Euclid’s Elements – Book 1

Definitions:

1. A point is that which has no part.

2. A line is breadth-less length.

3. The ends of a line are points

4. A straight line is a line which lies evenly with the points on itself

5. A surface is that which has length and breadth only

6. The edges of a surface are lines

7. A plane surface is a surface which lies evenly with the straight lines on itself

8. A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line

9. And when the lines containing the angle are straight, the angle is called rectilinear

10. When a straight line standing on a straight line makes the adjacent angles equal to one another, each of the equal angles is right and the straight line standing on the other is called a perpendicular to that on which it stands.

11. An obtuse angle is an angle greater than a right angle

12. An acute angle is an angle less than a right angle

13. A boundary is that which is an extremity of anything

14. A figure is that which is contained by any boundary or boundaries

15. A circle is a plance figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure equal to one another

16. And the point is called the center of the circle

17. A diameter of the circle is any straight line drawn through the center and terminated in both directions by the circumference of the circle, and such a straight line also bisects the circle.

18. A semicircle is the figure contained by the diameter and the circumference cut off by it. And the center of the semicircle is the same as that of the circle.

19. Rectilinear figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral those contained by more than four straight lines.

20. Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle that which has its three sides unequal.

21. Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled triangle that which has an obtuse angle, and an acute-angled triangle that which has its three angles acute.

22. Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.

23. Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.

Euclid’s Five Postulates:

1. To draw a straight line from any point to any point.

2. To produce a finite straight line continuously in a straight line.

3. To describe a circle with any center and radius

4. That all right angles equal one another

5. That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

Propositions:

1. To construct an equilateral triangle on a given finite straight line.

2. To place a straight line equal to a given straight line with one end at a given point.

3. To cut off from the greater of two given unequal straight lines a straight line equal to the less.

4. If two triangles have two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, then they also have the base equal to the base, the triangle equals the triangle, and the remaining angles equal the remaining angles respectively, namely those opposite the equal sides.

5. In isosceles triangles the angles at the base equal one another, and, if the equal straight lines are produced further, then the angles under the base equal one another.

6. If in a triangle two angles equal one another, then the sides opposite the equal angles also equal one another.

7. Given two straight lines constructed from the ends of a straight line and meeting in a point, there cannot be constructed from the ends of the same straight line, and on the same side of it, two other straight lines meeting in another point and equal to the former two respectively, namely each equal to that from the same end.

8. If two triangles have the two sides equal to two sides respectively, and also have the base equal to the base, then they also have the angles equal which are contained by the equal straight lines.

9. To bisect a given rectilinear angle.

10. To bisect a given finite straight line.

11. To draw a straight line at right angles to a given straight line from a given point on it.

12. To draw a straight line perpendicular to a given infinite straight line from a given point not on it.

13. If a straight line stands on a straight line, then it makes either two right angles or angles whose sum equals two right angles.

14. If with any straight line, and at a point on it, two straight lines not lying on the same side make the sum of the adjacent angles equal to two right angles, then the two straight lines are in a straight line with one another.

15. If two straight lines cut one another, then they make the vertical angles equal to one another. Corollary: If two straight lines cut one another, then they will make the angles at the point of section equal to four right angles.

16. In any triangle, if one of the sides is produced, then the exterior angle is greater than either of the interior and opposite angles.

17. In any triangle the sum of any two angles is less than two right angles.

18. In any triangle the angle opposite the greater side is greater.

19. In any triangle the side opposite the greater angle is greater.

20. In any triangle the sum of any two sides is greater than the remaining one.

21. If from the ends of one of the sides of a triangle two straight lines are constructed meeting within the triangle, then the sum of the straight lines so constructed is less than the sum of the remaining two sides of the triangle, but the constructed straight lines contain a greater angle than the angle contained by the remaining two sides.

22. To construct a triangle out of three straight lines which equal three given straight lines: thus it is necessary that the sum of any two of the straight lines should be greater than the remaining one.

23. To construct a rectilinear angle equal to a given rectilinear angle on a given straight line and at a point on it.

24. If two triangles have two sides equal to two sides respectively, but have one of the angles contained by the equal straight lines greater than the other, then they also have the base greater than the base.

25. If two triangles have two sides equal to two sides respectively, but have the base greater than the base, then they also have the one of the angles contained by the equal straight lines greater than the other.

26. If two triangles have two angles equal to two angles respectively, and one side equal to one side, namely, either the side adjoining the equal angles, or that opposite one of the equal angles, then the remaining sides equal the remaining sides and the remaining angle equals the remaining angle.

27. If a straight line falling on two straight lines makes the alternate angles equal to one another, then the straight lines are parallel to one another.

28. If a straight line falling on two straight lines makes the exterior angle equal to the interior and opposite angle on the same side, or the sum of the interior angles on the same side equal to two right angles, then the straight lines are parallel to one another.

29. A straight line falling on parallel straight lines makes the alternate angles equal to one another, the exterior angle equal to the interior and opposite angle, and the sum of the interior angles on the same side equal to two right angles.

30. Straight lines parallel to the same straight line are also parallel to one another.

31. To draw a straight line through a given point parallel to a given straight line.

32. In any triangle, if one of the sides is produced, then the exterior angle equals the sum of the two interior and opposite angles, and the sum of the three interior angles of the triangle equals two right angles.

33. Straight lines which join the ends of equal and parallel straight lines in the same directions are themselves equal and parallel.

34. In parallelogrammic areas the opposite sides and angles equal one another, and the diameter bisects the areas.

35. Parallelograms which are on the same base and in the same parallels equal one another.

36. Parallelograms which are on equal bases and in the same parallels equal one another.

37. Triangles which are on the same base and in the same parallels equal one another.

38. Triangles which are on equal bases and in the same parallels equal one another.

39. Equal triangles which are on the same base and on the same side are also in the same parallels.

40. Equal triangles which are on equal bases and on the same side are also in the same parallels.

41. If a parallelogram has the same base with a triangle and is in the same parallels, then the parallelogram is double the triangle.

42. To construct a parallelogram equal to a given triangle in a given rectilinear angle.

43. In any parallelogram the complements of the parallelograms about the diameter equal one another.

44. To a given straight line in a given rectilinear angle, to apply a parallelogram equal to a given triangle.

45. To construct a parallelogram equal to a given rectilinear figure in a given rectilinear angle.

46. To describe a square on a given straight line.

47. In right-angled triangles the square on the side opposite the right angle equals the sum of the squares on the sides containing the right angle.

48. If in a triangle the square on one of the sides equals the sum of the squares on the remaining two sides of the triangle, then the angle contained by the remaining two sides of the triangle is right.

Non-Euclidean Geometry

I found these excellent lectures on Youtube about Non-Euclidean Geometry. These are delivered by Prof. Judith V. Grabiner from Harvard University.

Aas Paas Hain Khuda

Came across this beautiful song few minutes ago and thought of sharing the lyrics with you all. From last hour or so, my heart has been feeling really heavy and wanted to run away from life and circumstances. And then, I don’t know how… came across this song. Instantly, it reminded… ‘yes, there is God nearby and He wouldn’t have wanted me to run away’.

These lyrics are in Urdu/Hindi, will be posting the lyrics in English sometime later.

Dhundhla jaayein jo manzilein

Ik pal ko tu nazar jhuka

Jhuk jaaye sar jahan wahi

Milta hai rab ka raasta

Teri kismat tu badal de

Rakh himmat bus chal de

Tere saathi mere kadmon ke hain nishaan

Tu na jaane aas pass hai khuda

Tu na jaane aas paas hai khuda

Tu na jaane aas pass hai khuda

tu na jaane aas paas hai khuda

Khud pe daal tu nazar

Haalaton se haar kar

Kahan chala re

Haath ki lakeer ko

Modhta marodta

Hai hausla re

Toh khud tere khwabon ke rang mein

Tu apne jahan ko bhi rang de

Ke chalta hoon mein tere sang mein

Ho shaam bhi toh kya

Jab hoga andhera

Tab paayega dar mera

Uss dar pe phir hogi teri subah

Tu na jaane aas pass hai Khuda

Tu na jaane aas pass hai Khuda

An open letter to all Muslim brothers and sisters

Dear Brothers and Sisters,

As-Salam AlaiKum (may peace be upon you). As most of you are aware, a Florida Pastor named Terry Jones has called for burning the holy book of Islam, the Quran on September 11^th 2010. Even though several Americans including prominent leaders of US have condemned the announcement of the Pastor, however the constitution of US grants the right to Terry Jones to carry out this extremist act. Understandably, this is causing a lot of anger among Muslims across the world and already triggered violent protests in some parts of Afghanistan.

I am neither an expert in Islamic studies nor a representative of any group. But as a practicing Muslim I would like to share some of my thoughts. When I was a little kid, my father told me about an incident in Islamic History that narrates the power of the Holy Quran and the will of Allah. I humbly would like to share it. Umar (Razi’Allah-O-Taalah’Anhu), before converting to Islam was severely against Prophet Mohammed (P.B.U.H) and his teachings. His hostility reached to an extent that he decided to murder Prophet Mohammed (P.B.U.H) and end Islam. On his way to kill Prophet Mohammed (P.B.U.H), Umar found that his sister and brother-in-law had converted to Islam. He reaches the house of his sister and starts arguing on why they have converted to Islam. His sister points Umar to the verses of Ta Ha, the 20^th chapter of Holy Quran and pleads him to read it once. Reluctantly Umar reads the verses and instantly his heart goes through a revolution. On the very same day, Umar meets Prophet Mohammed (P.B.U.H) and accepts Islam. Here was a person, who was on his way to murder Prophet Mohammed (P.B.U.H) and end Islam but his heart changed by reading the verses of Quran and Allah granted him wisdom. He went on to become a very dear friend of Prophet Mohammed (P.B.U.H) and contributed highly towards spreading Islam. This is the power of Holy Quran and the will of Allah. Allah states in Holy Quran Chapter no. 2, verse no. 269 “He (Allah) gives wisdom unto whom He will, and He unto whom wisdom is given, he truly hath received abundant good. But none remember except men of understanding”.

So, when a Pastor decides to burn the copies of Holy Quran, our duty as Muslims is pray to Allah to grant the Pastor with wisdom and bring a revolution in his heart. He has already purchased hundreds of copies of Holy Quran to burn. The only thing required is, he opens a copy of Holy Quran and read it. And if Allah decides to grant him wisdom, there would be a revolution in his heart. My dear Muslim brothers and sisters, I call upon you to pray to Allah for granting wisdom and bringing a revolution in the heart of Pastor Terry Jones. This is the best thing we can do.

Also, let me remind you, Allah has promised to protect Holy Quran. In Chapter 15, verse 9, Allah says ‘We have sent down the Reminder [Quran], and We will preserve it’. So, let there be no doubt in your mind, even if every copy of Holy Quran on the planet Earth is burnt by infidels, the Holy Quran would still be preserved by Allah.

May Allah grant us all with wisdom.

Regards,

Mohammed Abubakr,

Hyderabad India.

‘Giving up’ is not an option

Thousands of sleepless nights and hundreds of disturbed days, yet I see no direction in front of me. Someday I pondered over Riemann’s Zeros and someday whether P is equal to NP or not. That’s not all. Open problems in physics still continue to haunt me. Occasionally, when I sleep during nights, I wake up to dreams that demand to work harder. Someone advised me, as you grow old, you need to pick a subject and narrow your concentration towards it. Trust me, I tried following the advice very sincerely but it didn’t work out. I don’t have a kind of brain that would be disciplined enough to focus on a narrow field. It’s rather like a free bird that wants to explore as much as possible. There are days, when I get overjoyed with new ideas, I feel like the king of this world and there are days when I find myself alone in the dark corner of the room, isolated and lost.

Few days back, I was at International Congress of Mathematicians at Hyderabad. It was an experience of lifetime. Watching 3000 mathematics and their enthusiasm for advancing human knowledge was absolutely inspiring. When the Fields medals were announced, I clapped like a little kid and cheered for heroes of mathematics. There was something special about that moment. I can’t describe it in words. Four gentlemen who received the Fields Medal stood there with honour and pride… not many people in the crowd knew who they are and what they have done but all of us knew, they had done something exceptional and profound. While the 3000 people clapped, there were tears in the eyes of some of the family members of the Fields Medallists. Truly, without them, those Fields Medallists wouldn’t have been there. Behind every monumental work done by the researchers, there are sacrifices of the family members. I don’t know why I had the tears in my eyes too. I felt somehow connected yet isolated.

A month back, I turned 24 according to the conventional calendar followed across the world. Years have passed by at a rapid pace and the dreams still remain as dreams. I had complains for everything in life. My ideas are being rejected by journals, my hard work isn’t producing results, the girl I am madly in love with, might get married to someone else … nothing in life appeared going according to the plan. I questioned God… will my ideas ever reach out to the world? Will I ever be able to discover anything that will advance human knowledge? Whether am I destined to accept that I can’t marry her? I felt weak and terrible. I felt lost. I felt like giving up to the fate. I felt like a loser.

Then suddenly, a thought occurred. Why am I giving up? Just because few papers have been rejected and few ideas haven’t worked out, doesn’t mean its end of my research. Just because I haven’t discovered something phenomenal doesn’t mean, I am not going to discover in future. Just because, things are not working out between me and her doesn’t mean, things are not going to work out in future. The fact that, she is still unmarried is itself an indication that God’s is still on my side. May be, God’s help is on its way. There is always the pain in the initial stages of the life of an achiever. Why am I feeling like a loser? Why am I feeling weak? Why am I losing confidence? Why am I giving up?

Giving up is not an option. Let there be defeat. Let there be days of pain. Let there be years of isolation. Let there be tears of love. But someday … God’s help would arrive and it will bring victory. There is a verse in Holy Quran that says ‘Ijaza a Nasr 'Allah hi val fatah’ which means, ‘When there come the help of Allah and victory’. I shall wait for that help of Allah…I shall wait for the victory that Allah grants me.