ELEMENTARY NUMBER THEORY

By DONALD RODRIGUEZ

§5 MAIN SOURCES CONSIDERED §6 SQUARES TRANSMITTED TO THE LATIN WEST CHAPTER II ORDINARY MAGIC SQUARES A SQUARES OF ODD ORDERS §1 FIRST ATTEMPTS §2 METHOD OF DIAGONAL PLACING 1 DESCRIPTION 2 DISCOVERY OF THIS METHOD 3 ANOTHER WAY 4 A RELATED METHOD 5 MODIFYING THE SQUARE’S ASPECT §3 A METHOD BROUGHT FROM INDIA §4 SEPARATION BY PARITY §5 USE OF THE KNIGHT’S MOVE §6 PRINCIPLES OF THESE METHODS B SQUARES OF EVENLY-EVEN ORDERS §7 THE SQUARE OF ORDER 4 §8 METHOD OF DOTTING §9 EXCHANGE OF SUBSQUARES §10 GENERALIZATION §11 CONTINUOUS ?LLING §12 PRINCIPLES OF THESE METHODS §14 AN OLDER METHOD §15 CROSSING THE QUADRANTS §16 DESCENT BY THE KNIGHT’S MOVE 1 SQUARE OF ORDER 4 2 SQUARES OF HIGHER ORDERS §17 FILLING PAIRS OF HORIZONTAL ROWS §18 FOUR KNIGHT’S ROUTES §19 KNIGHT’S SHUTTLE §20 FILLING BY KNIGHT’S AND BISHOP’S MOVES §21 FILLING ACCORDING TO PARITY §22 FILLING THE SUBSQUARES OF ORDER 4 PARTICULAR CASES C SQUARES OF EVENLY-ODD ORDERS §23 EXCHANGES IN THE NATURAL SQUARE §24 AN OLDER METHOD §25 PRINCIPLES OF THESE METHODS §26 METHOD OF THE CROSS §27 CONSTRUCTION OF A BORDER §28 METHOD OF THE CENTRAL SQUARE CHAPTER III COMPOSITE MAGIC SQUARES A EQUAL SUBSQUARES DISPLAYING DI?ERENT SUMS §1 COMPOSITION USING SQUARES OF ORDERS LARGER THAN 2 §2 COMPOSITION USING SQUARES OF ORDER 2 B EQUAL SUBSQUARES DISPLAYING EQUAL SUMS §3 EXAMPLES C DIVISION INTO UNEQUAL PARTS §4 CROSS IN THE MIDDLE §5 CENTRAL SQUARE CHAPTER IV BORDERED MAGIC SQUARES §1 PRELIMINARY OBSERVATIONS §2 EMPIRICAL CONSTRUCTION §3 GROUPING THE NUMBERS BY PARITY §4 PLACING TOGETHER CONSECUTIVE NUMBERS §5 METHOD OF STIFEL §6 ZIGZAG PLACING §7 MATHEMATICAL BASIS OF THESE GENERAL METHODS §8 EQUALIZATION BY MEANS OF THE ?RST NUMBERS §9 ALTERNATE PLACING §10 METHOD OF STIFEL §11 GENERAL PRINCIPLES OF PLACING FOR EVEN ORDERS §12 EQUALIZATION BY MEANS OF THE ?RST NUMBERS §13 CYCLICAL PLACING §14 METHOD OF STIFEL §15 PRINCIPLES OF THESE METHODS CHAPTER V BORDERED SQUARES WITH SEPARATION BY PARITY §1 THE MAIN SQUARE AND ITS PARTS §2 FILLING THE INNER SQUARE §3 FILLING THE REMAINDER OF THE SQUARE BY TRIAL AND ERROR A METHODICAL ?LLING OF THE OBLIQUE SQUARE §4 COMPLETING THE PLACING OF ODD NUMBERS B METHODICAL PLACING OF THE EVEN NUMBERS §5 SITUATION AFTER ?LLING THE OBLIQUE SQUARE 1 DETERMINING THE NUMBER OF CELLS REMAINING EMPTY 2 DETERMINING THE SUM REQUIRED §6 RULES FOR PLACING THE EVEN NUMBERS 1 FIRST MAIN EQUALIZATION RULE 2 SECOND MAIN EQUALIZATION RULE 3 NEUTRAL PLACINGS 1 FIRST BORDER 2 OTHER BORDERS §8 CASE OF THE ORDER N = 4T + 3 (WITH T ?1) C PARTICULAR CASE OF THE ORDER 5 CHAPTER VI MAGIC SQUARES WITH NON-CONSECUTIVE NUMBERS PARTICULAR CASE: THE GIVEN NUMBERS FORM ARITHMETICAL PROGRESSIONS 1 THE NUMBERS FORM A SINGLE PROGRESSION 2 THE NUMBERS FORM N PROGRESSIONS 3 MAGIC SQUARE WITH A SET SUM 4 MAGIC PRODUCTS GENERAL CASE: SQUARES WITH ARBITRARY GIVEN NUMBERS A SQUARES OF ODD ORDERS §1 SQUARE OF ORDER 3 1 GENERAL OBSERVATIONS 2 THE GIVEN NUMBERS ARE IN THE ?RST ROW 3 THE MIDDLE NUMBER IS IN THE MEDIAN LOWER CELL 4 THE GIVEN NUMBERS ARE IN THE DIAGONAL 5 THE GIVEN NUMBERS ARE IN THE MIDDLE ROW §2 SQUARE OF ORDER 5 1 THE GIVEN NUMBERS ARE IN THE ?RST ROW 2 THE GIVEN NUMBERS ARE IN THE SECOND ROW 3 THE GIVEN NUMBERS ARE IN THE DIAGONAL 4 THE GIVEN NUMBERS ARE IN OPPOSITE LATERAL ROWS 5 THE GIVEN NUMBERS ARE IN THE ?RST TWO ROWS §3 SQUARE OF ORDER 4 2 THE GIVEN NUMBERS ARE IN THE UPPER ROW 3 THE GIVEN NUMBERS ARE IN THE SECOND ROW 4 THE GIVEN NUMBERS ARE IN THE END CELLS OF THE ?RST ROW AND THE MEDIAN OF THE SECOND 6 THE GIVEN NUMBERS ARE IN THE DIAGONAL 7 THE GIVEN NUMBERS ARE IN THE CORNER CELLS 8 WRITING IN THE SUM AS A WHOLE A CASES OF IMPOSSIBILITY B FILLING ACCORDING TO ‘SUBSTANCE’ §4 SQUARE OF ORDER 8 A PARTICULAR CASE: METHOD OF DIVISION B GENERAL CASE C FILLING THE SECOND HALF 2 THE GIVEN NUMBERS ARE IN THE ?RST TWO ROWS, WITHIN THE QUADRANTS’ DIAGONALS 3 THE GIVEN NUMBERS ARE IN THE ?RST AND THIRD ROWS, WITHIN THE DIAGONALS 4 THE GIVEN NUMBERS ARE IN THE ?RST AND FOURTH ROWS, WITHIN THE DIAGONALS 5 WRITING IN THE GLOBAL SUM §5 SQUARE OF ORDER 6 2 THE GIVEN NUMBERS ARE EQUALLY DISTRIBUTED IN THE LATERAL ROWS §6 SQUARES OF HIGHER EVENLY-ODD ORDERS CHAPTER VII OTHER MAGIC ?GURES §1 LITERAL SQUARES 1 SQUARES OF ODD ORDERS A CASE OF A PRIME ORDER B CASE OF THE ORDER N = 9 C CASE OF THE ORDER N = 3 Z, WITH Z ?= 3 D CASE OF THE SQUARE ORDER N = Z2, WITH Z ?= 3 E CASE OF THE COMPOSITE ORDER N = T Z, WITH T, Z ODD ?= 3 2 SQUARES OF EVEN ORDERS §2 SQUARES WITH ONE EMPTY CELL 1 SQUARES OF ODD ORDERS WITH CENTRAL CELL EMPTY 3 SQUARES OF EVENLY-EVEN ORDERS §3 SQUARES WITH DIVIDED CELLS §4 MAGIC TRIANGLES §5 MAGIC CROSSES §6 MAGIC CIRCLES 1 FIRST TYPE 2 SECOND TYPE §7 MAGIC RECTANGLES 1 BOTH SIDES ARE ODD 2 BOTH SIDES ARE EVENLY EVEN 3 ONE SIDE EVENLY EVEN AND THE OTHER EVENLY ODD 4 BOTH SIDES ARE EVENLY ODD §8 MAGIC CUBES CHAPTER I INTRODUCTION § 1 DE?NITIONS A MAGIC SQUARE IS A SQUARE DIVIDED INTO A SQUARE NUMBER OF CELLS IN WHICH NATURAL NUMBERS, ALL DI?ERENT, ARE ARRANGED IN SUCH A WAY THAT THE SAME SUM IS FOUND IN EACH HORIZONTAL ROW, EACH VERTICAL ROW, AND EACH OF THE TWO MAIN DIAGONALS A SQUARE WITH N CELLS ON EACH SIDE, THUS N2 CELLS ALTOGETHER, IS SAID TO HAVE THE ORDER N THE CONSTANT SUM TO BE FOUND IN EACH ROW IS CALLED THE MAGIC SUM OF THIS SQUARE THUS THE SQUARES OF THE ?GURES 1 AND HAVE THE ORDER 6 BUT THEIR CONSTANT SUM IS DI?ERENT, WITH 111 FOR THE ?RST AND 591 FOR THE OTHER THE ?RST BEING ?LLED WITH THE ?RST 36 NATURAL NUMBERS, ITS SUM IS THE SMALLEST POSSIBLE FOR A 6 × 6 MAGIC SQUARE IN THE SECOND, THE NUMBERS ARE NOT CONSECUTIVE THOUGH BOTH ARE MAGIC SQUARES, WHAT IS USUALLY CONSIDERED IS THE ?RST KIND, THUS A SQUARE OF ORDER N ?LLED WITH THE ?RST N2 NATURAL NUMBERS IN THIS CASE THE CONSTANT SUM IS EASILY DETERMINED: SINCE THE SUM OF ALL THESE NUMBERS EQUALS N2 (N2 + 1) , THE SUM IN EACH ROW (LINE, COLUMN, MAIN DIAGONAL), THUS THE MAGIC SUM FOR SUCH A SQUARE, WILL BE MN = N(N2 + 1) 131 100 192 123 175 FIG A SQUARE DISPLAYING THE SAME SUM IN THE 2N + 2 AFORESAID ROWS IS AN ORDINARY OR COMMON MAGIC SQUARE IT MEETS THE MINIMAL NUMBER OF REQUIRED CONDITIONS, AND SUCH A SQUARE CAN BE CONSTRUCTED FOR ANY GIVEN ORDER N ?3 (THERE CANNOT BE A MAGIC SQUARE OF ORDER 2 WITH DI?ERENT DE?NITIONS NUMBERS, AS WE SHALL SEE BELOW ) BUT THERE ARE MAGIC SQUARES WHICH DISPLAY FURTHER PROPERTIES A BORDERED MAGIC SQUARE IS ONE WHERE REMOVAL OF THE SUCCESSIVE BORDERS LEAVES EACH TIME A MAGIC SQUARE (FIG 3) WITH AN ODD-ORDER SQUARE, AFTER REMOVING EACH (ODD-ORDER) BORDER IN TURN, WE SHALL ?NALLY REACH THE SMALLEST POSSIBLE SQUARE, THAT OF ORDER 3 WITH AN EVEN-ORDER SQUARE, THAT WILL BE ONE OF ORDER 4 (THE BORDER CANNOT BE REMOVED SINCE THERE IS NO MAGIC SQUARE OF ORDER 2) FOR ORDER N ?5, BORDERED SQUARES ARE ALWAYS POSSIBLE AS SEEN ABOVE, THE MAGIC SUM FOR A SQUARE OF ORDER N ?LLED WITH THE N2 ?RST NATURAL NUMBERS IS MN = N(N2 + 1) ACCORDINGLY, FOR M CELLS, THIS AVERAGE SUM SHOULD BE M TIMES THAT QUANTITY; THIS WILL BE CALLED THE SUM DUE FOR M CELLS THUS, THE INNER SQUARE OF MTH ORDER (M ?5) WITHIN A BORDERED SQUARE OF ORDER N, BEING A SEPARATE ENTITY, WILL CONTAIN IN EACH ROW ITS SUM DUE, NAMELY M(M) N = M(N2 + 1) IF THE MAIN SQUARE IS ?LLED WITH THE N2 ?RST NATURAL NUMBERS THE MAGIC SUM IN ANY INNER SQUARE WILL THEREFORE DI?ER FROM THE SUM IN THE NEXT ONE, SURROUNDING IT, BY N2 + 1 WITH THIS IN MIND, WE CLEARLY SEE THE STRUCTURE OF BORDERED SQUARES: THE ELEMENTS OF THE SAME BORDER WHICH ARE OPPOSITE INTRODUCTION (HORIZONTALLY, VERTICALLY, OR DIAGONALLY FOR THE CORNER CELLS), WHICH WE SHALL CALL COMPLEMENTS, ADD UP TO N2 + 1 IF THE MAIN SQUARE IS OF ORDER N FOR EXAMPLE, IN EACH BORDER OF FIG 3, THE SUM OF OPPOSITE ELEMENTS IS 101, AND THE MAGIC SUMS OF THE SUCCESSIVE SQUARES ARE 505, 404, 303, AND PARTICULAR CASES OF ORDINARY MAGIC SQUARES IT IS A CONDITION FOR MAGIC SQUARES THAT THE MAIN DIAGONALS WILL CON- TAIN THE MAGIC SUM BUT THERE ARE ORDINARY MAGIC SQUARES IN WHICH THE MAGIC SUM IS ALSO FOUND IN THE BROKEN DIAGONALS —THUS PAIRS OF DIAGONAL ROWS, ON EITHER SIDE OF AND PARALLEL TO A MAIN DIAGONAL, COMPRISING N CELLS ALTOGETHER SUCH SQUARES ARE CALLED PANDIAGONAL (FIG 4 WHERE, FOR EXAM- PLE, THE BROKEN DIAGONALS 47, 31, 20, , 21, 26, 42 AND 47, 59, 18, 8, , MAKE THE MAGIC SUM) THEY ARE POSSIBLE FOR ANY ODD ORDER FROM 5 ON, AND FOR ANY EVEN ORDER DIVISIBLE BY 4, FROM 4 ON SUCH SQUARES REMAIN MAGIC IF WE MOVE ANY LATERAL ROW TO THE OTHER SIDE: THE MAIN DIAGONALS OF THE NEW SQUARE WILL BE MAGIC SINCE THEY WERE ALREADY MAGIC AS BROKEN DIAGONALS BY REPEATING SUCH VERTICAL AND/OR HORIZONTAL MOVES WE ARE ABLE TO PLACE ANY CHOSEN ELEMENT IN ANY GIVEN CELL OF THE SQUARE; FOR EXAMPLE, THE CELL CONTAINING 1 CAN BE MADE TO OCCUPY ANY PLACE IN THE SQUARE THIS FEATURE SEEMS TO HAVE BEEN HIGHLY PRIZED IN THE EARLY DAYS OF MAGIC SQUARES THIS PROPERTY MAY BE SIMPLY REPRESENTED IN THE CASE OF THE SMALLEST POSSIBLE PANDIAGONAL SQUARE, THAT OF ORDER 4, ONE FORM OF WHICH IS SEEN IN FIG 5 SUPPOSING IT REPEATED IN THE PLANE, AS IN FIG 6, ANY SQUARE OF ORDER 4 TAKEN IN IT WILL BE MAGIC (AND PANDIAGONAL) CATEGORIES OF ORDER ANOTHER CLASS OF ORDINARY SQUARES IS THAT OF COMPOSITE SQUARES (FIG 7): THE SUBSQUARES ARRANGED WITHIN THE MAIN MAGIC SQUARE ARE THEMSELVES MAGIC WITH EACH HAVING ITS OWN MAGIC SUM CLEARLY, THE POSSIBILITY OF SUCH AN ARRANGEMENT DEPENDS ON THE DIVISIBILITY OF THE ORDER OF THE MAIN SQUARE MOREOVER, IT WILL APPEAR THAT THE SMALLEST POSSIBLE COMPOSITE SQUARE IS THAT OF ORDER 9 COMPOSITE MAGIC SQUARES OF LESSER ORDER ARE NOT POSSIBLE OR PARTICULAR: THAT OF ORDER 8 MAY WELL BE DIVIDED INTO FOUR SUBSQUARES, BUT, BECAUSE OF THE IMPOSSIBILITY OF A 2 × 2 MAGIC SQUARE, THEY CANNOT HAVE DI?ERENT MAGIC SUMS; AS TO THAT OF ORDER 6, ITS FOUR SUBSQUARES ?LLED WITH DI?ERENT NUMBERS COULD NOT EVEN MAKE EQUAL SUMS FIG § 2 CATEGORIES OF ORDER AS MENTIONED ABOVE, NO MAGIC SQUARE OF ORDER 2 IS POSSIBLE INDEED (FIG 8) THE CONDITIONS WOULD BE A + D = B + C = B + D = C + D = C + A = A + B INTRODUCTION FROM WHICH WE INFER THAT A = B = C = D A MAGIC SQUARE OF ORDER 2 IS THEREFORE NOT POSSIBLE WITH NUMBERS WHICH ARE TO BE DI?ERENT A B C D ? † ? ? R ? ? † ? FIG CONSIDER NOW A SQUARE OF ORDER 3 (FIG 9) SURROUNDING THE CENTRAL CELL IS A BORDER COMPRISING FOUR PAIRS OF OPPOSITE CELLS, HORIZONTALLY (?), VERTICALLY (†) OR DIAGONALLY (?, ?) CLEARLY, WITH M DESIGNATING THE MAGIC SUM AND R THE CENTRAL ELEMENT, THE SUM IN EACH PAIR OF OPPOSITE CELLS MUST EQUAL M ?R THE SUM IN THE WHOLE SQUARE MUST THEREFORE EQUAL 4 (M ?R) + R; IT ALSO EQUALS 3M EQUATING THESE TWO QUANTITIES GIVES M = 3R LET US NOW ?LL THE 3 × 3 SQUARE WITH THE NINE ?RST NATURAL NUMBERS THEIR SUM BEING 45, THE MAGIC SUM M3 WILL BE 15; THEN R = 5 AND EACH PAIR OF OPPOSITE CELLS MUST ADD UP TO M3 ?5 = 10 LET US BEGIN BY CONSIDERING THE PLACE OF 1 IF IT IS IN A CORNER, 9 WILL BE IN THE OPPOSITE CORNER AND WE SHALL NEED, TO COMPLETE THE TWO LATERAL ROWS MEETING IN 1, TWO PAIRS OF NUMBERS ADDING UP TO 14 NOW THERE IS JUST ONE SUCH PAIR AVAILABLE, 6 AND 8 THEREFORE 1 MUST NECESSARILY BE IN THE MIDDLE OF A LATERAL ROW, WITH 6 AND 8 AS ITS NEIGHBOURS WITH THESE FOUR NUMBERS IN PLACE THE REMAINDER OF THE SQUARE IS DETERMINED (FIG 10) WE COULD EXCHANGE THE PLACES OF 6 AND 8, BUT THIS WOULD JUST INVERT THE ?GURE AS A RULE, WE SHALL NOT CONSIDER AS DI?ERENT THE SEVEN SQUARES OBTAINED FROM A GIVEN ONE BY ROTATION OR BY INVERSION, AS HERE, FROM FIG 10, FIG 11-13 AND FIG 14-17 WE INFER FROM THAT AND THE ABOVE STEPWISE CONSTRUCTION THAT THERE MUST BE JUST ONE FORM OF THE MAGIC SQUARE OF ORDER BANAL TRANSFORMATIONS OF ORDINARY MAGIC SQUARES 3 ?LLED WITH THE NINE ?RST NATURAL NUMBERS THIS IS AN EXCEPTION: FOR ALL OTHER SQUARES THE POSSIBILITIES ARE NUMEROUS FIRST, EACH NUMBER I MAY BE REPLACED BY N2 + 1 ?I THE RESULT CORRESPONDS TO PLACING THE NUMBERS IN REVERSE ORDER FOR A SYMMETRICAL MAGIC SQUARE, IN WHICH THE SUM OF TWO DIAGONALLY OPPOSITE ELEMENTS IS N2+ 1 WE OMIT HERE THE PARTICULAR FEATURE OF PANDIAGONAL SQUARES SEEN ABOVE (P 3) INTRODUCTION 1, THE SQUARE WILL JUST BE ROTATED BY 180? SEE FIG 18A & B OTHERWISE, THE SQUARE WILL BE DI?ERENT, WITH SOME OF THE NUMBERS HAVING NEW NEIGHBOURS SEE FIG 19A & B FIG 18A FIG 18B FIG 19A FIG 19B SECOND, WE MAY, FOR AN EVEN ORDER, EXCHANGE THE QUADRANTS DIAGONALLY (FIG 20A & B) IF THE ORDER IS ODD, PART OF THE MEDIAN ROWS MUST BE EXCHANGED AS WELL (FIG 21A & B) ? ? ? ? III I IV II ? ? ? ? II IV I III ? ? ? ? III I IV II † × ? ? ? ? ? II IV I III × † ? FIG 20A FIG 20B FIG 21A FIG 21B THIRD, CONSIDERING TWO PAIRS OF ROWS, WITH BOTH THE HORIZONTAL AND VER- TICAL ONES AT THE SAME DISTANCE FROM THEIR MEDIAN AXIS, WE MAY EXCHANGE THEM IN TURN (FIG 22A & B) × † ? ? † × FIG 22A FIG 22B THE MAGIC PROPERTY WILL HOLD SINCE LINES AND COLUMNS KEEP THEIR ELEMENTS, HISTORICAL OUTLINE AND DIAGONALS AS WELL FOR A SYMMETRICAL SQUARE, WE MAY PERFORM JUST ONE EXCHANGE OF TWO SYMMETRICAL ROWS: THE TWO PAIRS OF DIAGONALLY OPPOSITE ELEMENTS WILL MOVE TO THE OTHER DIAGONAL, BUT THIS WILL NOT CHANGE THE SUM IN THE DIAGONALS SINCE EACH SUCH PAIR ADDS UP TO N2 + 1 APPLYING THE SECOND TRANSFORMATION TO BORDERED SQUARES WILL NOT PRE- SERVE THEIR CHARACTER AS TO THE ?RST, IT WILL MODIFY THE PLACES OF ELEMENTS WITHIN THE BORDER ROWS (FIG 23A & B) NOW THIS IS A BANAL CHANGE: WE MAY ARBITRARILY MODIFY THE PLACE OF ELEMENTS WITHIN THE BORDER ROWS OF SUCH A SQUARE PROVIDED EACH OPPOSITE UNDERGOES THE SAME CHANGE FIG 23A FIG 23B § 4 HISTORICAL OUTLINE WHEN THE SCIENCE OF MAGIC SQUARES BEGAN IS NOT KNOWN NO ANCIENT GREEK TEXT DEALING WITH MAGIC SQUARES IS STILL EXTANT, NO ALLUSION WHAT- SOEVER IS MADE TO THEM IN ANTIQUITY, AND THE ONLY GREEK TEXT PRESERVED IS A BYZANTINE WORK BY MANUEL MOSCHOPOULOS (CA 1300), WHO APPEARS TO DRAW HIS KNOWLEDGE FROM SOME ARABIC OR PERSIAN SOURCE 2 FURTHERMORE, ALLUSIONS TO GREEK STUDIES BY LATER ARABIC AUTHORS SEEM UNRELIABLE, SUCH AS THE ATTRIBUTION OF THE DISCOVERY OF MAGIC SQUARES TO ARCHIMEDES BY THE 13TH-CENTURY GEOGRAPHER QASW¯IN¯I3 DESPITE THE FACT THAT THE LIST OF ARCHIMEDES’ WORKS IS WELL KNOWN SINCE GREEK TIMES STILL, IT NOW APPEARS CERTAIN THAT THERE EXISTED GREEK BOOKS ON THE SUBJECT, AND AT A REMARKABLE STAGE OF ADVANCEMENT THIS WE KNOW FROM A TEXT ATTRIBUTED TO THE PROLI?C TRANSLATOR OF GREEK WORKS TH¯ABIT IBN QURRA (836-901) HE TELLS US AT THE BEGINNING HOW HE BECAME ACQUAINTED WITH THE SCIENCE OF MAGIC SQUARES, OBVIOUSLY HARDLY KNOWN AT THAT TIME, AND THE DI?CULTY HE HAD TO INTERPRET THE TEXT OF THIS (APPARENTLY ANONYMOUS) WORK 4 MY ?RST ACQUAINTANCE WITH THE SUBJECT OF MAGIC SQUARE WAS THE 2 MOSCHOPOULOS’ TEXT HAS BEEN EDITED AND TRANSLATED BY TANNERY; ON ITS ORIGIN, SEE OUR STUDY OF IT 3 SEE WIEDEMANN’S BEITRAG V, PP 451-452 4 THIS AND OTHER PARTS OF TH¯ABIT’S TRANSLATION ARE FOUND IN THE ANONYMOUS MS INTRODUCTION 3 × 3 ?GURE MENTIONED BY NICOMACHOS IN THE ARITHMETIC 5 THEN AB¯U AL- Q¯ASIM AL-H IJ¯AZ¯I CAME ACROSS A 4×4 SQUARE, (?LLED WITH THE NUMBERS) FROM 1 TO 16 WITH CONSTANT DI?ERENCE, WHICH CAUSED HIS ADMIRATION 6 NEXT I FOUND A 6 ×6 ?GURE ON THE BACK OF EUCLID’S BOOK (OF THE ELEMENTS) IN THE HANDWRITING OF ISH ¯AQ 7 AFTERWARDS I CAME ACROSS A BOOK CONTAINING THREE OR FOUR ?GURES LESS THAN (THE ORDER) 10 THEN I FOUND IN THE ROYAL LIBRARY (IN BAGHDAD) TWO BOOKS, FOR THE GREATER PART DAMAGED BY INSECTS SO THAT ONE COULD UNDERSTAND JUST A SMALL PART OF THEM; THE NOTICE ABOUT THEM WAS IN AL-M¯AH¯AN¯I’S HANDWRITING AND THE ?RST PAGE FOR THE MOST PART IN H USAYN IBN M¯US¯A AL-NAWBAKHT¯I’S HANDWRITING 8 THEN I EXAMINED THEM, FOUND ELUCIDATING THEM VERY LABORIOUS, AND IT SEEMED TO ME THAT IT SHOULD BE POSSIBLE TO MAKE SENSE OF THOSE PARTS WHICH HAD BEEN DAMAGED IN ONE BY WHAT HAD BEEN PRESERVED IN THE OTHER WITH TH¯ABIT’S TRANSLATION GENERAL METHODS FOR CONSTRUCTING BORDERED AND COMPOSITE SQUARES MAY THUS BE CONSIDERED AS AVAILABLE IN THE LATE NINTH CENTURY FOR ORDINARY SQUARES, HOWEVER, WE DO NOT KNOW OF ANY ANCIENT SOURCE, AND THE DOCUMENTS WE HAVE ARE ALL, AS IT SEEMS, ORIGINALLY LONDON BL DELHI ARABIC 110, A 12TH-CENTURY COMPILATION OF WORKS BY VARIOUS AUTHORS FROM THE 9TH TO 12TH CENTURIES FOR THE ARABIC TEXT OF THIS PASSAGE, SEE FOL 60V IN THIS MS OR BELOW (P 281), APPENDIX 1 (NOTE OUR USE OF SLANTED WRITING, NOT ITALICS, THROUGHOUT FOR QUOTATIONS ) 5 NICOMACHOS’ INTRODUCTION TO ARITHMETIC (??????????? ????????), WRITTEN TOWARDS THE END OF THE ?RST CENTURY, IS —AS INDICATED BY ITS TITLE— AN ELEMENTARY INTRODUCTION TO NUMBER THEORY (SUCH IS INDEED THE GREEK MEANING OF ‘ARITHMETIC’) IT DOES, THOUGH, DEPART FROM THE SOUND MATHEMATICAL WORKS OF THE CLASSICAL GREEK PERIOD IN THAT ITS AUTHOR TENDS AT TIMES TO TAKE HIS CONJECTURES FOR THEOREMS STILL, THE INTRODUCTION WAS ONE OF THE MAIN CHANNELS OF TRANSMISSION OF GREEK MATHEMATICS, AND FOR EARLY MEDIAEVAL EUROPE THE ONLY ONE, NAMELY THROUGH THE LATIN ADAPTATION BY BOËTIUS, WHILE IN THE EAST IT WAS TRANSLATED TWICE: INTO SYRIAC AND, BY TH¯ABIT IBN QURRA ALSO, INTO ARABIC (ED KUTSCH) THERE IS NO MENTION OF A MAGIC SQUARE IN IT (OR IN THE ARABIC TRANSLATION), BUT A 3 × 3 SQUARE MAY HAVE APPEARED AS A MARGINAL ADDITION IN A RECENT STUDY, N VINEL HAS QUOTED (PP 551-552) A FRAGMENT BY IAMBLICHOS, A 4TH-CENTURY COMMENTATOR ON NICOMACHOS, WHICH SEEMS TO ALLUDE TO THE 3 × 3 MAGIC SQUARE 6 AB¯U’L-Q¯ASIM AL-H IJ¯AZ¯I WAS A THEOLOGIAN OF THE ?RST HALF OF THE 9TH CENTURY THE DESCRIPTION BEING RATHER NAIVE, TH¯ABIT MUST BE QUOTING HIM HIS NAME AND THOSE OF THE OTHER SCHOLARS APPEAR IN THE FIHRIST OF IBN AL-NAD¯IM, WRITTEN TOWARDS THE END OF THE 10TH CENTURY, WHICH CONTAINS IN PARTICULAR A LIST OF ARABIC BOOKS OR TRANSLATIONS 7 ISH ¯AQ IBN H UNAYN, A CONTEMPORARY OF TH¯ABIT IBN QURRA, TRANSLATED THE ELEMENTS, A TRANSLATION THEN REVISED BY TH¯ABIT HIMSELF 8 MUH AMMAD IBN ‘¯IS¯A IBN AH MAD AL-M¯AH¯AN¯I WAS A PERSIAN MATHEMATICIAN AND ASTRONOMER WORKING IN BAGHDAD AROUND THE MIDDLE OF THE 9TH CENTURY; H ASAN (AS COMMONLY) IBN M¯US¯A AL-NAWBAKHT¯I WAS A THEOLOGIAN AND PHILOSOPHER OF THE SECOND HALF OF THE 9TH CENTURY, PERSONALLY ACQUAINTED WITH TH¯ABIT (FIHRIST, ED FLÜGEL ET AL , I, 177, 11) INCIDENTALLY, IBN AL-NAD¯IM ALSO MENTIONS THE DAMAGE CAUSED BY ‘INSECTS’ (OR TERMITES: ARAD A) TO MANUSCRIPTS (I, 243, 27) HISTORICAL OUTLINE ARABIC GENERAL METHODS ARE SEEN TO APPEAR IN THE LATE TENTH OR EARLY ELEVENTH CENTURY FOR ODD AND EVENLY-EVEN ORDERS, AND THEN, AT THE END OF THE ELEVENTH, FOR EVENLY-ODD ONES (WITH THE EVENLY-ODD ORDER PRESENTING MOST DI?CULTY, SUCH A STEPWISE DEVELOPMENT WAS ONLY TO BE EXPECTED ) THIS THEN LEAVES OPEN THE QUESTION OF THE EXISTENCE OF ORDINARY MAGIC SQUARES IN GREECE GIVEN THE SUPERIOR LEVEL OF THE GREEK METHODS REPORTED IN TH¯ABIT’S TRANSLATION, IT WOULD BE SURPRISING, TO SAY THE LEAST, TO ?ND THE GREEKS NOT ACQUAINTED WITH THEIR CONSTRUCTION MY SUGGESTION OF TWENTY YEARS AGO MAY STILL BE CONSIDERED: THAT THIS SCIENCE REMAINED MORE OR LESS CON?NED TO (NEO-PYTHAGOREAN?) CIRCLES, JUST AS, MORE THAN A THOUSAND YEARS BEFORE, THE TEACHING OF THE PYTHAGOREAN SCHOOL HAD BEEN RESTRICTED TO ITS MEMBERS, AND ONLY LATER BEGAN TO SPREAD 9 THIS, INCIDENTALLY, MIGHT EXPLAIN WHY SO LITTLE OF THIS GREEK SCIENCE HAS BEEN TRANSMITTED ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? RETURNING TO ARABIC TIMES, WE DO SEE CONTINUOUS DEVELOPMENT IN THE 11TH AND EARLY 12TH CENTURIES, WITH THE DISCOVERY OF VARIOUS OTHER CON- STRUCTIONS FOR ORDINARY MAGIC SQUARES, SOMETIMES PANDIAGONAL AS WELL SQUARES ?LLED WITH NON-CONSECUTIVE NUMBERS ALSO BEGIN TO APPEAR THE ORIGIN OF THAT HAS TO DO WITH THE ASSOCIATION OF ARABIC LETTERS WITH NUMER- ICAL VALUES —AN ADAPTATION OF THE GREEK NUMERICAL SYSTEM (FIG 24); THIS ADAPTATION APPEARED IN EARLY ISLAMIC TIMES, BEFORE THE ADOPTION OF THE IN- DIAN NUMERALS, BUT REMAINED IN USE LATER SINCE TO EACH LETTER OF A WORD, OR TO EACH WORD OF A SENTENCE, WAS ATTRIBUTED A NUMERICAL QUANTITY, THUS TO THE WORD OR THE SENTENCE A CERTAIN SUM, THIS WORD OR SENTENCE COULD BE 9 LES CARRÉS MAGIQUES, P 270; MAGICHESKIE KVADRATY, PP 277-278 INTRODUCTION THE ?GURE OF JUPITER IS SQUARE, FOUR BY FOUR, WITH 34 ON EACH SIDE IF YOU WISH TO OPERATE WITH IT: MAKE A SILVER PLATE IN THE DAY AND THE HOUR OF JUPITER, PROVIDED JUPITER IS PROPITIOUS,11 AND ENGRAVE UPON IT THE ?GURE; YOU WILL FUMIGATE IT WITH ALOES WOOD AND AMBER WHEN YOU CARRY IT WITH YOU, PEOPLE WHO SEE YOU WILL LOVE YOU AND YOU WILL OBTAIN FROM THEM WHATEVER YOU REQUEST IF YOU PLACE IT IN THE STORE-HOUSE OF A MERCHANT, HIS TRADE WILL INCREASE IF YOU PLACE IT IN A DOVECOTE OR IN A HIVE, A ?OCK OF BIRDS OR A SWARM OF BEES WILL GATHER THERE IF SOMEONE UNLUCKY CARRIES IT, HE WILL PROSPER AND BE ALWAYS MORE SUCCESSFUL IF YOU PLACE IT IN THE SEAT OF A PRELATE, HE WILL ENJOY A LONG PRELATURE AND WILL NOT FEAR HIS ENEMIES, BUT BE SUCCESSFUL AMONG THEM THE ?GURE OF MARS WHEN UNFAVOURABLE MEANS WAR AND EXACTIONS IT IS A SQUARE ?GURE, ?VE BY ?VE, WITH 65 ON EACH SIDE IF YOU WISH TO OPERATE WITH IT, TAKE A COPPER PLATE IN THE DAY AND HOUR OF MARS WHEN MARS IS DECREASING IN NUMBER AND BRIGHTNESS, OR MALE?C AND RETROGRADING, OR IN 10 THIS AND OTHER EXAMPLES IN OUR MAGIC SQUARES FOR DAILY LIFE FOR THE ORIGINAL LATIN TEXT, SEE BELOW, APPENDIX 2 THE TWO MAGIC SQUARES REPRESENTED ARE THOSE OF OUR ?GURES 27 AND 29 BELOW 11 THUS ON THURSDAY, 1ST AND 8TH HOURS OF THE DAY AND 3RD AND 10TH OF THE NIGHT, AND WITH JUPITER ON A DIRECT COURSE AND INCREASING IN BRIGHTNESS HISTORICAL OUTLINE ANY WAY UNFAVOURABLE, AND ENGRAVE THE PLATE WITH THIS ?GURE; AND YOU WILL FUMIGATE IT WITH THE EXCREMENT OF MICE OR CATS IF YOU PLACE IT IN A NEW BUILDING, IT WILL NEVER BE COMPLETED IF YOU PLACE IT IN THE SEAT OF A PRELATE, HE WILL SU?ER DAILY HARM AND MISFORTUNE IF YOU PLACE IT IN THE SHOP OF A MERCHANT, IT WILL BE WHOLLY DESTROYED IF YOU MAKE THIS PLATE WITH THE NAMES OF TWO MERCHANTS AND BURY IT IN THE HOUSE OF ONE OF THEM, HATRED AND HOSTILITY WILL COME BETWEEN THEM IF YOU HAPPEN TO FEAR THE KING OR SOME POWERFUL PERSON, OR ENEMIES, OR HAVE TO APPEAR BEFORE A JUDGE OR A COURT OF JUSTICE, ENGRAVE THIS ?GURE AS SAID ABOVE WHEN MARS IS FAVOURABLE, IN DIRECT MOTION, INCREASING IN NUMBER AND BRIGHTNESS; FUMIGATE IT WITH ONE DRACHMA (= 8 OUNCE) OF CARNELIAN STONE IF YOU PUT THIS PLATE IN A PIECE OF RED SILK AND CARRY IT WITH YOU, YOU WILL WIN IN COURT AND AGAINST YOUR ENEMIES IN WAR, FOR THEY WILL ?EE AT THE SIGHT OF YOU, FEAR YOU AND TREAT YOU WITH DEFERENCE IF YOU PLACE IT UPON THE LEG OF A WOMAN,12 SHE WILL SU?ER FROM A CONTINUOUS BLOOD ?OW IF YOU WRITE IT ON PARCHMENT ON THE DAY AND THE HOUR OF MARS AND FUMIGATE IT WITH BIRTHWORT AND PLACE IT IN A HIVE, THE BEES WILL ALL ?Y AWAY IT WAS THUS THE ARRIVAL OF SUCH TEXTS IN LATE MEDIAEVAL EUROPE WHICH ?RST AROUSED INTEREST IN, AND LED TO THE STUDY OF, SUCH SQUARES THERE (THE ?GURES OF THESE SQUARES ARE GIVEN BELOW, § 6) THIS INCIDENTALLY EXPLAINS THE USE OF THE TERM ‘MAGIC’ —FORMERLY ALSO ‘PLANETARY’, WHICH WE ?ND STILL EMPLOYED BY FERMAT 13 THE ORIGINAL ARABIC (PERHAPS ORIGINALLY GREEK) DENOMINATION, ‘HARMONIOUS ARRANGEMENT OF NUMBERS’ (WAFQ AL-A‘D¯AD), WHICH HAD A MORE MATHEMATICAL CONNOTATION, REMAINED UNKNOWN, AS WELL AS THE VARIOUS CONSTRUCTIONS ABUNDANTLY DESCRIBED IN ARABIC AND PERSIAN MANUSCRIPTS WE MAY INFER FROM THIS THAT ‘MAGIC’ APPLICATIONS DO NOT SEEM TO HAVE PLAYED A SIGNI?CANT RÔLE AT THE OUTSET AS THE ORIGINAL NAME INDICATES, SUCH SQUARES WERE THEN BEING STUDIED AS A BRANCH OF NUMBER THEORY, JUST AS WERE THE PERFECT AND AMICABLE NUMBERS SO HIGHLY ESTEEMED BY PYTHAGOREANS AND THEIR FOLLOWERS IN LATE ANTIQUITY IN SHORT, SUCH RE- SEARCH WAS AT THE TIME CONSIDERED AS BEING PERFECTLY SERIOUS AND HAD NOT YET BEEN TAINTED WITH THE SAD REPUTATION LATER ACQUIRED THROUGH POPULAR USE OF THESE SQUARES FINALLY, WE MAY NOTE THAT MAGIC SQUARES ARE ALSO FOUND ELSEWHERE THE MAGIC SQUARE OF ORDER 3 APPEARS IN CHINA AT THE BEGINNING OF OUR ERA; TO OUR KNOWLEDGE, HIGHER-ORDER SQUARES DO NOT OCCUR THERE BEFORE THE 12 REVERTING TO EVIL USES 13 VARIA OPERA MATHEMATICA, P 176; OR ŒUVRES COMPLÈTES, II, P 194 INTRODUCTION ANOTHER FAMOUS MATHEMATICIAN WAS AB¯U ‘AL¯I AL-H ASAN IBN AL-HAY- 14 EDITED IN OUR MAGIC SQUARES IN THE TENTH CENTURY 15 THUS IN OUR EDITION WE COULD ONLY SURMISE THAT THE WHOLE OF IT WAS TAKEN FROM ANCIENT GREEK SOURCES (ABOVE, P V) OUR A II 7 - A II 35 ARE FOUND ON FOL 80V-84V OF THE LONDON MANUSCRIPT DELHI ARABIC 110, WHILE A II 44 - A II 54 OCCUR ON FOL 99R- 100V, FOLLOWED BY LARGER SQUARES NOT REPRODUCED BY ANT ¯AK¯I (FOL 102V-107R) OTHER SOURCES ARE ALSO USED IN THIS MANUSCRIPT, INCLUDING THREE OF THE AUTHORS MENTIONED BELOW (B¯UZJ¯AN¯I, IBN AL-HAYTHAM, AS?Z¯AR¯I); THIS MAY ACCOUNT FOR SECTIONS OMITTED FROM TH¯ABIT’S TRANSLATION, WITH THE AUTHOR SELECTING HIS EXCERPTS 16 THE MOST SIGNI?CANT PARTS ARE FOUND IN OUR MAGIC SQUARES IN THE TENTH CENTURY; FOR THE COMPLETE EDITION, SEE OUR TRAITÉ D’AB¯U’L-WAF¯A’ HIS COMPATRIOT JAM¯AL AL-ZAM¯AN ‘ABD AL-JABB¯AR KHARAQ¯I (D 1138/9) DOES NOT CLAIM ANY ORIGINALITY, BUT HIS TREATISE IS THE ?RST TO TEACH GENERAL METHODS FOR ALL THREE TYPES OF THE TWO MAIN CATEGORIES OF MAGIC SQUARES, THUS NOW INCLUDING THE PREVIOUSLY OMITTED CASE OF ORDINARY SQUARES OF EVENLY-ODD ORDERS HIS TEXT THUS SHOWS THAT THE PROBLEM OF CONSTRUCTING 17 PUBLICATION OF THIS SOURCE IN OUR UNE COMPILATION ARABE THIS SOURCE HAS OTHER EXCERPTS THAN THE LONDON MANUSCRIPT (ABOVE, N 4) 18 SEE OUR UN TRAITÉ MÉDIÉVAL 19 OUR L’ABRÉGÉ ENSEIGNANT LA DISPOSITION HARMONIEUSE DES NOMBRES 20 OBTAINING THIS SEPARATION IS EASY WITH ORDINARY SQUARES —UNLIKE IN THE CASE OF BORDERED ONES ALL THESE METHODS WILL BE DISCUSSED BELOW INTRODUCTION ANOTHER PERSIAN AUTHOR WRITING ABOUT MAGIC SQUARES IN THE NEXT CEN- TURY IS ‘ABD AL-WAHH¯AB IBN IBR¯AH¯IM ZANJ¯AN¯I HIS SHORT TREATISE IS LIMITED TO THE CONSTRUCTION OF BORDERED SQUARES AS WELL AS THAT OF ORDER 4 CON- TAINING IN ITS UPPER ROW FOUR GIVEN NUMBERS OR, IF THIS ARRANGEMENT IS NOT POSSIBLE, THEIR SUM ITS IMPORTANCE IS NOT DUE TO ANY ORIGINALITY BUT TO ITS SUCCESS, AS WITNESSED BY THE NUMBER OF MANUSCRIPT COPIES EXTANT ZANJ¯AN¯I HIMSELF TELLS US IN THE INTRODUCTION THAT HE WROTE IT AT THE REQUEST OF SOME FRIENDS, WHO JUST WANTED TO KNOW THE RULES FOR CONSTRUCTING SQUARES OF ANY ORDER 22 THIS INDEED ILLUSTRATES THE LATER CHARACTERISTICS OF TREATISES ON MAGIC SQUARES: WRITTEN IN RESPONSE TO PUBLIC DEMAND, THEY ARE TO TEACH METHODS AIMING AT RESULTS, WITHOUT BOTHERING THE READER WITH QUESTIONS OF FUNDAMENTS OR FEASIBILITY THE ?RST TRACE OF MAGIC SQUARES IN THE WEST IS THE TREATISE OF THE BYZANTINE MANUEL MOSCHOPOULOS, WRITTEN AT THE VERY BEGINNING OF THE 14TH CENTURY (ABOVE, P 8) HE CONSTRUCTS ORDINARY SQUARES OF ODD ORDERS, OF EVENLY-EVEN ORDERS (TAKEN BY HIM, AS IT SEEMS, TO MEAN N = 2K), BUT NONE OF EVENLY-ODD ORDERS IT WOULD SEEM THAT HE ATTEMPTED TO RECON- STRUCT THE METHODS FROM EXAMPLES OF SQUARES HE SAW IN SOME MANUSCRIPT, EITHER PERSIAN OR ARABIC THE SAME HAPPENED LATER, FROM THE 14TH CENTURY ON, IN WESTERN EU- ROPE, WHEN THE ?RST EXAMPLES OF SQUARES ARRIVED THROUGH LATIN TRANSLA- TIONS OF MAGIC TEXTS (SEE ABOVE, PP 11-12, AND BELOW, § 6) THEY CONTAINED TWO SETS OF SEVEN SQUARES ATTRIBUTED TO THE SEVEN PLANETS THEN KNOWN (IN THE GEOCENTRIC SYSTEM, THUS INCLUDING SUN AND MOON), AND THE ONLY EX- PLANATIONS WERE, AS WE HAVE SEEN, ABOUT THEIR MAGIC USE NO ARABIC TEXT DESCRIBING METHODS WAS THEN TRANSLATED, OR STUDIED, AND THE ?RST ATTEMPTS TO EXPLAIN OR GENERALIZE THEM WERE BASED SOLELY ON THESE EXAMPLES WITH THESE ?RST STUDIES ARE ASSOCIATED THE NAMES OF CORNELIUS AGRIPPA, GIRO- LAMO CARDANO AND CLAUDE-GASPAR BACHET DE MÉZIRIAC IN THE 16TH AND EARLY 17TH CENTURIES A CURIOUS EXCEPTION IS THE CASE OF MICHAEL STIFEL (1487-1567), WHO GIVES INSTRUCTIONS FOR CONSTRUCTING BORDERED SQUARES OF ALL ORDERS;23 THERE IS NO MENTION OF ANY SOURCE, NOR DOES HE GIVE HIMSELF ANY CREDIT, AND HIS METHODS ARE NOT EXACTLY LIKE THE ONES WE KNOW FROM ARABIC SOURCES MEANWHILE, TREATISES EXPOUNDING THE CONSTRUCTION OF MAGIC SQUARES 21 SEE OUR HERSTELLUNGSVERFAHREN III 22 SEE HERSTELLUNGSVERFAHREN II, II ? 23 IN HIS ARITHMETICA INTEGRA, FOL 24V-30R THE 3×3 SQUARE OF FIG 25 APPEARS BOTH AS SUCH AND IN INVERTED (RIGHT- TO-LEFT) FORM AS TO THE 4 × 4 SQUARE, THAT OF FIG 26 APPEARS MOSTLY IN THE INVERTED FORM WHILE THE SECOND (WHICH IS THE PREVIOUS ONE ROTATED AND WITH MEDIAN COLUMNS SWITCHED) WILL BE USED BY DÜRER IN HIS MELENCOLIA IN ORDER TO DATE IT (1514) THE TWO 5 × 5 SQUARES OF FIG 28 AND 29 RESULT FROM WELL-KNOWN CON- STRUCTIONS, AS WE SHALL SEE LATER (II, §§ 2 AND 4) HERE THEY ARE, HOWEVER, QUITE DI?ERENT IN ASPECT, FOR THE SECOND SQUARE SEPARATES THE ODD NUMBERS FROM THE EVEN ONES, ALL PLACED IN THE CORNERS 24 T AW¯ALI‘ AL-ISHR¯AQ F¯I WAD ‘ AL-AWF¯AQ, MS PARIS BNF ARABE 2698 25 MS 65496 OF THE LONDON SCHOOL OF ORIENTAL AND AFRICAN STUDIES 26 THEIR OCCURRENCES IN MANUSCRIPTS IS GIVEN IN FOLKERTS’ ZUR FRÜHGESCHICHTE INTRODUCTION THE CON?GURATION OF THIS 7×7 SQUARE (FIG 32), ALSO FOUND IN INVERTED FORM, IS THE MOST COMMONLY SEEN IN MEDIAEVAL ARABIC TEXTS (II, § 2) 27 IN THE WAY DESCRIBED IN OUR LES CARRÉS MAGIQUES DANS LES PAYS ISLAMIQUES, P (RUSSIAN EDITION, PP 266-267) FINALLY, THE 9 × 9 SQUARE OF FIG 35 IS ALSO VERY COMMON, AND EASY TO CONSTRUCT (II, § 2)