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diameter AB is equivalent to the squares of the two segments AI and BH. For FD : AD :: AD: EG, whence (V. 6.) FD.EG AD, or 2FD.EG = 2AD2; but (IV. 15. cor.) 2AD AB or DE, and consequently 2FD.EG=DE'; wherefore (VI. 2.) 2AH.IB HI, and hence, by the first additional proposition to Book II., the segments AI, BH are the sides of a right-angled triangle, of which AB is the hypotenuse, or AB AI2+BH2.

PROP. IV. THEOR.

A chord of a circle is divided in continued proportion, by straight lines inflected to any point in the opposite circumference from the extremities of a parallel tangent, which is limited by another tangent applied at the origin of the chord.

Let AB, AC be two tangents applied to a circle, CD a chord drawn parallel to AB, and AE, BE straight lines inflected to a point E in the opposite circumference; then will the chord CD be cut in continued proportion at the points F and G, or CF: CG:: CG: CD,

For join BD, BC, and CE. Because the tangent AB is equal to AC (III. 22. cor.),

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CBG and CBE are similar, for they have a common angle CBE, and the angle BCG or BCD is equal to BDC or BEC (III. 16.): Wherefore BG: BC:: BC: BE, and BC2=BG.BE. Hence AB.CD=BG.BE, and AB: BE:: BG: CD ; but FG being parallel to AB, AB BE :: FG: GE (VI. 2.), and

consequently FG: GE:: BG: CD; therefore (V. 6.) FG.CD= BG.GE; and since (III. 26.) BG.GE=CG.GD, it follows that CG.GD=FG.CD, and FG: CG:: GD: CD, and hence (V. 10.) CF : CG : : CG ; CD,

PROP. V. THEOR.

If, from the vertex of a triangle, two straight lines be drawn, making equal angles with the sides and cutting the base; the squares of the sides are proportional to the rectangles under the adjacent segments of the base.

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Because the angles DBF and EBG are equal, they stand

(III. 16. cor.) on equal arcs DF and EG, and consequently (III. 18. cor.) FG is parallel to DE. Whence (VI. 1.) AB: BC:: AF: CG, and therefore (V. 13.) AB2: BC2 ;; AB.AF:BC.CG ; but (III. 26.) AB.AFDA.AE, and BC.CG EC.CD. Wherefore AB2: CD2 :: DA:AE: EC.CD.

B

D

E

If the triangle ABC be right-angled at C, and the vertical

lines BD and BE cut the base internally; then BC2+AC.CE: BC2 : ; AE:CD. For make AH equal to EC. Because AB2: BC:: DA.AE: EC.CD, and (II. 10.) AB2=AC12+BC2, therefore AC+ BC2: BC: DA.AE: EC.CD, and, by division, AC2: BC2:: DA.DELEC.CD:

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EC.CD. But, by successive decomposition, DA.AE-EC.CD= DA.AC-DA.EC-EC.CD-DA.AC-EC.AC-AC.HD; whence AC BC2:: AC. HD: EC.CD, and (V. 13. and cor.) AC.EC: BC2: EC.HD: EC.CD, or (V. 3.): : HD: CD; consequently (V. 9.) BC2+AC·EC: BC2 :: HC: CD; but, AH being equal to EC, HC is equal to AE; wherefore BC2+ AC.EC: BC2:: AE: CD..

If the vertical lines BD, BE cut the base AC of a rightangled triangle ACB externally; then will BC2-AC.EC: BC2 :: AE CD. For make

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A

HA

CE

DA.AE-EC.CD: EC.CD; but DA.AE-EC.CD=DA.AC+ DA.EC-EC.CD-DA.AC-EC.AC=AC.HD: wherefore

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AC2: BC2:: AC.HD: EC.CD, and AC.EC: BC2:; EC.HD: EC.CD : : HD : CD, and consequently BC-AC.EC: BC2:: HC or AE: CD.

PROP. VI. THEOR.

The perpendicular within a circle, is a mean proportional to the segments formed on it by straight lines, drawn from the extremities of the diameter, through any point in the circumference.

Let the straight lines AEC and BCG, drawn from the ex

tremities of the diameter of a circle through a point C in the circumference, cut the perpendicular to AB; the part DF within the circle is a mean proportional between the segments DE and DG.

For the angle ACB, being in a semicircle, is a right angle (III. 19.), and the angle ABG is common to the two triangles ABC and GBD, which are, therefore, similar (VI. 11.). Hence the remaining angle BAC is equal to BGD, and consequently the triangles ADE and GDB are similar; wherefore AD: DE : DG: DB, and (V. 6.) AD.DB= DE.DG. But (III. 26. cor.), the rectangle under AD and DB is equivalent to the square of DF; whence DE.DG=DF, and (V. 6.) DE: DFDF: DG.

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The Appendix to the books of Geometry cannot fail, by its novelty and singular beauty, to prove highly interesting. The first part is taken from a scarce tract of Schooten, who was Professor of Mathematics af Leyden, early in the seventeenth century. But the second and most important part is chiefly selected from a most ingenious work of Mascheroni, a celebrated Italian mathematician; which in 1798 was translated into French, under the title of Geometrie du Compas. It will be perceived, however, that I have adapted the arrangement to my own views, and have demonstrated the propositions more strictly in the spirit of the ancient geometry.

NOTES TO TRIGONOMETRY.

1. THE French philosophers have, at the instance of Borda, lately proposed and adopted the centesimal division of the quadrant, as easier, more consistent, and better adapted to our scale of arithmetic. On that basis, they have also constructed their ingenious system of measures. The distance of the Pole from the Equator was determined with the most scrupulous accuracy, by a chain of triangles extending from Calais to Barcelona, and since prolonged to the Balearic Isles. Of this quadrantal arc, the ten millionth part, or the tenth part of a second, and equal to 39.371 English inches, constitutes the metre, or unit of linear extension. From the metre again, are derived the several measures of surface and of capacity; and water, at its greatest degree of contraction, furnishes the standard of weights.

It would be most desirable, if this elegant and universal system were adopted, at least in books of science. Whether, with all its advantages, it be ever destined to obtain a general currency in the ordinary affairs of life, seems extremely questionable. At all events, its reception must necessarily be very slow and gradual; and, in the meantime, this innovation is productive of much inconvenience, since it not only deranges our habits, but lessens the utility of our delicate instruments and elaborate tables. The fate of the centesimal division may fi. nally depend on the continued merit of the works framed after that model.

2. The remarks contained in the preliminary scholium, will obviate an objection which may be made against the succeeding demonstrations, that they are not strictly applicable, except when the arcs themselves are each less than a quadrant. But this in fact is the only case absolutely wanted, all the derivative arcs being at once comprehended under the definition. of the sine or tangent. To follow out the various combinations, would require a fatiguing multiplicity of diagrams; and

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