Llandaff
Thursday evening. November 9

My dear Father.

I send you a neat proof of the theorem. That if you join the angles of a triangle to the points of bisection of the opposite sides there [sic] lines meet in a point; proved by the doctrine of limits.

[illustration]

In the first place I assume, that if a rectilineal figure becomes indefinitely small it may be considered as a mathematical point

Now let A B C be a triangle and D.E F the centres of its sides.

If D E. & .D F be joined

D.E C F is a parallelogram. & �� D C bisects the other diagonal E F. in H

Similarly, B F bisects D E in C

and, A E bisects D F in K.

�� the 3 lines from the angles to the centres of sides of the large triangle are reduced to 3 lines from the centres of sides to the angles angles of the small triangle

The same holds in the case of the still smaller triangle, G H K. & for each successive triangle which can be inscribed in the preceding.

�� it is true in the limit when the no of inscribed ∆ ^s is infinite but the ∆ ^s are then indefinitely small & become a mathematical point.

�� these 3 lines pass through one point.

Q. E D.

I examined Tintern Abbey ^<1> carefully and made out the arrangement without difficulty.

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Notes:

1. Monmouthshire, south Wales; founded in 1131