Today in History...

Julius Lugwig Weisbach
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Born: 10 Aug 1806 in Mittelschmiedeberg (near Annaberg), Germany
Died: 24 Feb 1871 in Freiberg, Germany
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Of his nine siblings, Julius was the second youngest.
Weisbach was educated at the lyceum in Annaberg near his home town, then at the Bergschule in Freiberg. Having completed the courses at these schools by 1822, Weisbach wanted to continue his education. However, his parents could not afford to finance him so he borrowed money to enable him to attend the Bergakademie in Freiberg.
In 1827 Weisbach was advised by Friedrich Mohs to study at Göttingen. Mohs was the famous German mineralogist who, in 1812, devised a scale, now known as the Mohs scale, to determine the hardness of a mineral by observing whether its surface is scratched by various substances of known hardness. Weisbach followed Mohs' advice and spent two years in Göttingen before moving to the University of Vienna. The reason for this move was that Mohs had gone to Vienna. In Vienna, Weisbach studied mathematics, physics and mechanics.
During 1830 Weisbach travelled on foot for six months through Hungary, the Tyrol, Bavaria, and Bohemia. From 1831 he taught mathematics at the Freiberg Gymnasium and, starting in 1832, he also taught mathematics at the Bergakademie in Freiberg. In this same year of 1832, Weisbach married Marie Winkler and they would have a son who was to became the Professor of Mineralogy at the Bergakademie in Freiberg.
In 1836 Weisbach was promoted to professor, and his expertise was now such that in addition to mathematics he was professor of mine machinery and of surveying. He had, at this stage only one publication, but this would soon change. From around 1839 he became interested in hydraulics and some of his most important work would be done on this topic. This interest in hydraulics seems to have been as a result of Weisbach visiting the Paris Industrial Exposition in 1839.
His interests were always wide and this is reflected in the range of courses that Weisbach was teaching around this time: descriptive geometry, crystallography, optics, mechanics and machine design.
In 1855 Zürich Polytechnikum opened, and in the year prior to that they made a serious attempt to appoint quality staff. For example they appointed Clausius as professor of physics, but Weisbach was not tempted by the offer that was made to him and preferred to remain in Freiberg. In 1855 Weisbach was back in Paris, this time visiting the World Exposition which was held there.
We have indicated the range of Weisbach's interests and this can be seen from the topics of the fourteen books and 59 papers he wrote on mechanics, hydraulics, surveying, and mathematics. It is in hydraulics that his work was most influential, with his books on the topic continuing to be of importance well into the 20th century.
Among the honours Weisbach received for his contributions to science were honorary degrees from Leipzig, and election to membership of various learned societies such as the St Petersburg Academy of Sciences, the Royal Swedish Academy of Sciences and the Accademia dei Lincei.

Circles Revision

Circle Theorems
Circles
A circle is a set of points which are all a certain distance from a fixed point known as the centre.
A line joining the centre of a circle to any of the points on the circle is known as a radius.
The circumference of a circle is the length of the circle. The circumference of a circle = 2 × π × the radius.


The red line in the second diagram is called a chord. It divides the circle into a major segment and a minor segment.
Theorems
Angles Subtended on the Same Arc

Angles formed from two points on the circumference are equal to other angles, in the same arc, formed from those two points.
Angle in a Semi-Circle

Angles formed by drawing lines from the ends of the diameter of a circle to its circumference form a right angle. So c is a right angle.
Higher Proof
We can split the triangle in two by drawing a line from the centre of the circle to the point on the circumference our triangle touches.

We know that each of the lines which is a radius of the circle (the green lines) are the same length. Therefore each of the two triangles is isosceles and has a pair of equal angles.

But all of these angles together must add up to 180°, since they are the angles of the original big triangle.
Therefore x + y + x + y = 180, in other words 2(x + y) = 180.
and so x + y = 90. But x + y is the size of the angle we wanted to find.
Tangents
A tangent to a circle is a straight line which touches the circle at only one point (so it does not cross the circle- it just touches it).
A tangent to a circle forms a right angle with the circle's radius, at the point of contact of the tangent.

Also, if two tangents are drawn on a circle and they cross, the lengths of the two tangents (from the point where they touch the circle to the point where they cross) will be the same.

Angle at the Centre

The angle formed at the centre of the circle by lines originating from two points on the circle's circumference is double the angle formed on the circumference of the circle by lines originating from the same points. i.e. a = 2b.
Higher Proof
You might have to be able to prove this fact:

OA = OX since both of these are equal to the radius of the circle. The triangle AOX is therefore isosceles and so ∠OXA = a
Similarly, ∠OXB = b


Since the angles in a triangle add up to 180, we know that ∠XOA = 180 - 2a
Similarly, ∠BOX = 180 - 2b
Since the angles around a point add up to 360, we have that ∠AOB = 360 - ∠XOA - ∠BOX
= 360 - (180 - 2a) - (180 - 2b)
= 2a + 2b = 2(a + b) = 2 ∠AXB
Higher Alternate Segment Theorem

This diagram shows the alternate segment theorem. In short, the red angles are equal to each other and the green angles are equal to each other.
Proof
You may have to be able to prove the alternate segment theorem:

We use facts about related angles:
A tangent makes an angle of 90 degrees with the radius of a circle, so we know that ∠OAC + x = 90.
The angle in a semi-circle is 90, so ∠BCA = 90.
The angles in a triangle add up to 180, so ∠BCA + ∠OAC + y = 180
Therefore 90 + ∠OAC + y = 180 and so ∠OAC + y = 90
But OAC + x = 90, so ∠OAC + x = ∠OAC + y
Hence x = y
Cyclic Quadrilaterals
A cyclic quadrilateral is a four-sided figure in a circle, with each vertex (corner) of the quadrilateral touching the circumference of the circle. The opposite angles of such a quadrilateral add up to 180 degrees.
Higher Area of Sector and Arc Length

If the radius of the circle is r,
Area of sector = πr2 × A/360
Arc length = 2πr × A/360

In other words, area of sector = area of circle × A/360
arc length = circumference of circle × A/360