[[[[[
Problem. Prop^n: 18. To describe a figure. similar &
similarly situate to another figure.
Resolve the figure into Triangles. ACH. AHB.
Make the polygon CEFD = angular. to it by 22.1.
Therefore because the Triangles are = angular
AB
:BH::CD:DF.
& AG:GH::CE:EF likewise AG:AH::CE:CF & AH:
AB
::CF:CD.
whence by equality AG:
AB
::EF:CB. & after the same manner
HB:GH::DF:FE. Therefore AGHB is alike & like situate to CEFD.
[Standard Euclidean diagrams, though usually polygons composed of a greater number
of triangles are represented]
------
Prop^n: 19. Theor: Like Triangles ABC; DEF are in Duplicate Ratio
of their Homologous sides BC. EF
Let there be made BC:EF::EF:BG. & let
AG be drawn Then Triangle ABG = EDF. by prop: 15.
But \triangle ABC.ABG::BC:BG & \frac{BC}{BG} is twice \frac{BC}{EF.} 10 D. 5.
Therefore \frac{ABC}{ABG} = twice \frac{BC}{EF} QDE
[Standard Euclidean diagrams, same lettering as Simson]
------
20 Similar polygons. ABCD. FGHK may be divided into the same
N^o of similar Triangles having the same Ratio to one the other
which the polygons have. & the polygons have to one the other
the Duplicate Ratio of their Homologous sides---AB.FG.
Join AD FK. & because of similar: Def 1.6.
AB
:BD::FG:GK
& \angle^e Bis = therefore the Triangles ABD. FGK are similar E E [E E: Each to
each]
For the same Reason \angle^s ACD. FHK are similar---
And Because \triangle ABD is similar to FGK ABD has to FGK the Duplicate
Rat: that AD has to BK & \triangle ACD has to FKH the duplicate Ratio of AD. FK [FK
is blurred on the photograph, I can only infer them due to what makes mathematical
sense]
Therefore as \triangle ABD to FGK::ACD:FHK::2AD:FK::2AB:FG. [Note: I believe
those are 2's here, as this is what makes mathematical sense, but as always it is
ambiguous to the similarity between Davy's 2 and Q (or square symbol)]
Then as 2 ABD + ACD \sqrt{FGH + FHK}::2.
AB
:FG. [\sqrt: radical sign. Davy is mixing notations from various traditions
here.]
That is CB has to GH. the Duplicate Ratio that
AB
has to FG.
[Standard Euclidean diagrams, though usually polygons composed of a greater number
of triangles are represented]
------
Prop^n: 21. Rectilineal Figures A. B. which are similar to
the same Rectilineal Fig. C
Because A is similar to C it is equiangular to it
& because B is similar to C it is equiangular to it wherefore
A & B are also = angular & have the sides about = \angle^s
similar to C ie Homologous. therefore A & B are similar 11.5 .1 Def. 6
[Standard Euclidean diagrams, identical to Simson's]
------
]]]]]
✕
AnnaBeddoes
Anna Beddoes, née Edgeworth, the wife of Thomas Beddoes who befriended Davy when he
arrived in Bristol. Critics have speculated that they had an affair because Davy wrote
poems to Anna and she wrote poems for him that he copied out in his notebooks.