lSt9.] 



THE CIVIL EXGINEER ANP ARCHITECT'S JOURNAL, 



293 



for which the sinking must be taken into consideration, less alti- 

 tudes. 



The estimate of the quantity of discharge for greater altitudes 

 may be effected in the following manner : — When H signifies the 

 altitude of pressure above the upper edge of the orifice, and / the 



TT 



sides of the square, seek first for m = - . For this value of m, 



find in Table I. the corresponding value of the co-eflScient k', and 

 then the quantity of discharge niiiy be estimated by equations (g) 

 or (?) §. 11. For practical purposes equation (i) suffices; and the 

 altitude will be the whole perpendicular distance measured above 

 the orifice, and designated by H". 



When H'' = 1-2U8; Z=-2meter; 

 equation (i), 



logV(H" + JO = 



I 



= m = -eOSi; and by 



]ogP 



log ^/ (_ig) = 



log: k' = 



•05943 

 •60206—2 

 •64633 

 •78061 — 1 



logQ 

 or 122-580 liter. 

 1 22-659 liter. 



= •08843—1 or Q = •12258 cubic meter. 

 The experimental result (Tab. III. No. 4) gives 



Table II. 



Comparison of the Co-efficients for Greater Altitudes, with experiments 



with Quadrilateral Orifices. 



Compare the co-efficients here found by experiment by means of 

 the ratio m with those computed in Table I., and it will be found 

 that the greatest error does not exceed J per cent. 



(To be conlinued.J 



REVIS'WS. 



An Algebra of Batios, founded on simple and general Definitions; with 

 a Theory of Exponents extended to Incommensurable Ratios, and 

 the Propositions of the Fifth Book of Euclid easily and symbolically 

 deduced. By Henry B. Beownino, Architect, Stamford. Cam- 

 bridge: Macmillan, 1849; 8vo. pp. xiv + 133. 

 In the olden time many of our great architects were also pro- 

 found geometers: — yet, not many of oicr architects, but many of 

 the continental ones. Indeed, with the solitary exception of the 

 builder of St. Paul's, we look in vain for anything approaching to 

 even moderate acquaintance with the principles of geometry 

 amongst English architects. The "riile-of-thumb" is the univer- 

 sal guide; and mere taste is considered infinitely more valuable 

 than any amount of science, Far be it, however, from us to 

 undervalue taste: though much that is obtruded on the public as 

 the very testhetic of architectural taste, is only the wild arabesque 

 of a prurient imagination. We give honour where honour is due 

 for all developed conceptions that are worthy of the name of 

 "tasteful:" but we shall continue, as our pages will prove we have 

 hitlierto done, to censure with the utmost freedom those unculti- 

 vated vagrancies of fancy that are so often obtruded upon us as 

 the emanations of superior genius. 



Though many architects have been able goofiieters, th«r atten- 

 tion has been for the most part (indeed almost wholly) confined to 

 those forms of it that had a more or less direct bearing on archi- 

 tectural problems — that is, to "Descriptive Geometry," or to the 

 statical conditions which were essential to the safety of a struc- 

 ture. Few of them wandered into the regions of pure abstraction 

 —into the philosophy upoa which geometiical evidence is based. 



To do this bespeaks a still higher order of logical intellect than has 

 been evinced in the cases of De I'Orme, Jousse, Derande, Lan- 

 dau, Frezier, or even by the great Buonarotti, or the universal Da 

 Vinci. Uesargues, indeed, is the only marked exception — one of 

 tlie most marked exceptions even in mathematical history. He, 

 from wliom Pascal acknowledged that he learnt almost all he 

 knew, whom one of the most original writers of the age has called 

 "the Monge of his centurj'," and whose researches have in some 

 important matters anticipated discoveries of our own age, — such a 

 man is an honour to the profession, independently of any profes- 

 sional works he executed in his native city of Lyons. 



We confess that to find an architect publishing a work on such 

 an abstract subject as that of Euclid's fifth book, notwithstanding 

 the single precedent of Desargues, came upon us by surprise; and 

 from what we happen to know of the geometrical character of 

 English architectural writers and professors, it was not without 

 some misgivings that we opened it — fearing to encounter a heap of 

 crude conundrums, that would confer little honour on the class to 

 which Mr. Browning belongs. A slight glance, however, over its 

 pages, w ith a pause here and there, sufficed not only to remove our 

 apprehensions, but to convince us that it deserved a more deliberate 

 and systematic examination. Such an examination we have made; 

 and though we take exception to an important step of his investi- 

 gation, and remit the subject back to his consideration, we yet form 

 an exceedingly high opinion of the skill and address of several 

 detailed parts of his subsequent investigations. Let him remove 

 this one objection (fatal to his whole system as it at present stands), 

 and we shall gladly acknowledge that he has conferred a gieat 

 boon on mathematical science. As it is, he has made a step in 

 advance: but till his foundations are better laid, he has but an 

 insecure footing. 



As a great deal of misapprehension on the subject of proportion 

 exists, even in "high places," we have thought it desirable, for the 

 sake of clearing away a little of this, to enter into some account 

 of the real character and difficulties of the subject, at greater 

 length than we usually devote to a review. 



We have little doubt that the geometry of a succeeding and not 

 distant age will wear a different face from its present one ; and no 

 one branch of it will be more changed in its physiognomy than the 

 doctrine of proportion. The gioniHrie rccente of the French is 

 finding its way even into the higher class of our elementary books: 

 but any reference to this would be foreign to our present purpose. 

 We only purpose to point out what are the difficulties, and the 

 present degree of success attained in dealing with them, that 

 attach to the one specific subject — that of Ratio. 



All men of even common observation without pretension to sci- 

 ence, have a rude notion of proportion: but it is rude indeed with 

 the greater part of them, if we accept their language as an index 

 of their conceptions. We hear them talk of the "proportion 

 between two" things, or of "one thing to another;" or one thing 

 being "so many more times as large" as another. These are really 

 neither more or less than illiterate vulgarisms of phraseology; 

 whilst at the same time they point to an imperfectly expressed 

 conception of what constitutes ratio. Let them be slightly modi- 

 fied, and we get Euclid's definition {lib. v. def. 4) of ratio — viz. 

 "Ratio is the relation of one magnitude to another of the same 

 kind, with respect to quantuplicity" — that is, how often (how many 

 times, parts of a time, or times and parts of a time) one magni- 

 tude contains another of the same kind. This, we say, is the rude 

 idea; and were all magnitudes commensurable (that is, such that 

 every two of the same species had any finite common measure, 

 however small) it would be adequate to all the purposes of geome- 

 trical science : — as, for instance, did one magnitude contain the 

 other 10, 25^, lOOfg^, etc. times (as these have the common mea- 

 sure 1, ■^, Tffu, etc.\ the entire doctrine of proportion could be 

 established with respect to such magnitudes with simplicity and 

 facility, as will be shown presently. When, however, we come 

 (to take a familiar instance) to compare the side of a square 

 with its diagonal, the latter is /v2 times the former — that is, 



412136 figures ad infinitum ,. , „ xt . 



1 J- ^ .. •- times the former. Now, view 



lo''''- how we will, ^ cannot be a finite number; and hence all 

 10^" 



the reasonings which involve an expression of the relation infinite 



terms, must be nugatory in respect to such a ratio as that of the 



side of a square to its diagonal. This method, though the earliest 



and most obvious, is clearly an insufficient one on which to build a 



universal system of such relations. It nevertheless suggests an 



idea — and that is much. 



