- DIAGONAL 
pofe, or as any machine executed by human {kill could be 
expected to perform with accuracy. 
The engine is wrought exatly as that of the clock-maker ; 
excepting that the operator, between every cutting, fhifts the 
fituation of the cutter, by turning the {crol] the number of 
‘times pert ed. 
To conftru& wheels for orodeang diagonal retilinear 
motion i this engine, and the table already given 
Ifa diagonal line, of one nen inlength, and forming an 
angle of 24° with the bafe or horizontal line be pair, by 
‘one-half revolution of the two whcels; an t the fame 
line fhould be again d-fcribed in a contrary dire&tion, by the 
revolution of the other half. 
As every fhift is eae to be .3,d part of an inch, the 
femi-circumference of the wheel mutt be divided into 32 
€qual parts, or the aoe into 64, and radi eapile from the 
circumference towards the centre. hen, to cut the hori- 
when 
k 
Bafe at the top, 16 
I, cut it as near to the circumference a 
to 2, and turn the fcroll 164 turns ; to yale the cutter fo 
a 4 will be on the fame circle as I. 
o cut the whee ves the vertical motion.— 
Obferve, after the wheel i is “divided, as before, into 64 cqual 
parts, that the di ivifion numbered 1 mut be at right angles to 
that upon the horizont re becaufe the point of motion is 
ainder the centre, (fs. and 15. Plate V.) and that of the 
cutter on alevel with it. eat confulting the table, again ft 
24°, in the ea marked Perp. is 7.3. Cut as before the firft 
divifion, as near to the rim as convenient, and advance the 
cutter 7,3, turns at ack divifion to 32- Bring it back the 
saa number of joie to 645 and the wheels will be ready for 
o the body to be 
common trigonometry. 
Of Curvilinear Motion. 
The mere produ@tion of diagonal flraight lines would be 
too limited for the ornamental varieties to which machinery of 
i fe) 
feems fig to Waly his affertion. 
this eae eee the deviation of the line deferibed from one 
angle t be made fo frequent and minute, as to 
deferibe ce Gale the ellipfe, the parabola, or any other 
curve, fa or irregular, with a very great degree of accu- 
racy. I cribing circles, or circular curves, the plan 
adopted wae to confider them as in{cribed polygons, ofa num- 
ber of fides Gropotisedte to the circumference of the circle 
“ 
al red. The a ae Reel ses fs 7 PlaeV. I peeia 
only containing 9g fides in the quadrant, or 36 in whole 
circle, does not uae very far: and ce each of thefe {paces 
in the circumference been again divided into 10, or even into 
5 equal parts, the deviation of the polygon from the 
circle would have been totally imperceptible to any human 
eye. 
To affif in calculating the circles a formed the curved 
lines, it was neceflary to afce:tain the diameters, from whic 
ce pds of the circumferences Sone found by the com- 
f cana in the 
principle, and as the meafure w aétion 
was made the flandard of unity, and all the fubfequent calcu- 
lations made in thirty-fecond par 
Rules.—1. Diside 90°, peau - meafure of the quadrant 
of a circle, by the number of fides of a polygon oo in 
that quadrant, or by the fourth part of the fides which c 
pleat the whole inferibed polygon. a oes will ae 
the — of the acute angle at the.ce 
acute angle being found by ie preceding tule, is to 
be fubt raGied from 180°, and the remainder, divided by . 
un 
. Asail the triangles are flan the angles 0 of all at oe 
ci concicte ce are equal. An e twenty-ninth 
pofition of the firft book of Fuclid’s Elements, ae 
oO fI the wu of the above rules, when go° is 
We — 9s (ike number of polygonic fides in the quadrant) 
e quotient gives 10 the fecond rule 180° — 10° 
uh and 170 . The meafure of the angle 
A B1 is therefore 85°. Again, the angle, BA 2, is 20%, 
and the angle, A 1 2, is equal to the angle A B1, ao con- 
o- 2= 
fequently 85°; but the angles, A Br, and Ara, .bei cing 
alternate angles, are equal to each other. Gonicaceniy 85° 
— 10 °, the meafure of the angle a12. In the fame 
way allt the other angles will be found, and when found, the 
table already given will fhew the turns of the fcroll required 
to cut the bafe and perpendicular lines neceflary to produce 
a diagonal or hypothenufe line equal to 18, viz. bale 1.5, 
and perpendicular 17.9. a different meafure is wanted, it 
may be got by common oe For, if the hypothe. 
nufe is to be 15, then as 1 15 3: 1.5 : 1.25, for the 
Pat = asi8: 15 :: 17.9 : 14.805, for the oe 
dicu 
The following table gives the diameters and circumferences 
of a feries of circles reprefente by polygons, from a dia- 
meter of .9,d of aninch, to one of $4d, or two inches. The 
firft column contains the meafure of the diameter; the fecond, 
that of the circumference; the third, the ohe le umber 
of fides of an infcribed p olygon ; ; and the fourth, the nume 
ber of fides contained i in a quadrant of the circle. 
6 ABLE 
