 
 2-285368 
 
 1-410118 
 
 9884 
 
 99419*9 
 
TABLES OF PAEABOLIC CURVES. 33 
 
 Angle 
 ABD 
 or 
 DBC. 
 
 Curve 
 
 AEC. 
 
 Tangent 
 A B or B C. 
 
 Radius at 
 Apex. 
 
 Half-Chord 
 A D or D C. 
 
 45 
 
 2-295587 
 
 1-414214 
 
 1-0000 
 
 1-000000 
 
 IO 
 
 2-305878 
 
 1-418345 
 
 1-0117 
 
 1-005835 
 
 20 
 
 2-316242 
 
 1-422513 
 
 1-0235 
 
 1-011704 
 
 30 
 
 2-326678 
 
 1-426718 
 
 1-0355 
 
 1-017607 
 
 40 
 
 2-337188 
 
 1-430960 
 
 1-0476 
 
 1-023546 
 
 50 
 
 2-347773 
 
 1-435239 
 
 1-0599 
 
 1-029520 
 
 46 
 
 2-358433 
 
 1-439556 
 
 1-0723 
 
 1-035530 
 
 10 
 
 2-369169 
 
 1-443912 
 
 1-0849 
 
 1-041577 
 
 20 
 
 2-379982 
 
 1-448306 
 
 1-0976 
 
 1-047660 
 
 3 
 
 2-390874 
 
 1-452740 
 
 1-1105 
 
 1-053780 
 
 40 
 
 2-401845 
 
 1-457213 
 
 1-1235 
 
 1-059938 
 
 5 
 
 2-412895 
 
 1-461726 
 
 1-1366 
 
 1-066134 
 
 47 
 
 2-424025 
 
 1-466279 
 
 1-1500 
 
 1-072369 
 
 10 
 
 2-435237 
 
 1-470874 
 
 1-1635 
 
 1-078642 
 
 20 
 
 2-446531 
 
 1-475509 
 
 1-1771 
 
 1-084955 
 
 30 
 
 2-457909 
 
 1-480187 
 
 1-1910 
 
 1-091308 
 
 40 
 
 2-469371 
 
 1-484907 
 
 1-2049 
 
 1-097702 
 
 5 
 
 2-480919 
 
 1-489670 
 
 1-2191 
 
 1- 104136 
 
 ! 
 
34 TABLES OF PARABOLIC CURVES. 
 
 Angle 
 ABD 
 or 
 DBC. 
 
 Curve 
 
 A EC. 
 
 Tangent 
 A B or B C. 
 
 Radius at 
 Apex. 
 
 Half-Chord 
 A D or D C. 
 
 48 
 
 2-492553 
 
 1-494476 
 
 1-2335 
 
 1-110612 
 
 10 
 
 2-504274 
 
 1-499327 
 
 1-2480. 
 
 1-117130 
 
 20 
 
 2-516083 
 
 1-504221 
 
 1-2627 
 
 1-123691 
 
 30 
 
 2-527981 
 
 1-509160 
 
 1-2776 
 
 1-130294 
 
 4 
 
 2-539969 
 
 1-514145 
 
 1-2926 
 
 1-136941 
 
 5 
 
 2-552049 
 
 1-519176 
 
 1-3079 
 
 1-143633 
 
 49 
 
 2-564222 
 
 1-524253 
 
 1-3233 
 
 1-150368 
 
 10 
 
 2-576489 
 
 1-529377 
 
 1-3390 
 
 1-157149 
 
 20 
 
 2-588850 
 
 1-534549 
 
 1-3548 
 
 1-163976 
 
 3 
 
 2-601307 
 
 1-539769 
 
 1-3709 
 
 1-170850 
 
 40 
 
 2-613862 
 
 1-545038 
 
 1-3871 
 
 1-177770 
 
 50 
 
 2-626515 
 
 1-550356 
 
 1-4036 
 
 1-184738 
 
 50 
 
 2-639267 
 
 1-555724 
 
 1-4203 
 
 1-191754 
 
 10 
 
 2-652120 
 
 1-561142 
 
 1-4372 
 
 1-198818 
 
 20 
 
 2-665074 
 
 1-566612 
 
 1-4543 
 
 1-205933 
 
 3 
 
 2-678131 
 
 1-572134 
 
 1-4716 
 
 1-213097 
 
 40 
 
 2-691293 
 
 1-577708 
 
 1-4892 
 
 1-220312 
 
 5 
 
 2-704561 
 
 1-583335 
 
 1-5070 
 
 1-227579 
 
TABLES OF PARABOLIC CURVES. 35 
 
 Angle 
 ABD 
 or 
 DBC. 
 
 Curve 
 AEC. 
 
 Tangent 
 A B or B C. 
 
 Radius at 
 Apex. 
 
 Haff-Chord 
 A D or D C. 
 
 51 
 
 2-717937 
 
 1-589016 
 
 1-5250 
 
 1-234897 
 
 IO 
 
 2-731421 
 
 1-594751 
 
 1-5432 
 
 1-242268 
 
 20 
 
 2-745014 
 
 1-600542 
 
 1-5617 
 
 1-249693 
 
 30 
 
 4 
 
 50 
 
 52 
 
 2-758718 
 2-772535 
 2-786467 
 2-800514 
 
 1-606388 
 1-612291 
 1-618251 
 1-624269 
 
 1-5804 
 1-5995 
 1-6187 
 1-6383 
 
 1-257172 
 1-264706 
 1-272296 
 1-279942 
 
 IO 
 
 2-814678 
 
 1-630346 
 
 1-6580 
 
 1-287645 
 
 20 
 
 2-828960 
 
 1-636483 
 
 1-6781 
 
 1-295406 
 
 30 
 40 
 
 5 
 53 
 
 2-843363 
 
 2-857888 
 2-872535 
 
 2-887308 
 
 1-642680 
 1-648938 
 1-655257 
 1-661640 
 
 1-6984 
 1-7190 
 1-7399 
 1-7610 
 
 1-303225 
 1-311105 
 1-319044 
 1-327045 
 
 IO 
 
 2-902208 
 
 1-668086 
 
 1-7825 
 
 1-335107 
 
 20 
 
 2-917236 
 
 1-674597 
 
 1-8043 
 
 1-843233 
 
 30 
 4 
 
 5 
 
 2-932394 
 2-947684 
 2-963108 
 
 1-681173 
 1-687815 
 1-694524 
 
 1-8263 
 
 1-8487 
 1-8714 
 
 1-351422 
 1-359676 
 1-367996 
 
 D 2 
 
36 TABLES OF PARABOLIC CUEVES. 
 
 Angle 
 ABD 
 or 
 DBG. 
 
 Curve 
 
 A EC. 
 
 Tangent 
 A B or B C. 
 
 Eadius at 
 Apex. 
 
 Half-Chord 
 A D or D C. 
 
 54 
 
 2-978667 
 
 1-701302 
 
 1-8944 
 
 1-376382 
 
 IO 
 
 2-994364 
 
 1-708148 
 
 1-9178 
 
 1-384835 
 
 20 
 
 3-010200 
 
 1-715064 
 
 1-9414 
 
 1-393357 
 
 3 
 
 3-026177 
 
 1-722051 
 
 1-9654 
 
 1-401948 
 
 40 
 
 3-042298 
 
 1-729110 
 
 1-9898 
 
 1-410610 
 
 50 
 
 3-058564 
 
 1-736241 
 
 2-0145 
 
 1-419343 
 
 55 
 
 3-074977 
 
 1-743447 
 
 2-0396 
 
 1-428148 
 
 IO 
 
 3-091539 
 
 1-750727 
 
 2-0650 
 
 1-437027 
 
 20 - 
 
 3-108253 
 
 1-758084 
 
 2-0909 
 
 1-445980 
 
 3 
 
 3-125121 
 
 1-765517 
 
 2-1171 
 
 1-455009 
 
 40 
 
 3-142144 
 
 1-773029 
 
 2-1436 
 
 1-464115 
 
 50 
 
 3-159326 
 
 1-780620 
 
 2-1706 
 
 1-473298 
 
 56 
 
 3-176668 
 
 1-788292 
 
 2-1980 
 
 1-482561 
 
 IO 
 
 3-194173 
 
 1-796045 
 
 2-2258 
 
 1-491904 
 
 20 
 
 3-211843 
 
 1-803881 
 
 2-2540 
 
 1-501328 
 
 30 
 
 3-229681 
 
 1-811801 
 
 2-2826 
 
 1-510835 
 
 4 
 
 3-247689 
 
 1-819806 
 
 2-3117 
 
 1-520426 
 
 5 
 
 3-265870 
 
 1-827898 
 
 2-3412 
 
 1-530102 
 
TABLES OF PARABOLIC CURVES. 37 
 
 Angle 
 ABD 
 or 
 DBC. 
 
 Curve 
 
 A EC. 
 
 Tangent 
 A B or B C. 
 
 Radius at 
 Apex. 
 
 Half-Chord 
 A D or D C. 
 
 57 
 
 3-284226 
 
 1-836078 
 
 2-3712 
 
 1-539865 
 
 10 
 
 3-302760 
 
 1-844348 
 
 2-4016 
 
 1-549715 
 
 20 
 
 3-321474 
 
 1-852707 
 
 2-4325 
 
 1-559655 
 
 3 
 
 3-340372 
 
 1-861159 
 
 2-4639 
 
 1-569686 
 
 40 
 
 3 359457 
 
 1-869704 
 
 2-4958 
 
 1-579808 
 
 5 
 
 3-378731 
 
 1-878344 
 
 2-5282 
 
 1-590024 
 
 58 
 
 3-398197 
 
 1-887080 
 
 2-5611 
 
 1-600334 
 
 10 
 
 3-417857 
 
 1-895914 
 
 2-5945 
 
 1-610742 
 
 20 
 
 3-437716 
 
 1-904847 
 
 2-6284 
 
 1-621247 
 
 30 
 
 3-457775 
 
 1-913881 
 
 2-6629 
 
 1-631852 
 
 4 
 
 3-478039 
 
 1-923017 
 
 2-6980 
 
 1-642558 
 
 50 
 
 3-498511 
 
 1-932258 
 
 2-7336 
 
 1-653366 
 
 59 
 
 3-519194 
 
 1-941604 
 
 2-7698 
 
 1-664279 
 
 10 
 
 3-540091 
 
 1-951058 
 
 2-8066 
 
 1-675299 
 
 20 
 
 3-561206 
 
 1-960621 
 
 2-8440 
 
 1-686426 
 
 30 
 
 3-582543 
 
 1-970294 
 
 2-8821 
 
 1-697663 
 
 4 
 
 3-604105 
 
 1-980081 
 
 2-9207 
 
 1-709012 
 
 5 
 
 3-625896 
 
 1-989982 
 
 2-9600 
 
 1-720474 
 
38 TABLES OF PARABOLIC CUEVES. 
 
 Angle 
 ABD 
 or 
 DBC. 
 
 Curve 
 AEC. 
 
 ^Tangent 
 A B or B C. 
 
 Radius at 
 Apex. 
 
 Half-Chord 
 A D or D C. 
 
 60 
 
 3-647919 
 
 2-000000 
 
 3-0000 
 
 1-732051 
 
 IO 
 
 3-670179 
 
 2-010136 
 
 3-0406 
 
 1-743745 
 
 20 
 
 3-692679 
 
 2-020393 
 
 3-0819 
 
 1-755559 
 
 30 
 
 3-715423 
 
 2-030772 
 
 3-1240 
 
 1-767494 
 
 4 
 
 3-738416 
 
 2-041276 
 
 3-1668 
 
 1-779552 
 
 50 
 
 3-761662 
 
 2-051906 
 
 3-2103 
 
 1-791736 
 
 61 
 
 3-785165 
 
 2-062665 
 
 3-2546 
 
 1-804048 
 
 10 
 
 3-808930 
 
 2-073556 
 
 3-2996 
 
 1-816489 
 
 20 
 
 3-832961 
 
 2-084579 
 
 3-3455 
 
 1-829063 
 
 3 
 
 3-857263 
 
 2-095738 
 
 3-3921 
 
 1-841771 
 
 40 
 
 3-881841 
 
 2-107036 
 
 3-4396 
 
 1-854616 
 
 5 
 
 3-906699 
 
 2-118474 
 
 3-4879 
 
 1-867600 
 
 62 
 
 3-931843 
 
 2-130054 
 
 3-5371 
 
 1-880726 
 
 IO 
 
 3-957277 
 
 2-141781 
 
 3-5872 
 
 1-893997 
 
 20 
 
 3-983007 
 
 2-153655 
 
 3-6382 
 
 1-907415 
 
 3 
 
 4-009039 
 
 2-165681 
 
 3-6902 
 
 1-920982 
 
 4 
 
 4-035377 
 
 2-177859 
 
 3-7431 
 
 1-934702 
 
 5o 
 
 4-062027 
 
 2-190195 
 
 3-7970 
 
 1-948577 
 
TABLES OF PAEABOLIO CURVES. 39 
 
 Angle 
 ABD 
 or 
 DBG. 
 
 Curve 
 
 A EC. 
 
 Tangent 
 A B or B C. 
 
 Radius at 
 Apex. 
 
 Half-Chord 
 A D or D C. 
 
 63 
 
 4-088995. 
 
 2-202689 
 
 3-8519 
 
 1-962610 
 
 IO 
 
 4-116287 
 
 2-215346 
 
 3-9078 
 
 1-976805 
 
 20 
 
 4-143910 
 
 2-228168 
 
 3-9647 
 
 1-991164 
 
 30 
 
 4-171869 
 
 2-241158 
 
 4-0228 
 
 2-005690 
 
 4 
 
 4-200170 
 
 2-254320 
 
 4-0820 
 
 2-020386 
 
 5 
 
 4-228821 
 
 2-267657 
 
 4-1423 
 
 2-035256 
 
 64 
 
 4-257827 
 
 2-281172 
 
 4-2037 
 
 2-050304 
 
 IO 
 
 4-287196 
 
 2-294868 
 
 4-2664 
 
 2-065532 
 
 20 
 
 4-316935 
 
 2-308750 
 
 4-3303 
 
 2-080944 
 
 3 
 
 4-347051 
 
 2-322820 
 
 4-3955 
 
 2-096544 
 
 40 
 
 4-377551 
 
 2-337083 
 
 4-4620 
 
 2-112335 
 
 5 
 
 4-408444 
 
 2-351542 
 
 4-5297 
 
 2-128321 
 
 65 
 
 4-439737 
 
 2-366202 
 
 4-5989 
 
 2-144507 
 
 10 
 
 4-471438 
 
 2-381065 
 
 4-6695 
 
 2-160896 
 
 20 
 
 4-503555 
 
 2-396137 
 
 4-7415 
 
 2-177492 
 
 30 
 
 4-536095 
 
 2-411421 
 
 4-8149 
 
 2-194300 
 
 4 
 
 4-569069 
 
 2-426922 
 
 4-8899 
 
 2-211323 
 
 5 
 
 4-602486 
 
 2-442645 
 
 4-9665. 
 
 2-228568 
 
TABLES OF PARABOLIC CURVES. 
 
 Angle 
 ABD 
 or 
 DBG. 
 
 Curve 
 
 AEC. 
 
 Tangent 
 A B or B C. 
 
 Radius at 
 Apex. 
 
 Half-Chord 
 A B or D C. 
 
 66 
 
 4-636355 
 
 2-458593 
 
 5-0447 
 
 2-246037 
 
 10 
 
 4-670686 
 
 2-474773 
 
 5-1245 
 
 2-263736 
 
 20 
 
 4-705487 
 
 2-491187 
 
 5-2060 
 
 2-281669 
 
 3 
 
 4-740768 
 
 2-507843 
 
 5-2893 
 
 2-299842 
 
 40 
 
 4-776540 
 
 2-524744 
 
 5-3743 
 
 2-318261 
 
 5 
 
 4-812813 
 
 2-541896 
 
 5-4612 
 
 2-336929 
 
 67 
 
 4-849599 
 
 2-559305 
 
 5-5500 
 
 2-355852 
 
 10 
 
 4-886909 
 
 2-576975 
 
 5-6408 
 
 2-375037 
 
 20 
 
 4-924754 
 
 2-594914 
 
 5-7336 
 
 2-394489 
 
 30 
 
 4-963145 
 
 2-613126 
 
 5-8284 
 
 2-414214 
 
 40 
 
 5-002097 
 
 2-631618 
 
 5-9254 
 
 2-434217 
 
 50 
 
 5-041622 
 
 2-650396 
 
 6-0246 
 
 2-454506 
 
 68 
 
 5-081731 
 
 2-669467 
 
 6-1260 
 
 2-475087 
 
 10 
 
 5-122437 
 
 2-688837 
 
 6-2298 
 
 2-495966 
 
 20 
 
 5-163756 
 
 2-708514 
 
 6-3360 
 
 2-517151 
 
 3 
 
 5-205701 
 
 2-728504 
 
 6-4447 
 
 2-538648 
 
 40 
 
 5-248287 
 
 2-748814 
 
 6-5560 
 
 2-560465 
 
 5 
 
 5-291529 
 
 2-769453 
 
 6-6699 
 
 2-582609 
 
TABLES OF PARABOLIC CTHtVES. 41 
 
 Angle 
 ABD 
 or 
 DBG. 
 
 Curve 
 A EC. 
 
 Tangent 
 A B or B C. 
 
 Radius at 
 Apex. 
 
 Half-Chord 
 A Dor DC. 
 
 69 
 
 5-335443 
 
 2-790428 
 
 6-7865 
 
 2-605089 
 
 10 
 
 5-380046 
 
 2-811747 
 
 6-9059 
 
 2-627912 
 
 20 
 
 5-425353 
 
 2-833418 
 
 7-0282 
 
 2-651087 
 
 30 
 
 5-471381 
 
 2-855451 
 
 7-1536 
 
 2-674621 
 
 40 
 
 5-518148 
 
 2-877853 
 
 7-2820 
 
 2-698525 
 
 5 
 
 5-565674 
 
 2-900635 
 
 7-4137 
 
 2-722808 
 
 70 
 
 5-61398 
 
 2-92380 
 
 7-5487 
 
 2-74748 
 
 10 
 
 5-66307 
 
 2-94737 
 
 7-6870 
 
 2-77254 
 
 20 
 
 5-71299 
 
 2-97135 
 
 7-8289 
 
 2-79802 
 
 30 
 
 5-76374 
 
 2-99574 
 
 7-9745 
 
 2-82391 
 
 40 
 
 5-81535 
 
 3-02057 
 
 8-1238 
 
 2-85023 
 
 50 
 
 5-86784 
 
 3-04584 
 
 8-2771 
 
 2-87700 
 
 71 
 
 5-92124 
 
 3-07155 
 
 8-4344 
 
 2-90421 
 
 10 
 
 5-97556 
 
 3-09774 
 
 8-5960 
 
 2-93189 
 
 20 
 
 6-03084 
 
 3-12440 
 
 8-7619 
 
 2-96004 
 
 30 
 
 6-08710 
 
 3-15155 
 
 8-9322 
 
 2-98868 
 
 4 
 
 6-14436 
 
 3-17920 
 
 9-1073 
 
 3-01783 
 
 5 
 
 6-20266 
 
 3-20737 
 
 9-2872 
 
 3-04749 
 
42 TABLES OF PARABOLIC CURVES. 
 
 Angle 
 ABD 
 or 
 DBG. 
 
 Curve 
 A EC. 
 
 Tangent 
 ABorBC. 
 
 Radius at 
 Apex. 
 
 Half-Chord 
 A D or D C. 
 
 72 
 
 6-26202 
 
 3-23607 
 
 9-4721 
 
 3-07768 
 
 10 
 
 6-32247 
 
 3-26531 
 
 9-6623 
 
 3-10842 
 
 20 
 
 6-38405 
 
 3-29512 
 
 9-8578 
 
 3-13972 
 
 30 
 
 6-44678 
 
 3-32551 
 
 10-0590 
 
 3-17159 
 
 40 
 
 6-51069 
 
 3-35649 
 
 10-2660 
 
 3-20406 
 
 50 
 
 6-57583 
 
 3-38808 
 
 10-4791 
 
 3-23714 
 
 73 
 
 6-64223 
 
 3-42030 
 
 10-6985 
 
 3-27085 
 
 10 
 
 6-70993 
 
 3-45317 
 
 10-9244 
 
 3-30521 
 
 20 
 
 6-77896 
 
 3-48671 
 
 11-1571 
 
 3-34023 
 
 30 
 
 6-84937 
 
 3-52094 
 
 11-3970 
 
 3-37594 
 
 40 
 
 6-92119 
 
 3-55587 
 
 11-6442 
 
 3-41236 
 
 50 
 
 6-99447 
 
 3-59154 
 
 11-8991 
 
 3-44951 
 
 74 
 
 7-06926 
 
 3-62796 
 
 12-1620 
 
 3-48741 
 
 10 
 
 7-14561 
 
 3-66515 
 
 12-4333 
 
 3-52609 
 
 20 
 
 7-22356 
 
 3-70315 
 
 12-7133 
 
 3-56557 
 
 30 
 
 7-30317 
 
 3-74198 
 
 13-0024 
 
 3-60588 
 
 40 
 
 7-38449 
 
 3-78166 
 
 13-3010 
 
 3-64705 
 
 50 
 
 7-46757 
 
 3-82223 
 
 13-6094 
 
 3-68909 
 
TABLES OF PARABOLIC CURVES. 43 
 
 Angle 
 A13D 
 or 
 DEC. 
 
 Curve 
 A EC. 
 
 Tangent 
 A B or B C. 
 
 Radius at 
 Apex. 
 
 Half-Chord 
 A D or D C 
 
 75 
 
 7-55248 
 
 3-86370 
 
 13-9282 
 
 3-73205 
 
 10 
 
 7-63928 
 
 3-90612 
 
 14-2578 
 
 3-77595 
 
 20 
 
 7-72802 
 
 3-94952 
 
 14-5987 
 
 3-82083 
 
 30 
 
 7-81878 
 
 3-99393 
 
 14-9515 
 
 3-86671 
 
 4 
 
 7-91164 
 
 4-03938 
 
 15-3166 
 
 3-91364 
 
 5 
 
 8-00666 
 
 4-08591 
 
 15-6947 
 
 3-96165 
 
 76 
 
 8-10392 
 
 4-13357 
 
 16-0864 
 
 4-01078 
 
 10 
 
 8-20349 
 
 4-18238 
 
 16-4923 
 
 4-06107 
 
 20 
 
 8-30547 
 
 4-23239 
 
 16-9131 
 
 4-11256 
 
 30 
 
 8-40994 
 
 4-28366 
 
 17-3497 
 
 4-16530 
 
 4 
 
 8-51701 
 
 4-33621 
 
 17-8028 
 
 4-21933 
 
 So 
 
 8-62676 
 
 4-39012 
 
 18-2731 
 
 4-27471 
 
 77 
 
 8-73931, 
 
 4-44541 
 
 18-7617 
 
 4-33148 
 
 10 
 
 8-85475 
 
 4-50216 
 
 19-2694 
 
 4-38969 
 
 20 
 
 8-97320 
 
 4-56041 
 
 19-7973 
 
 4-44942 
 
 30 
 
 9-09478 
 
 4-62023 
 
 20-3465 
 
 4-51071 
 
 40 
 
 9-21963 
 
 4-68167 
 
 20-9181 
 
 4-57363 
 
 5 
 
 9-34786 
 
 4-74482 
 
 21-5133 
 
 4-63825 
 
44 
 
 TABLES OF PARABOLIC CURVES. 
 
 Angle 
 ABD 
 or 
 
 DbC. 
 
 Curve 
 A EC. 
 
 Tangent 
 A B or B C. 
 
 Radius at 
 Apex. 
 
 Half-Chord 
 AD or DC. 
 
 78 
 
 9-47964 
 
 4-80973 
 
 22-1335 
 
 4 70463 
 
 10 
 
 9-61511 
 
 4-87649 
 
 22-7802 
 
 4-77286 
 
 20 
 
 9-75441 
 
 4-94517 
 
 23-4547 
 
 4-84300 
 
 30 
 
 9-89772 
 
 5-01585 
 
 24-1588 
 
 4-91516 
 
 4 
 
 10-04522 
 
 5-08863 
 
 24-8941 
 
 4-98940 
 
 5 
 
 10-19709 
 
 5-16359 
 
 25-6627 
 
 5-06584 
 
 79 
 
 10-35354 
 
 5-24084 
 
 26-4664 
 
 5-14455 
 
 10 
 
 10-51477 
 
 5-32049 
 
 27-3076 
 
 5-22566 
 
 20 
 
 10-68101 
 
 5-40263 
 
 28-1884 
 
 5-30928 
 
 30 
 
 10-85250 
 
 5-48740 
 
 29-1116 
 
 5-39552 
 
 40 
 
 11-02949 
 
 5-57493 
 
 30-0798 
 
 5-48451 
 
 5 
 
 11-21225 
 
 5-66533 
 
 31-0961 
 
 5-57638 
 
 80 
 
 11-40107 
 
 5-75877 
 
 32-1634 
 
 5-67128 
 
 10 
 
 11-59626 
 
 5-85539 
 
 33-2856 
 
 5-76937 
 
 20 
 
 11-79814 
 
 5-95536 
 
 34-4663 
 
 5-87080 
 
 3 
 
 12-00707 
 
 6-05886 
 
 35-7098 
 
 5-97576 
 
 40 
 
 12-22343 
 
 6-16607 
 
 37-0204 
 
 6-08444 
 
 50 
 
 12-44763 
 
 6-27719 
 
 38-4032 
 
 6-19703 
 
TABLES OF PAEABOLIO CUEVES. 45 
 
 Angle 
 ABD 
 or 
 DBC. 
 
 Cur^e 
 AEC. 
 
 Tangent 
 ABorBC. 
 
 Radius at 
 Apex. 
 
 Half-Chord 
 A D or D C. 
 
 81 
 
 12-68010 
 
 6-39245 
 
 39-8635 
 
 6-31375 
 
 IO 
 
 12-92131 
 
 6-51208 
 
 41-4072 
 
 6-43484 
 
 20 
 
 13-17175 
 
 6-63633 
 
 43-0409 
 
 6-56055 
 
 30 
 
 13-43196 
 
 6-76547 
 
 44-7716 
 
 6-69116 
 
 40 
 
 13-70255 
 
 6-89979 
 
 46-6072 
 
 6-82694 
 
 50 
 
 13-98415 
 
 7-03962 
 
 48-5563 
 
 6-96823 
 
 82 
 
 14-27745 
 
 7-18530 
 
 50-6285 
 
 7-11537 
 
 10 
 
 14-58318 
 
 7-33719 
 
 52-8344 
 
 7-26873 
 
 20 
 
 14-90216 
 
 7-49571 
 
 55-1857 
 
 7-42871 
 
 30 
 
 15-23528 
 
 7-66130 
 
 57-6955 
 
 7-59575 
 
 40 
 
 15-58350 
 
 7-83443 
 
 60-3783 
 
 7-77035 
 
 5 
 
 15-94787 
 
 8-01564 
 
 63-2506 
 
 7-95302 
 
 83 
 
 16-32954 
 
 8-20551 
 
 66-3304 
 
 8-14435 
 
 IO 
 
 16-7298 
 
 8-4047 
 
 69-6383 
 
 8-3450 
 
 20 
 
 17-1500 
 
 8*6138 
 
 73-1974 
 
 8-5555 
 
 3 
 
 17-5917 
 
 8-8337 
 
 77-0338 
 
 8-7769 
 
 40 
 
 18-0566 
 
 9-0651 
 
 81-1770 
 
 9-0098 
 
 5 
 
 18-5466 
 
 9-3092 
 
 85-6606 
 
 9-2553 
 
46 TABLES OF PAR AS OLIO CURVES. 
 
 Angle 
 ABD 
 or 
 DEC. 
 
 Curve 
 
 AEC. 
 
 Tangent 
 A B or B C. 
 
 Radius at 
 Apex. 
 
 Half-Chord 
 A D or D C. 
 
 84 
 
 19-0638 
 
 9-5668 
 
 90-5231 
 
 9-5144 
 
 10 
 
 19-6104 
 
 9-8391 
 
 95-8083 
 
 9-7882 
 
 20 
 
 20-1891 
 
 10-1275 
 
 101-5667 
 
 10-0780 
 
 30 
 
 20-8029 
 
 10-4334 
 
 107-8565 
 
 10-3854 
 
 40 
 
 21-4549 
 
 10-7585 
 
 114-7450 
 
 10-7119 
 
 50 
 
 , 22-1490 
 
 11-1045 
 
 122-3110 
 
 11-0594 
 
 85 
 
 22-8892 
 
 11-4737 
 
 130-646 
 
 11-4301 
 
 10 
 
 23-6805 
 
 11-8684 
 
 139-858 
 
 11-8262 
 
 20 
 
 24-5282 
 
 12-2913 
 
 150-075 
 
 12-2505 
 
 30 
 
 25-4386 
 
 12-7455 
 
 161-448 
 
 12-7062 
 
 40 
 
 26-4190 
 
 13-2347 
 
 174-158 
 
 13-1969 
 
 5 
 
 27-4776 
 
 13-7631 
 
 188-423 
 
 13-7267 
 
 86 
 
 28-6246 
 
 14-3356 
 
 204-509 
 
 14-3007 
 
 10 
 
 29-8711 
 
 14-9579 
 
 222-738 
 
 14-9244 
 
 20 
 
 31-2309 
 
 15-6368 
 
 243-509 
 
 15-6048 
 
 30 
 
 32-7201 
 
 16-3804 
 
 267-318 
 
 16-3499 
 
 40 
 
 34-3582 
 
 17-1984 
 
 294-786 
 
 17-1693 
 
 5 
 
 36-1684 
 
 18-1026 
 
 326-705 
 
 18-0750 
 
TABLES OF PARABOLIC! CURVES. 47 
 
 Angle 
 ABD 
 or 
 DBC. 
 
 Curve 
 AEG. 
 
 Tangent 
 ABor BC. 
 
 Radius at 
 Apex. 
 
 Half-Chord 
 A D or D C. 
 
 87 
 
 38-1799 
 
 19-1073 
 
 364-090 
 
 19-0811 
 
 10 
 
 40-428 
 
 20-230 
 
 408-265 
 
 20-206 
 
 20 
 
 42-956 
 
 21-494 
 
 460-978 
 
 21-470 
 
 30 
 
 45-821 
 
 22-926 
 
 524-582 
 
 22-904 
 
 40 
 
 49-097 
 
 24-562 
 
 602-298 
 
 24-542 
 
 5 
 
 52-876 
 
 26-451 
 
 698-629 
 
 26-432 
 
 88 
 
 57-284 
 
 28-654 
 
 820-035 
 
 28-636 
 
 10 
 
 62-494 
 
 31-258 
 
 976-036 
 
 31-242 
 
 20 
 
 68-745 
 
 34-382 
 
 1181-14 
 
 34-368 
 
 3 
 
 76-386 
 
 38-202 
 
 1458-36 
 
 38-188 
 
 40 
 
 85 936 
 
 42-976 
 
 1845-91 
 
 42-964 
 
 50 
 
 98-215 
 
 49-114 
 
 2411-19 
 
 49-104 
 
 89 
 
 114-586 
 
 57-299 
 
 3282-14 
 
 57-290 
 
 10 
 
 137-505 
 
 68-757 
 
 4726-57 
 
 68-750 
 
 20 
 
 171-883 
 
 85-946 
 
 7385-65 
 
 5-940 
 
 30 
 
 229-181 
 
 114-593 
 
 13130-6 
 
 1 i4-589 
 
 4 
 
 343-774 
 
 171-888 
 
 29544-6 
 
 1 1-885 
 
 50 
 
 687-549 
 
 343-775 
 
 118180 ] 343-774: 
 
 90 
 
 Infinite 
 
 Infinite 
 
 Infinite 
 
 finite 
 
 UNIVERSITY 
 
LONDON: 
 
 PRINTED BY WILLIAM CLOWES AND SONS, LIMITED, STAMFORD STREET 
 AND CHARING CROSS. 
 
 v v ; 
 5 
 
YA