The Sum of the propos'd Figures is 136; and 11434 iseach Rank (fuppofe) s: Then, by the Nature of the Question,

1 {a+b+c+d+e+f+g+b=a+e+i+n-\-b-\-ƒ-|-k+o

13

···•c+d-\-g-\-b=i+n+k+-0.

In like Manner a+b+e-\f=l-\-p+m-\-q [4 [al-b-\-c\d+n+o+p+q=d+sik+n\a\f+i+q · · · b + c -|- o + p = g+kiƒ+l.

15

16

But the Sum of the two Parts of the 5th Equation is =25; confequently either Part of it is

534

a-\-b-\-c+d+n+o+p+q=a+e+i+n+d+b+m+q

18 |···b-\-c-|-0-|-pe+i+k\m=s, per 6°.

19 |a|-b+c_l-d-\-n+o+p-\-q=b-\f+k\-o+c-|-g+1+B [10]• • •a+d-\-n+q=ƒ+k\-g+l=s, per 6°. [11]: a + d + n \-q=d-j-b-j-m-l-q

{12:a+n=b+m

In like Manner d--q=e+i 14 Alfo n+q=b+c

15 And a-l-dote

|16|c|i\-b-\-m—=i-|-k+l-\-m, per 8°.

|17|• :e4-b=k_|-|

18 Alfo c-\-p=f+k

19 Alfo b+0=g+l

20 And f+gi-|-m

21|f+k-\-g1l-l—a+f-l-l+q per 6o. |22|•••k-1-ga-1-q

23Likewife fl=d+n,

Having thus far proceeded, and the Queftion propos'd being (probably) capable of a great many different Solutions; 'tis to be prefumed that a may be equal to any of the given Numbers: Beginning therefore with the leaft of them, viz. 1, and putting a equal to it viz. 1; the next Thing to be done is to find the Value of another corner Letter as n. 22 cannot be equal to 2; for, if it was; then --m being

Среда

(per 12°.) = a+n (=1+2) fhould be 3; but there are no remaining two Numbers of the given ones whofe Sum is 3; confequently n cannot be = 2 (a being 1.)

=4:

Neither can be equal to 3: For fuppofing 23, then h+m being (per 12°.) = a+n2 (13) fhou'd be But there are no remaining two Numbers of the given ones whofe Sum is 4; therefore, &c.

But (for ought can be feen yet) n may be equal to 4; putting therefore 24; then bm (= a+n, per 120. =1+4)=5; that is h, m are equal to 2, 3 which are the only two Numbers remaining whofe Sum is 5; and therefore d+q=34—5—29=e+i (per 13); that is d, q are equal to 13, 16 or 14, 15, and accordingly e, i are equal to 14, 15 or 13, 16 only; for no other couple amounts

to 29.

Let us now fee what the Confequence is of putting q=13; then dis 16: And then the Square may be defigned, in part, thus

I

b

16

The four corner Letters being thus defign'd; and confequently e, i equal to 15, 14, as alfo h, m being equal to 2, 3, it is manifeft f cannot be equal to 5, 6 or 7; for, if it was, เ fhould be equal to 15, 14 or 13 which are Numbers already difpos'd off: "But (perhaps) ƒ may be, and therefore fuppofe it, 8, and then I will be 12.

Again g+k14; and there remain no two Numbers whofe Sum is 14, only 5 and 9: But k+1 or k+12 is likewife (per 17°) =e+b, viz. equal to 16 (2+14), 17 (2+ 15 or 3-1-14), or 18 (3+15); confequently k is equal to 4, 5 or 6 (Not equal to 9)); and therefore it (viz. k) is — 5. k+1(=5+12) being thus found 17 must likewise bee+b (per 170.), which may be effected two Ways; viz. by putting e15, and then b will be 2; or, putting e14, will be 3 Let us chofe the former; and then i will be 14, and m will be =3: And then the Square will be farther defignable, thus

It remains to difpofe of the four Numbers 6, 7, 10 and II instead of b, c, o and p, fo as b+c may be 17, as alfo op 17, which may be done by coupling 6, 11 as alfo 7, 10: But c+p must be (by 18°) k + ƒ= 13, which will be effected by 6 +7: From whence p being =6, c will be=7; and then o=11, and confequently 10: And then the Square will be fully compleated, thus

Or putting p7; then c=6, then o = 10, and b = 11: And then the Square will stand, thus

Note, Book II. begins Page 321. Signature * Y

A TREATISE

A

TREATISE

OF

ALGEBRA.

BOOK II.

Note, Befides the Characters explain'd in Book I. Pages 2, 3, and 4. thefe following are added; viz.

E

A

Circle is fuppos'd to be divided into 360 equal Parts, called Degrees; and each Degree into 60 equal Parts, called Minutes; and each Minute into 60 equal Parts, called Seconds; &c. Any Portion of whofe Circumference is called an Arch, and is measured by the Number of Degrees it

contains.

2. A Chord or Subtenfe is a ftraight Line, connecting the Extremities of an Arch; as

BE is the Chord of the Arches BAE, BDE.

3. A Sine (or Right-fine) is a ftraight Line.drawn from one End of an Arch perpendicular to that Diameter paffing thro' the other End; or it is half the Chord of twice the Arch; fo BF is the Sine of the Arcs BA, BD. And here it is evident, that the Sine of 90 Degrees (which is equal to the Radius or Semi-Diameter of the Circle) is the greatest of all Sines, the Sine of an Arc greater than a Quadrant being lefs than the Radius.

4. The Difference of an Arc from a Quadrant, whether it be greater or lefs, is called its Complement; fo HB is the Complement of the Arcs BA, BD; BI is the Sine of that Complement, aud therefore it is called the Co-fine, or SineComplement af the Arcs BA, BD.

5. The Secant of an Arc is a ftraight Line drawn from the Center thro' one End of the Arc till it meet with the Tangent, which is a ftraight Line touching the Circle at the Extremity of that Diameter which cuts the other End of the Arc; fo CG is the Secant, and AG the Tangent of the Arcs BA, BD: And CK is the Co-fecant, and HK the Co-tangent of the faid Arcs.

6. A