परिकर्माष्टके वर्गः
In परिकर्माष्टकम् the second part of लीलावती the process of obtaining squares of numbers is detailed in two stanzas.
अथ वर्गः Now squares
वर्गे करण-सूत्रम् वृत्त-द्वयम् Method of squaring in two stanzas Nos. 19 and 20.
Stanza No. 19 is in उपजाति-वृत्तम्
सम-द्वि-घातः कृतिरुच्यतेऽथ
स्थाप्योऽन्त्य-वर्गो द्वि-गुणान्त्य-निघ्नाः
स्व-स्वोपरिष्टाच्च तथापरेऽङ्कास्-
त्यक्त्वान्त्यमुत्सार्य पुनश्च राशिम् ||१९||
पदच्छेदैः –
समद्विघातः कृतिः उच्यते अथ स्थाप्यः अन्त्यवर्गः द्विगुणान्त्यनिघ्नाः |
स्वस्वोपरिष्टात् च तथा अपरे अङ्काः त्यक्त्वा अन्त्यम् उत्सार्य पुनः च राशिम्
अन्वयेन –
- अथ "समद्विघातः"-(इति) कृतिः उच्यते = Now कृतिः i.e. procedure called as समद्विघातः number raised to two, or number multiplied by itself, i.e. square of a number is stated
- द्विगुणान्त्यनिघ्नः अन्त्यवर्गः स्थाप्यः =
- द्विगुणान्त्यनिघ्नः = द्वि-गुण-अन्त्य-निघ्नः = Note, अन्त्य = left-most digit, द्विगुणान्त्य = double of left-most digit, द्विगुणान्त्यनिघ्नः = multiplied by double of left-most digit
- अन्त्यवर्गः = square of the left-most digit
- स्थाप्यः = put up
- तथा च = likewise
- पुनः च अन्त्यम् त्यक्त्वा राशिम् उत्सार्य = Again, starting with a new cycle, on eliminating the left-most digit,
- स्वस्वोपरिष्टात् अपरे अङ्काः = other numbers to be put up respectively
From the interpretation of the श्लोक linked here, one gets to summarize the procedure as –
- Take the square of the left-most (first) digit and write the square above the digit.
- Then multiply the next (second) digit by double of the first digit and write the result on the top.
- The multiply the third digit by double of the first digit and write the result on the top.
- Do this up to the digit at unit's place.
- Then cross the first digit and move the remaining digits one place to the right. Then repeat the same procedure.
- Finally add the products written at the top. The sum is the desired square.
An illustration of this procedure as linked here is –
To find square of 297 –
- Its digits 2, 9 and 7 entered in Row X in cols. 1, 2 and 3 respectively.
- For cycle A, left-most digit is 2. Its square is 4 This is entered above the digit 2, hence in Row A-1, Col 1
- Next (second) digit is 9. This is to be multiplied by double of first digit. Hence 9*2*2 = 36. The digits 3 and 6 are entered into Row A-2, cols. 1 and 2
- Next (third) digit is 7. This is to be multiplied by double of first digit. Hence 7*2*2 = 28. The digits 2 and 8 are entered into Row A-3, cols. 2 and 3. This completes cycle A.
- Now for cycle B, in Row Y, we enter the digits 2, 9 and 7, of the given number by shifting them one place rightward, hence in Row Y cols. 2, 3 and 4. We also consider the left-most digit (2) as eliminated and hence in Row Y, col. 2, we put it in brackets. By that the number of the second cycle is 97 and the left-most digit for the second cycle is 9.
- Square of this new left-most digit is 9^2 = 81. Digits 8 and 1 are entered above number 9 hence in Row B-1 (for the second cycle) and in cols. 2 and 3
- Now we have to multiply the last digit 7 by double of new left-most digit. Then we get 7*2*9 = 126. We enter digits 1, 2 and 6 in Row B-2, cols. 2, 3, and 4.
- Now for the last cycle C, we move digits 2, 9 and 7 of the given number 297 once more right-ward and hence in Row Z in cols. 3, 4 and 5.
- We also both (2) and (9) as eliminated. We are left with 7 as the left-most digit. Its square is 49. Digits 4 and 9 are entered in Row C in cols 4 and 5.
- Now in the "Answer" row we write the vertical totals of numbers in Cols 1, 2, 3, 4 and 5.
- The answer of 297^2 is 88209.
Table 1
| Col 1 | Col 2 | Col 3 | Col 4 | Col 5 | |
| Answer | 8 | 8 | 2 | 0 | 9 |
| Row C | 4 | 9 | |||
| Row B-2 | 1 | 2 | 6 | ||
| Row B-1 | 8 | 1 | |||
| Row A-3 | 2 | 8 | |||
| Row A -2 | 3 | 6 | |||
| Row A-1 | 4 |
Table 2
| Row X | 2 | 9 | 7 | ||
| Row Y | (2) | 9 | 7 | ||
| Row Z | (2) | (9) | 7 |
Two more optional procedures are mentioned in the next 20th श्लोक –
पदच्छेदैः –Stanza No. 20 is in इन्द्रवज्रा-वृत्तम्
खण्ड-द्वयस्याभिहतिर्द्वि-निघ्नी तत् खण्ड-वर्गैक्य-युता कृतिर्वा |
इष्टोन-युज्राशि-वधः कृतिः स्यादिष्टस्य वर्गेण समन्वितो वा ||२०||
पदच्छेदैः –
खण्डद्वयस्य अभिहतिः द्विनिघ्नी तत् खण्ड-वर्ग-ऐक्य-युता कृतिः वा |
इष्ट-ऊन-युज्-राशि-वधः कृतिः स्यात् इष्टस्य वर्गेण समन्वितः वा ||२०||
अन्वयेन –
वा कृतिः (स्यात्) = Or the procedure can be
- खण्डद्वयस्य अभिहतिः द्विनिघ्नी
- Make two parts of the given number, say, n = a + b
- get product of the two parts, a*b
- double the product, 2*ab
- तत्-खण्ड-वर्ग-ऐक्य-युता |
- make squares of each of the two parts a^2 and b^2
- make sum of the two squares a^2 + b^2
- add the (previous) product, i.e. a^2 + b^2 + 2*ab
वा कृतिः स्यात् = Or the procedure can be
- इष्ट-ऊन-युज्-राशि-वधः
- Assume some number x
- Add and subtract it from the given number (n-x) and (n+x)
- Get product of these (n-x)*(n+x)
- इष्टस्य वर्गेण समन्वितः
- add square of the assumed number to the above product, i.e. (n-x)*(n+x) + x^2
अत्र उद्देशकः – Here (in Stanza No. 21), an example (addressed to the girl लीलावती)
Stanza No. 21 is in उपजाति-वृत्तम्
सखे नवानाञ्च चतुर्दशानां ब्रूहि त्रि-हीनस्य शत-त्रयस्य |
पञ्चोत्तरस्याप्ययुतस्य वर्गं जानासि चेद्वर्ग-विधान-मार्गम् ||२१||
पदच्छेदैः –
सखे नवानां च चतुर्दशानां ब्रूहि त्रि-हीनस्य शत-त्रयस्य |
पञ्च-उत्तरस्य अपि अयुतस्य वर्गं जानासि चेत् वर्ग-विधान-मार्गम् ||२१||
अन्वयेन –
सखे वर्ग-विधान-मार्गम् जानासि चेत् = Dear, if you know the method of finding squares
नवानां च = of 9
चतुर्दशानां = of 14
त्रि-हीनस्य शत-त्रयस्य = of number 3-less than 300
पञ्च-उत्तरस्य अयुतस्य अपि = of number 5 more than 10000
वर्गं ब्रूहि = tell (me) square(s) of these.
न्यासः (Discussion) –
९/ १४/ २९७/ १०००५/ एषाम् (of these) यथोक्तकरणेन (by procedures as told) जाताः वर्गाः (squares are) ८१/ १९६/ ८८२०९/ १००१०००२५//
अथ वा OR
- नवानाम् for square of 9
- खण्डे ४/ ५/ Take two parts of 9, say 4 and 5
- अनयोः आहतिः २०/ Their product 4*5 = 20
- द्वि-घ्नी ४०/ Twice of the product = 2*20 = 40
- तद्-खण्ड-वर्गौ १६/ २५/ Squares of the two parts 4^2 = 16 and 5^2 = 25
- अनयोः ऐक्येन ४१/ Sum of the two squares 16 + 25 = 41
- युता = with (i.e. added to) previous result, i.e. 40 + 41 = 81
- जाता सा एव कृतिः ८१// = Answer is same 81
अथ वा OR
- चतुर्दशानाम् for square of 14
- खण्डे ६/ ८/ Take its two parts 6 and 8
- अनयोः आहतिः ४८/ Their product 48
- द्वि-घ्नी ९६/ Double the product, hence 96
- तद्-खण्ड-वर्गौ ३६/ ६४/ Squares of the two parts 36 and 64
- अनयोः ऐक्येन १००/ Sum of the two squares 36 + 64 = 100
- युता added to 96, hence 100 + 96
- जाता सा एव कृतिः १९६// = Answer is same 196
- अथ वा खण्डे ४/ १०/ If we take the two parts as 4 and 10
- तथा अपि सा एव कृतिः १९६// = Then also by same procedure, same answer 196
अथ वा OR
- राशिः २९७/ Take the number 297
- अयम् त्रिभिः ऊनः २९४/ Assume a number 3, deduct it from the given number 297 – 3 = 294
- अयम् त्रिभिः युतः च ३००/ Add the assumed number to the given number 297 + 3 = 300
- अनयोः घातः (२९४*३०० =) ८८२००/ Product of 294 * 300 = 88200
- त्रि-वर्गः ९/ Square of the assumed number 3^2 = 9
- युतः added to product 88200 + 9
- जातः वर्गः सः एव ८८२०९// That is the square (of the given number) 88209
एवम् सर्वत्र | = Likewise always
इति वर्गः | = Here ends getting squares of numbers.
As can be noted the first procedure is applicable to any number howsoever large. Procedures 2 and 3 appeal to be simple at first sight but would not be very simple for large numbers.
There would be many tricks of the trade. For example in Vedic Mathematics there is mention of an aphorism सूत्रम् – एकाधिकेन पूर्वेण which is a solution for finding square of any number having 5 in unit's place. For example, for 175^2,
- here पूर्व i.e. number previous to 5 is 17.
- Take एकाधिक one more than that, i.e. 17 + 1 = 18.
- Multiply these two, hence 17*18 = 306.
- Now place 25 ahead of this 306. So 175^2 = 30625
Focus here will however be only on लीलावती.
शुभमस्तु !
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