805 



HYPOTHECATION; 



HYPOTHENUSE. 



806 



phraseology of this system are drawn directly from the Roman juris- 

 prudence; 4thly, that the Scotch right of hypothec is not to be deemed 

 a right of property, but a right arising from a tacit contract, inseparable 

 from the contract of lease ; Sthly, that this right is purely legal, and 

 cannot be created by convention, the only conventional hypothecs 

 known to the law of Scotland being bottomry and respondentia ; and, 

 lastly, that the rules relative to hypothec have arisen from the common 

 law without the intervention of statute. Another right of a different 

 description is called a hypothec in Scotland namely, the right of a 

 law agent to take his client's decree for expenses, or judgment for costs, 

 in his own name, in order that he may recover payment of his account 

 as taxed by the auditor of court. This right cannot be defeated by a 

 collusive settlement. In Scotland, a law agent, whether employed to 

 conduct a litigation or in other professional business, such as con- 

 veyancing, is entitled to retain his employer's title-deeds and papers 

 until his just account is paid. This right has also been called a 

 hypothec ; but it is clearly a lien. For further information on this 

 part of the Scotch law of hypothec, the reader is referred to Bell's 

 ' Commentaries,' vol. i. 



In France there is a distinction between privileges and hypotheques. 

 All tacit hypothecs, according to the division above kept in view, are 

 included under the former, which are subdivided into a general prefer- 

 ence over all the moveables or personal property in the debtor's 

 possession, and limited preferences over particular articles of property 

 for particular obligations. This last named, in so far as it affects 

 moveable property, is the class of rights which has been spoken of 

 above as tacit hypothec, and it includes the landlord's security for his 

 rent. There is also a classification of privileges sur les immeubles, con- 

 listing of tacit preferences over what is in England called real property, 

 and of privileges which extend to both moveable and immoveable pro- 

 perty. The term hypotheque is applied to conventional securities over 

 immoveable or landed property, and is the object of much useful legis- 

 lation ; such securities being, from the efforts to give virtual effect to 

 the law for partition of successions, without reducing them below the 

 proper extent for agricultural operations, more common in France than 

 perhaps in any other part of the world. See on the matter of the 

 immediately preceding remarks, ' Code Civil,' h'b. iji., tit. 18 j and 

 Troplong, Droit Civil explique', ' Privileges et Hypotheques.' 



HYPOTHECATION. [MORTGAGE.] 



HYPOTHENUSE or HYPOTENUSE (fao-rwovaa, subtending), 

 is a term which has always been applied since the time of Euclid to 

 the side of a right-angled triangle which subtends, or is opposite to, 

 the right angle. 



The property of the hypothenuse of a right-angled triangle being 

 one of the most important elementary propositions in the whole of 

 mathematics, it will be worth while to devote some space to its con- 

 sideration. We shall proceed to give some demonstrations, derived 

 from different principle*. 



The property in question, in a limited form, is this : that the tqttare 

 on the hypothenuse is equal to the sum of the trjuarct on the sides. 

 The introduction of the square, however, in preference to any other 

 figure, arises from the fact of the property of the hypothenuse with 

 respect to the square being demonstrated before that with respect to 

 ny other figure. The general proposition is this: if three similar 

 figures (that is, figures of the game shape, differing only in size) be 

 described upon the three sides of a right-angled triangle, the content of 

 that which is described upon the hypothenuse will be equal to the 

 Bum of the contents of the figures described upon the sides. Thus, all 



D B 



semicircles being similar figures, let AXCB, ATC, and CZB, be the 

 semicircles described on the hypothenuse and sides of the triangle 

 ACS, right angled at c : then A Y and c Z B are together equal to 

 A X B. Hence was obtained the first instance in which a curvilinear 

 pace was reduced to an equivalent rectilinear one. Take away the 

 segments A X c and c v B from both sides of the preceding equation, 

 and th# remainders of the smaller semicircles, namely, the lunulct YX 

 and z v, are together equal to the remainder of the larger one, namely, 

 the triangle A C B. This proposition is attributed to Hippocrates. 

 [GEOMETRY.] 



80 soon, however, as the proposition is demonstrated with respect 

 to squares, all the rest follows easily, after the doctrine of proportion 

 has been established. It is the property of similar figures described on 

 two lines to be in the same proportion as the squares on those lines ; 

 if then the squarej on two lines be together equal to that on a third, 



then any two similar figures described on the first two linea are 

 together equal to the corresponding figure described on the third. 



We shall now sketch four different demonstrations of this funda- 

 mental proposition, desiring it to be remembered that we suppose the 

 reader to have already become acquainted with it in an elementary 

 course of geometry. 



1 . Let c D (in the preceding figure) be drawn perpendicular to A B. 

 Suppose that (after the manner of some writers on geometry) the 

 theory of proportion and of similar triangles is established before any- 

 thing is proved relatively to the areas of figures. Then it is easily 

 shown that A c D and CDS are triangles similar to one another, and 

 to the whole ACB. Now in such a system of geometry, it can 

 easily be shown, without the aid of our theorem, that any two similar 

 figures, described on two straight lines, are to one another in the pro- 

 portion of the squares on those lines. Consequently, A c B, A D c, B D c, 

 being similar triangles described on A B, A c, B c, are to one another 

 as the squares on A B, A c, B c. But the first triangle is evidently equal 

 to the sum of the other two : consequently, the square on A B ia equal 

 to the sum of the squares on A c and c B. This demonstration may be 

 objectionable in a geometrical point of view, but it contains one of the 

 most useful modes of illustrating the proposition to a person unac- 

 quainted with geometry. Let such a one be made to remark the very 

 visible fact, that two similar figures described on two straight lines are 

 always of the same relative magnitude, each to the square described on 

 the same line : he will then, seeing that the right-angled triangle is 

 made up of two right-angled triangles similar to itself, each having 

 one of the sides for its hypothenuse, be able to see that the square 

 on the hypothenuse is equal to the sum of the squares on the sides. 



2. The next method shall be ocular demonstration, made by cutting 

 the square on the hypothenuse into the squares on the sides. Let A B 



be the triangle, right-angled at ; and on A B describe the square A D E B, 

 and on A c and B the squares A K a c and c H K B. From K draw E L per- 

 pendicular to B H, and from D draw D Q perpendicular to A 0. It is easily 

 proved that the triangles A c B, B L E, D <J A, are equal in all respects ; 

 whence (1) the line F o must pass through D, since D Q = A ; (2) E L = 

 BO=BK. Hence, by the parallels, the triangles NKB and MEL are 

 altogether equal, so that EM = BN, whence MD = NA, and, by the 

 parallels, DOM and A H N are altogether equal. And A F D is in all 

 respects equal to B L E. Out of the square A D E B take B L E, and 

 remove it to A F D ; remove H L E to NKB, and A H N to D a H. Then 

 the square A D B will be formed into the two squares A F o o and 

 HCBK. 



In a paper by Professor Kelland, ' On Superposition ' (' Edinb. Trans.' 

 vol. xxi. p. 2) are tvxlve different ways of dissecting a square into two 

 squares, one of which shall be three times the other. This paper 

 suggested to Mr. Airy, the Astronomer Royal, the following very 

 simple mode of making any two squares into one ; from which a demon- 

 stration of i. 47, may be derived'of a more elementary character than 

 that given by Euclid himself. Place the squares A o, K o, side by side, 

 as in the preceding diagram ; and make a L equal to B c, as before. 

 Join F, L and L, K, then F L, L K are equal and perpendicular to each 

 other. Turn the triangles F OL, LB K, round the points F, K until F O 

 coincides with F A, and K B with K H. The two squares will thus be 

 turned into the square on F L or L K. The same tiling may be done by 

 translation of the triangles, without rotation. It is rather singular 

 that, many as have been the eyes which have rested on the preceding 

 diagram, no one should ever have made it yield the dissection just 

 given. 



Each mode of demonstrating a geometrical proposition usually 

 belongs to its own particular generalisation of that proposition. The 

 one last before us suggests the following generalisation : If any two 

 triangles have a pair of supplemental angles, and if the sides opposite 

 to those angles be called hypothenuses, and if r, Q, be the remaining 

 sides of one triangle, and P , <j', of the other, then two parallelograms 



