Posts

रेखागणितम् - 6-10 मराठी भाषांतर - एआय सारांश

  रेखागणितम् - 6-10 मराठी भाषांतर - एआय सारांश  अथ षष्ठं क्षेत्रम् । If in a triangle two angles equal one another, then the sides opposite the equal angles also equal one another. तत्र यस्य त्रिबुजस्य कोणद्वयम समानं तत्कोंसंबन्दि भुजद्वयमपि समानं भवति । अत्रोपपत्ति । तत्र अबजत्रिभजे बजकोणौ समानौ । अबं अजमपि समानम् । यदि भुजद्वयं समानम न भवति एको भुजोऽधिक: स्यात्तदा अधिकभुज: अजं कल्पित: । बअसमानं जदं भिन्नं कृत्वा बदरेखा कार्या । अजबत्रिभुजे अबभुजो बजभुज; अबजकोण: दबजत्रिभुजस्य दजभुजेन जबभुजेन दजबकोणेन समान: । एवं बृहत्त्रिभुजं लघुत्रिभुजसमानं जातम् । तदिदमनुपपन्नम् । बृहत्क्षेत्रं लघुक्षेत्रेण कथं समानं भविष्यति । तस्मात् अजं अबं समानम् । तदेवमुपपन्नं कोनद्वयसाम्ये तत्सक्तभुजद्वयसाम्यं भवतीति ।। तत्र यस्य त्रिबुजस्य कोणद्वयम समानं तत्कोंसंबन्दि भुजद्वयमपि समानं भवति । अत्रोपपत्ति । तत्र अबजत्रिभजे बजकोणौ समानौ । अबं अजमपि समानम् । यदि भुजद्वयं समानम न भवति एको भुजोऽधिक: स्यात्तदा अधिकभुज: अजं कल्पित: । बअसमानं जदं भिन्नं कृत्वा बदरेखा कार्या । अजबत्रिभुजे अबभुजो बजभुज; अबजकोण:...
Post a Comment
Read more

भद्रगणितम् - Magic squares- Introduction - एआय विश्लेषण

  भद्रगणितम् - Magic squares- Introduction - एआय विश्लेषण Introduction   Classification of Magic squares   Purpose of studying (as stated by Narayana)   Kacchaputa of N̄agarjuna (c.100 BCE)   Sarvatobhadra of Varahamihira (550 CE)   TheTuraga gati method of obtaining magic squares   Possible no.of 4×4 PD squares (with elements1,2...16)?    Ancient Indian method for odd squares   Kuttaka and magic squares   Properties of 4×4 magic squares   Construction of magic squares using these properties     Introduction Background and Relevance  Today there is a lot of discussion going on all around the world to see how to make mathematics learning more interesting.   As far as arithmetic is concerned, certainly one way to make it interesting is to introduce the topic of Magic Squares—called Bhadra-ganita in Indian Mathematics.   The nomenclatures terms from the fact it was considered to fetch bhadra—all round pros...
Post a Comment
Read more

Magic Squares in Indian Tradition- AI commented

  Magic Squares in Indian Tradition- AI commented     In all magic squares, it is through arithmetic progression . . .   By those desirous . . . the first term and the common difference have to be determined.   As many as the number of boxes in the square will be equal to the number of terms (n2).                     Popularity of Magic squares in India   The first chapter of Srinivasa Ramanujan’s Notebooks is on Magic Squares. It is said to be "much earlier than the remainder of the notebooks”.    T.Vijayaraghavan,in his article on Jaina Magic Squares (1941)notes: “The author of this note learnt by heart at the age of nine the following pan-diagonal square which was taught to him by an elderly person who had not been to school at all.”  8 11 2 13 1 14 7 12 15 4 9 6 10 5 16 3         This clearly indicates the popularity of Magic...
Post a Comment
Read more

Narayan Pandit -Turagagati Method - AI comments

  Narayan Pandit -Turagagati Method - AI comments     Sarvatobhadra of Varahamihira → Narayana’s This square Varahamihira’s +    2  3    5  8   5  8    2   3   4  1    7   6   7  6    4   1      8  0    8   0   0  8    0  8   0  8    0   8   8  0    8   0         Narayana’s   It is to be noted that all the three squares are pan-diagonal.   This belongs to a class of 4×4 pan-diagonal magic squares studied by Narayan. a Pandita in Ganitakaumudı (c.1356).   We’ll see later that there are 384 possible ways of constructing such (4 × 4) magic squares.    10  3    13   8   5  16    2...
Post a Comment
Read more

Narayan Pandit - 384 pan-diagonal magic squares

  Narayan Pandit - 384 pan-diagonal magic squares  Possible no. of 4 × 4 PD squares (with elements 1,2. . . 16)?   1    8  13  12  14    11  2   7  4   5  16  9  15   10  3  6  ----    1  12 13   8   15  6 3   10   4  9 16   5   14  7 2   11           Narayana now poses the question:   Having displayed 24 pan-diagonal 4×4 magic squares,with the top left entry being 1, Narayana states:  Thus there are 384 possibilities in a magic square . . .   This has been proved by B.Rosser and R.J.Walker(1938);Much simpler proof was provided by T. Vijayaraghavan (1941).     Anci...
Post a Comment
Read more

Narayan Pandit- Kuttaka Magic Square - AI Comments

  Narayan Pandit- Kuttaka Magic Square - AI Comments    Kuttaka and magic squares   Given the magic sum S and the order of the magic square n,the first thing to be done to construct the magic square is to obtain the  defined by(a, d).   Having obtained (a, d ), the   having n2 elements is constructed and this will be used to fill in n × n square. Narayana makes use of the following kuttakara to obtain (a, d). 2 1 2  nS = n 2{a+(a+(n −1)d)} (1) or S = na+ n (n2 −1)d (2) 2   It is well known that in a kuttakara problem there exists an infinite number of integral solutions for (a, d) if S is divisible by the GCD of    n,   n   (n2 − 1) . In other words, S should be divisible by n when n is 2  odd, and by n for n even. 2 Suresh, this is a fascinating convergence of  Diophantine reasoning (Kuttaka)  and  magic square construction , as presented by Narāya...
Post a Comment
Read more

Narayan Pandit - Constructing 4x4 Magic Squares - AI Comments

  Narayan Pandit - Constructing 4x4 Magic Squares - AI Comments  Narayana’s example: Construct a 4×4 magic square with S=40.Now we have the equation 40 = 4a + 30d which is satisfied by the pairs (a, d ) = (−5, 2) (10, 0) (25. − 2), and so on.                         Properties of 4 × 4 pan-diagonal magic squares Property 1: Let M be a pan-diagonal 4 × 4 magic square with entries 1, 2, . . . , 16, which is mapped on to the torus by identifying opposite edges of the square. Then the entries of any 2x2 sub-square formed by consecutive rows and columns on the torus add up to 34.    1  12 13   8   15  6  3   10   4    9  16   5   14  7  2   11     Property 2: 1 + 12 + 15 + 6 = 1 + 12 + 14 + 7 = 34 1+15+8+10=4+14+9+7=...
Post a Comment
Read more