70 
news to tell you, which I should much rejoice at, were it 
not for one consideration, which is, that I am afraid it will 
in some measure prevent your exaltation to honour. Dr. 
Simson has restored Euclid's Porisms. They are printed off, 
and will shortly be published. This intelligence I have 
from the best authority, viz., the Earl of Stanhope, at whose 
expense they are printed.” Mr. Wildbore replied to this, 
“ August 18th, 1775,” and his letter contains the following 
passages : — “I much rejoice to hear that you have made an 
acquaintance with that learned and judicious nobleman, the 
Earl of Stanhope ; and his communicating to you the contents 
of Dr. Simson’s posthumous works is the reason of my being 
in such haste to answer your last agreeable favour. In 
contemplating what these Porisms of Euclid might be, and 
to what uses they were applied, I have hit upon something 
of which I have not met with the least hint in any author 
What I mean is this : that the Porisms quoted by Dio- 
phantus, and the other remarkable Propositions , assumed 
without proof by that author, which do so much honour to 
Antiquity, are no other than the Porisms of Euclid, and 
that this kind of Porisms composed Euclid’s seco7id Book 
But, as this may. require something more than bare assertion, 
I shall here show the method by which some of the most 
remarkable flow from the two Lemmata* which are the 
148th and 149th Propositions of the seventh Book of Pappus* 
From the first-named Lemma the following is evident. In 
any right-angled triangle the double rectangle contained 
under the sum of the hypothenuse and greater leg, and the 
difference of the hypothenuse and cathetus [the perpen- 
dicular] is equal to the square of the sum of the said 
difference and the greater leg ” He then deduces “the 
method which Diophantus makes use of in the 22nd Propo- 
* 1. If upon a right line AB there be taken two points C, D, such that 
2AB‘CD:ziCB 2 ; then AD 2 rrAC 2 -pBD 2 .— (Pappus, prop. 148.) 
2. IfBA*BC=BD 2 ; then 0), (AD+DC)BD=DA- DC ; 
(ft), (AD+DC)CB=rDC 2 ; (c), (AD+DC)AB=AD 2 . 
(Pappus, prop. 149.) 
