us 



THE CIVIL ENGINEKR AND ARCHITECT'S JOURNAL. 



[April, 



liiiep, fl = CO feet = 8640 lines, r = 318 feet =: 45792 lines, and .r 

 := 3(Ki, and d ^ 1441, giving 2| inches for the diminution of the 

 Adige. 



4tli. At the Fossa Bellina which is the lowest of the derivatives 

 with respect to the sea, it was found that z = 10.11.8= 1580 lines, 

 h = 4.4.2 = 626 lines, a = 60 feet = 8640 lines, c = 258 =37512 

 lines. Whence ,r= 301 and d= V ("^ -•»'.») = 1531, which 

 suhtracted from 1580, leaves 29 lines for the diminution of the 

 Adige, that is 2 inches and 5 lines. 



5th. But at Castagnaro, which is the first and farthest from the 

 sea of all, it was found that i-= 14.2.10 = 2050 lines, h= 1491 

 lines, ff = 35064 lines, e = 95040 lines ; dimensions taken above the 

 two falls on each side of the Cunetta, wliicli remains in the middle, 

 the result of which calculated separately will be .r — 250 lines and 

 d — 1816 lines, a sum which diminished by 2050 lines, leaves 234 

 lines, or 1.7.6, for the diminution of the Adige at tlie flood time, 

 by reason of the diversion, which tlie two falls are able to produce 

 on each side of the Cunetta. Calculating then the diminution of 

 this, we have z = 2050 lines, b = 2127 lines, a — 3816 lines, c = 

 95040 lines, as above, whence d — nearly 2000 lines, whicli, sub- 

 tracted from 2050, leaves 50 lines, or 4 inches 2 lines, wherefore 

 we have for the whole diminution of the Castagnaro 1.11.8, or 

 within 4 lines of 2 feet. 



XV. Scholium 2. The celebrated Abbate Guido Grandi, mathe- 

 matician of the Grand Duke of Tuscany, in his treatise on the 

 motion of water, professes "that if two horizontal rivers, LG, FG, 

 moved with a velocity, I G, G K, be united in one trunk, whose 

 velocity and direction will be G H ; and, on the other hand, sup- 

 ])Osing the said trunk H G, has the \elocity H G, it ought, with 

 the retrograde motion to divide itself into two branches, G L, G F, 

 they will not regain the velocity, I G, KG, equal to the first, unless 

 the angle, LG F, be a right angle," the which being different from 

 what we liave before established, we are obliged to e.\amine, 

 according to our power, the foundation on which the aforesaid pro- 

 position rests. Grandi resolves the total velocity, G H, which 

 arises from the two, G K, G I, by means of the complement of the 

 parellelogi-am with the two lines expressing the force, H E, G E, 

 of which H E is the perpendicular let fall on G K produced ; but 

 if conversely, says he, the trunk, H G, he resolved into branches, 

 whose velocity shall not be the same as on entering the trunk, it 

 may be greater or less, and will only be equal in the case when the 

 angle, L G F, is a right one. Tlie direction of the velocity, G H, 

 resulting from the conjunction of the two laterals, G I, G K, is 

 exactly what all staticians have laid down. To have a clear proof: 

 on the line GH raise the perpendiculars, K S, Irp, and the velocity, 

 G K will be obtained, resulting actually from the two G 5, sK, and 

 the velocity G I, in the two others, G <p, <p I, of which K 6, ^ I, 

 nowise contribute to the progressive motion, but only G 5, G t, 

 tlien G 8 -|- G ^ are equal to G H, as is more easy to demonstrate ; 

 then each quantity denotes really the velocity with which the 

 water in the trunk moves after receiving the influents, and it is to 

 be noted that the prevalence of one perpendicular K B, above the 

 other I (p will only oblige the brancli to bend a little from its 

 course. Wherefore the illustrious author then considers the 

 converse of the proposition, tliat is, wlien the trunk passes into 

 the branches, to resolve the velocity, H G, into two, H E, E G, and 

 .says, tliat in G F the water will run with the velocity GE, greater 

 by the acute angle than G K, the which will be true, whatever bend 

 ajid through whatever arm,GE, all the water of the trunk may flow, 

 whilst H G does not express all the velocity, the same quantity not 

 going through G E, which did when G F was considered as an 

 influent, it results that H G ought to resolve itself in another 

 .shape than tliat which is tlie case, that is, considering G (p by the 

 velocity G I, and G 5 by the velocity G K, wlience the original 

 velocity, G K, G I, in the two canals respectively, will be precisely 

 restored, now reputed as difl'erent branches, G F, (« L ; whence the 

 conversion of the influents into diffluents will not change the 

 velocity ; in either case it will be retained, provided it be not 

 ■changed by any external circumstance. 



■XVI. Sclwlhim 3. — I think it would not be superfluous to give 

 an example of the increase of height vvliicli a ri\er really acquires 

 from the reception of another. We will supjiose the velocity a 

 mean proportional of the height, using the preceding formula 

 z — ^V (<'' + 2rfj7 j/ rf.r -)- a^). The average deptli of the section 

 of the recipient = 3962 lines = d, its breadth 115200 lines = c. 

 The true section of the recipient is figured, in which A and B 

 -denote the profile of the hanks, C the bottom, D E the surface of 

 the water, P F the average depth ; the next figure is the section of 

 the influent in which the shoal E H appears much more elevated 

 tlian the bottom I and B M S the surface. The better to adapt it 

 40 jH-aotice and calculation, I shall divide ttie section into several 



jiarts, reducing them one by one to the section of the recipient, 

 wliich then added together, gives tlie amount of increase. In the 



section of the influent, D E H I L N RT, DE denotes the right 

 bank, R T V the left, B H the bottom of the shoal at the toe of 

 the riglit bank, L N 11 the bottom on the left bank, and H I L the 

 bottom of the influent. The portion B F E must be considered of 

 the mean height 3.0.4, that is taking half E F by reason of the 

 triangle BFE or B A E, the base BF is 11 feet, or 1584 line.s, 

 wherefore performing the necessary operation, we shall have z 

 — 3963 lines, from which subti-acting 3962 lines, the average height 

 of the section, there remains one line for the increase of that 

 portion B F E. Likewise through the portion F G H E, 17 feet 

 wide, and 6.0.9 feet high = 873 lines, we shall have z — 3968 lines, 

 from which subtracting 3962 lines, there remains 6 lines for the 

 increase of the recipient in height by reason of the aforesaid ad- 



dition. G H I L M will have a mean height of 13.5.3 = 1935 

 lines, and a width of 126 feet= 18144 lines, whence .c= 4102 lines, 

 and this third increment will be 11 inches 8 lines. M L NO formed 

 by the left lower shoal will have a mean height of 1333 lines and 

 100 feet 14400 lines, whence z will be 4026 lines, and the heiglit 

 required for the increase caused by its addition 5^ inches. The 

 shoal O N S R is 26 feet = 3744 lines wide, and the mean depth 

 3 feet 6 inclies 3 lines = 507 lines, and z = 3966, giving 4 lines 

 tor the increase. Finally, the portion comprising the escapement 

 of the bank may be considered 8 feet wide, and 1.9.1. Its reduced 

 height not giving any sensible increase, collecting together all the 

 aforesaid measures, we shall have the total increase of 1 foot 

 5 inches 1 1 lines. 



XVII. Scholium 4. — According to what is registered in the visi- 

 tation of tlie Po and Reno made in 1693, by Cardinal d'Adda and 

 Barberini, to calculate the increase produced in the Po by tlie 

 addition of the Reno, it will be necessary only to use tlie ]>receding 

 formula, as likewise to find the same effect at the general visitation 

 of 1720. Taking the data of 1693 aforesaid, supposing the 

 a\erage height of the Po without the Reno at Lagoscuro 31 feet 

 = 372 inches, the height of the Reno at the pass of Annegati, 

 that is 6 = 9 feet = 108 inches ; the width of the Reno there 189 = 

 a = 2261 inches; the width of the Po at Lagoscuro 760 feet = c 

 = 9120 inches, where ,r = 3 feet 6 inches, rf^ = 51478848 ; 2 dj; \/ 

 d,r = 3906000 and ar' =; 74088 numbers, which added together make 

 55458936, whose logarithm is 7-7439015, which divided by 3, to 

 obtain the cube root, gives log. 2-5813005, the number to wliich is 

 381 yVsli ; «'"'' since the fraction answers to 4 lines, if 372 is sub- 

 treated from 381.4, there remains 9 inches 4 lines for the increase 

 required according to the aforesaid supposition. 



XVIII. Scholium 5. — In a report presented by Guglielmini at the 

 time of the visitation, and which was registered in the Acts of it, 

 and printed in the Florence collection, in wliich he calculates the 

 rise at 8 inches 9 p. only, but the difference between us arises from 

 his having taken the nearest numbers neglecting fractions. Eustace 

 Manfredi, in answer to Giovanni Ceva, says in reply to the other 

 proposition, " To say truly we shall find that the 9^ inches found 

 by Ceva, is one inch more than what results from the former cal- 

 culation of Guglielmini, and that by a small error of a fraction," 

 &c. [See JManfredi's notes to Guglielmini's book on the nature of 

 rivers.] 



XIX. SclioUum 6. — In all the above examples we have supposed 

 for the calculation of the velocity that it is either a direct or mean 

 proportional to the height of the water, and that so as not to 

 difter from what has been laid down frequently by many renowned 

 authors ; and also to give a proof of the manner of employing the 

 formula we have discovered, when greater precision is required, 

 the velocity must be found by an instrument (the hydraulic pen- 



