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12

EXTRACTION OF SQUARE ROOT.

squares. Thus 3=√4-1, = √2o-1 so that if a right-angled Thus√3 triangle ABC be drawn, right angled at C and having AB = 2 inches, and AC-1 inch, BC represents √3, an inch being the unit. (The triangle may be constructed by drawing a semicircle on AB as diameter and making AC in it = 1 inch.) If the unit is half-aninch BC represents √AB-AC2, i.e. √42 - 2o or 12.

√5=√32-22, i.e. is the perpendicular of a right-angled triangle the hypotenuse of which is 3 and the base of which is 2, or it may be determined as the hypotenuse of a right-angled triangle one side of which is 2 and the other 1, since √5=√22 + 1o. If we halve the unit the same line would represent √20.

√6=√2a +√22, i.e. if √2 be first determined, √6 is the hypotenuse of a right-angled triangle the sides of which are 2 and √2, or it may be determined from /6=√√32-√32.

√7 = √22 + √32, and can be determined if √3 is known.

√8 has already been given.

√10 = √32 + 12.

2

√11 =√4a - √53, and can be determined if √5 is known.

√12 has been given above; and the method is probably sufficiently exemplified by the above, but we will take two examples of larger numbers.

√47 = √6a + √112, thus being made to depend on √11.

other ways.

2

2

√179=√132+ √103 thus being made to depend on √10, it might also be written =√11+√47 or could be determined in No definite instructions can be given as to the best mode of working in any particular case, but as a rule triangles having sides of nearly equal magnitude should be selected, since the intersections of lines cutting at very acute angles cannot be accurately determined.

DEFINITION. Three magnitudes are said to be in harmonic progression when the first is to the third as the difference between the first and second is to the difference between the second and third: and the second magnitude is said to be an harmonic mean between the first and third.

Thus if the magnitudes represented by the lengths of three lines (as AB, AC, AD, fig. 9) are in harmonic progression and the lines be superimposed with a common extremity as in that fig.: then AB : AD :: BC : CD.

The reciprocals of magnitudes in harmonic progression are in arithmetic progression and conversely :-for, if AB, AC, AD are in harmonic progression then by definition

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an identical expression with the above.

PROBLEM 11. (Fig. 9.) To find the harmonic mean between two given lines AB, AD, i.e. to find a line of length l such that

AB: AD: the difference between AB and l

the difference between AD and l.

Set off the given lengths from the same point (A) on any line and in the same direction along it, as AB, AD. Take any point E outside AD and join AE, DE. Through B draw FBG parallel

14

HARMONIC PROGRESSION.

to DE meeting AE in F and make BG = BF. Join EG cutting AD in C and AC will be the required harmonic mean. For by the

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AB: AD :: BF: DE.

Also by the similar triangles CBG, CDE,

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progression, the first two terms being given.

The above construction may be adapted to find the third term of a harmonic progression the first two terms being given. SupSuperpose them with a common exTake any point

pose AB and AC given. tremity as in the fig. 9.

FB and produce it to G making_BG = BF.

outside AC. Join Join AF and GC

producing them to meet in E and draw ED through E parallel to FB meeting AC (produced if necessary) in D. AD will be the required third term.

DEF. When four points in a straight line as ABCD in fig. 9 fulfil the condition

AB: AD :: BC : CD,

they constitute a Harmonic Range, and if through any point E outside the line the four straight lines EA, EB, EC, ED be drawn these four lines constitute a Harmonic Pencil, which is denoted by E{ABCD}. Any straight line drawn across the pencil is called a

Transversal, and every transversal of a harmonic pencil is divided harmonically in the points in which it intersects the lines of the pencil: i. e. the four points of intersection constitute a Harmonic Range. For in fig. 9 draw any transversal as HKLM, and through K draw fKg parallel to ED and therefore to FG, meeting EA, EC in fand g. Obviously since BF = BG, ... Kƒ= Kg.

By similar triangles HKƒ, HME

HK : HM :: ƒK : EM,

and by similar triangles KLg, MLE

but

::

KL LM gK: EM,

:

Kf = Kg,

HK: HM :: KL : LM,

or HKLM constitute a Harmonic Range.

A particular case of a Harmonic Pencil is furnished by the pencil formed of two straight lines and the bisectors of the angles between them, as shewn in fig. 10, where AD bisects the angle

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BAC and AE is drawn perpendicular to AD, and therefore bisecting the exterior angle between AC and BA produced. For draw any transversal as BFGE, and through F draw PFN parallel to AE and meeting AB, AC in P and N.

Then PF= FN and

BF BE PF: AE, by similar triangles BPF, BAE,

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FG GE FN: AE, by similar triangles FGN, EGA,

.. BF : BE :: FG : GE,

or the pencil is harmonic.

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HARMONIC RANGES AND PENCILS.

A line of given length may obviously be divided harmonically in an infinite number of ways, since a line of length HK=BE can be drawn from any point H on AB to terminate on AE and HL HK :: LM : MK.

Harmonic Properties of a complete Quadrilateral.

If FBeA, FDe,C be harmonic ranges (fig. 11), the straight lines AC, ee,, BD meet in a point, as also AD, BC and ee.

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For if BD, AC meet in E, draw Ee; then the pencil E (AeBF) is harmonic and FC is a transversal, so that e, must lie on Ee.

Similarly if AD and BC meet in O, the pencil O (AeBF) is harmonic and FC a transversal, so that e, must lie on Oe.

If ABCD is any quadrilateral, E the intersection of the sides. AC and BD, F of the sides AB and CD, O the intersection of the diagonals AD and BC; it follows conversely that EA, EO, EB, EF form a harmonic pencil, as also FE, FC, FO and FA. If EO meet AB in e and CD in e,, AeBF and Ce ̧DF are therefore harmonic ranges, and if FO meet AC in ƒf and BD in f,, AfCE and Bf,DE are both harmonic ranges.

Further if AD meet FE in a and BC meet it in b, BOCb is a harmonic range since it is a transversal of the pencil F(ECƒA), therefore AF, Aa, AE and Ab form a harmonic pencil, and therefore FaEb is a harmonic range, i. e. FE is divided harmonically in a and b.

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