HYPOTHENCSE. 



HYPOTHKSI8. 



baring the supplemental angle* for their angle*, and r and r 1 forthe 

 id** of one, and and q' for the aide* of the other, are together equal 

 to the parallelogram under the hypothenuM* inclined at one angle 

 equal to the mm of those opposite to p and Q', or to r 1 and q. 



3. The next demonstration i derived from the Hindu treatise* on 

 algebra: not that it U actually found there, for the Hindu work* 

 demonstrate nothing ; but attached to the statement of the proposition, 

 in the margin of come copies, is the following diagram, which is no 

 doubt that belonging to the demonstration, which in a* follow* : Let 



' 



A c B be the triangle, and describe the square A B D E on the hypo- 

 thenuse. Draw D u perpendicular to A c, and E u perpendicular to D H, 

 and produce B c to meet EQ in r. Then the square ia made up of the 

 four equal triangles ACB, BPE, KOD, DBA, and of the smaller square 

 o r H, which is the square on B c, the difference of A and c B. But 

 the four triangles make up twice the rectangle of A c and c B, and twice 

 the rectangle on two lines, together with the square on their difference, 

 is the sum of their squares : whence the square on A B is the sum of 

 the squares on A c and c B. Judging by the general character of 

 Hindu mathematics, it must be supposed that their demonstration 

 was arithmetical, supposing the sides of the triangle to be represented 

 by numbers, and using the equation 



(a - i) s + 2 ai = o= + V. 



The following is the method of obtaining right-angled triangles, of 

 which the sides shall be whole numbers. Take any two whole numbers 

 whatsoever, x and y, of which s is the greater ; then if J 3 .v* and 

 2 xy be the two sides of a right-angled triangle, the hypothenuse is 

 a + y'. For instance, let *=11, y = 7; then *> y' = 72, 2 ary = 154, 

 and a* + |i l =l70: whence 72 and 154 being sides of a right-angled 

 triangle, its bypothenuse is 170. It is a remarkable property of any 

 three numbers which represent the sides of a right-angled triangle, 

 that one of them must be divisible by 5. 



4. The last demonstration which we shall give is one which shows 

 the property in question to be but one simple and prominent case of a 

 property of great beauty and generality, common to all triangles. 

 This property was first noted by Pappus, and it shows that any 

 parallelograms whatsoever being described upon the two sides of a 

 triangle, a third parallelogram, equal to their sum, can immediately be 

 drawn upon the third side. 



il 



Let A B o be a triangle, on two sides of which, A c and c B, let any 

 parallelograms A r o c and B c B K be described. Produce r o and K H 

 to meet in t, and join z c, and produce it to w. Through A and B 

 draw A o and B E parallel to x w, whence it follow* that A D z c and 

 c s IB are parallelogram*, and, by equality of bases and altitudes, 

 severally equal to A r o o and c B K B. And A D and E B are equal and 

 parallel to z c, and therefore to one another ; whence A D K B is 

 a parallelogram made up of the parallelogram* ADTW and BWVE, 

 which, by equality of base* and altitude*, are everally equal to 

 A D I c and B o z K, that U, to A r o c and o H K B. Hence the parallel- 

 ogram on the aid* ABU equal to the sum of those on the sides A o 

 and OB. 



Now let the triangle be right-angled at o, and let the parallelogram* 

 cm A o and o B be square*, and repeat the preceding construction. 

 Then OCBI is a rectangle, and ozc is in all respect* equal to A c B, 

 wbeno* SC = AB, and thenc* A D and B I are equal to AB, and the 

 parallelogram DBA* is equilateral. But the angle BAG U equal to 

 ecu, which 1* equal to OBA, the triangle* CSH and BAO being 



altogether equal. But c B A and c A B are together equal to a right 

 angle ; whence D A o and c A B are the *ame, or D A B is a right angle. 

 Consequently, A D IB is an equilateral parallelogram, right angled at 

 A, or it is a square ; and the parallelogram A D E B, that is, the squar 



\, 



on A B, is equal to the sum of the parallelograms A r a c and c u K B, 

 that is, to the sum of the squares on A c and c B. 



HYPOTHESIS (inro-etaa, sub-potitio, supposition), literally, the act 

 of placing one thing under another, that the latter may stand U'xvn 

 and be supported by the former ; by extension, the assignment of 

 cause or reason sufficient to make it a consequence that an event or 

 phenomenon must happen. For instance, the sun would disappear if 

 it were deprived of its power of giving light, and also if an opaque 

 body came between us and it : either of these circumstances would 

 cause what we term a total eclipse, and either is therefore sufficient, as 

 on hypothesis, to explain a total eclipse. 



In the article CAUSE (in natural philosophy) will be found the dis- 

 cussion of several considerations connected with the use of hypotheses; 

 and in the article ATTRACTION an instance of the important distinction 

 between an hypothesis asserted because it is true, and one assumed 

 because it is sufficient to explain observed phenomena. We suppose 

 these articles to be known to the reader. 



The following mode of argument is known in logic by the name of a 

 hypothetical syllogism : If A exist, z exists ; but A does exist, there- 

 fore z does exist. Or, establish the absolute truth of an hypothesis, 

 and the phenomena which necessarily follow may be asserted even 

 without experiment. But this we are seldom in a condition to do. 

 The preceding process cannot be converted : if A exist, let z necessarily 

 follow ; z has appeared, are we then entitled to say that A exists ? By 

 no means ; for when we prove that z necessarily follows from A, we do 

 not therefore show that z follows from nothing but A. But if we can 

 establish the following : If A exist, z follows ; if B exist, z follows ; if 

 c exist, z follows ; and z cannot happen in any other way : then from 

 the arrival of z we ore entitled to assume that one of the three, A, B, 

 or c, must necessarily exist, perhaps two, and perhaps all three. At 

 the some time, if the existence of the consequence con be denied, the 

 hypothesis is overthrown. If A exist, z follows ; but z does not happen ; 

 then it is perfectly certain that A doe* not exist. The following sum- 

 mary of the four cases may be more worthy of our readers' considera- 

 tion than many of them will suspect 



1. When A is B, T is z. 



But A is B. 



2. When A is B, T is z. 



But A is not B. 



Therefore T is z. 



Nothing can be concluded : T may be z 

 on some other grounds, or T may not be 

 z precisely because A is not B, or for some other reason. 



At* M *"**- 



4. When A is B, Y is z. j Nothing can be concluded : A may be B, 

 But T is z. j and either because T is z, or for some 



other reason ; and A may not be B, and there may be some 

 other reason why T should be z. 



The establishment of an hypothesis in natural philosophy may be 

 considered as a process of which the following are the heads : 



1. The phenomenon observed is z, and it is shown to be a necessary 

 consequence either of A, B, or c, which seem natural and probable : 

 also of D, E, &c., which seem altogether out of the question. 



2. All the necessary consequences which can be shown to follow 

 A, B, or c, are deduced as far as that can be donu ; and if all their con- 

 sequences really happen, then there is no choice between A, B, and c ; 

 but if z', a necessary consequence; say of c, should be found not to 

 happen, then c cannot exist, and the choice can only lie between 

 A and B. 



8. Let A appear the more probable of the two, then A is assumed to 

 be the cause of z until something to the contrary appears. If A ami 11 

 should be inconsistent with one another, then if one be assumed, it 



