Sunday, October 26, 2014

Introductory rites and rubrics of the Quddasa of Mar Addai and Mar Mari



The Quddasa has a specific set of rites and rubrics as an introduction apart from treating the first G’hantha cycle as an introduction. This is sandwiched between the first and the second G’hantha cycles. It comes with two very important rites viz., the exchange of peace and dialogue prayer, the former is the traditional inaugural rite of every Mar Toma Nazrani liturgical ceremony and the latter is in all Eucharistic liturgies.

Kai-kasthoori – Exchange of Peace

The celebrant turns towards the congregation, blesses them with the sign of the Cross and says, “Peace be with you.” (Jn 20:19) This is immediately after the qanona of the first G’hantha cycle. The congregation replies, “with you and with your Spirit” (Gal 6:18; Phil 4:23) while drawing the symbol of the Cross on themselves. The first deacon then approaches the entrance of the Mad’baha. The celebrant goes up to him, places his folded palms into the open palms of the decaon. Both of them withdraw their palms, bring them close their own lips and kiss them in the folded position. These set of actions is known as kai-kasthoori. The deacon then turns towards the congregation and announces, “my brethren, give peace to one another in the love of Christ.”

The celebrant then gives kai-kasthoori to the Archdeacon or to the one close to him in the Mad’baha. As a chain this is repeated among the others. Simultaneously, the first deacon gives kai-kasthoori to the second deacon and both of them go to their respective sides in the Haikala and give kai-kasthoori to the first ones in the first rows. It then is continued as a chain reaching the last ones in the last rows. In some places, the deacons give kai-kasthoori to each of the first ones in each of the rows in either sides of the Haikala. Kai-kasthoori is a beautiful external expression of the Christian unity among the entire gathering in the church (1 Cor 10:17). This also helps the people to forget all ill-feelings and to forgive the evil-doings of others (Mt 5:23-24). Moreover, this helps them to seek forgiveness from their brethren (1 Cor 11:27-28). In the beginning and at the end of each of the Liturgies of the Hours and in ecclesiastical gatherings, Mar Toma Nazranis give kai-kasthoori.

Diptychs

Diptychs is the book that contains the names of all those who are to be specially remembered during the Holy Qurbana. Deacon reads out from the diptychs the names of those who are dead and living while the congregation continue with the exchange of peace. It is to be remembered that no special intentions are read out at this time. As the Holy Qurbana is for “our life and for the peace of the world, and for the crowning of the year that it may be blessed and filled with abundance through God’s goodness,” neither at the beginning of the Holy Qurbana nor during its celebration, particular intentions are placed. Moreover, it is to be noted that the Holy Qurbana is not to be traded for temporal favours, gifts and goodness.

Deacon now helps the congregation to prepare themselves physically and mentally for the participation in the “awe-inspiring Mysteries that are being sanctified” by making a detailed announcement. He says that through the intercession of the celebrant peace is to be flourished in the world. The community well bound by the exchange of peace, is further made aware that the ultimate peace is through God, and the Holy Qurbana is the perfect instrument for that in the world. The deacon calls for thankful and prayerful mind, pure and contrite heart, attentive and reverential disposition, closed eyes and opened heart and internal and external silence from each of the participants. He concludes his announcement by declaring that, “peace be with us.” This assures the community of the flourishing of peace among them through the Holy Qurbana.

Unveiling the Mysteries

The celebrant prays in a low voice while the deacon makes the announcement. The celebrant reiterates his weak and frail nature and seeks God’s blessings to proceed with the celebration. He thanks God for sending His Grace on him for making him “worthy to offer before Him the Holy Qurbana which is a living and holy sacrifice.” Through this prayer he reaffirms that the Holy Qurbana is for the “praise of the Holy Trinity and for the sanctification of the whole congregation.”

The celebrant takes the soseppa that veiled the Body and Blood, rolls it and keeps it around the Body and Blood. As soseppa symbolizes the stone that covered the Holy Sepulchre, the above action symbolizes the Great Resurrection itself (Jn 20:1). Soseppa also is the symbol of the sacred linen cloth that covered the Body of Jesus when he was buried. Hence, the additional significance of the above action is the rolled up line cloth at the place of the burial of Jesus (Jn 20:5-9).

The celebrant then prays with deep gratitude, “O Lord, by Your grace, You have made me worthy of Your Body and Blood.” He then devotedly implores for the blessings to “be present before God with confidence on the day of judgement.” The celebrant generously incenses the Eucharistic gifts and the altar which now displays the resurrection scene.

Pauline Salutation

The formal commencement of the Quddasa of Mar Addai and Mar Mari is with the recitation of the Pauline Salutation (2 Cor 13:14). The celebrant recites it with a loud voice and blesses the Eucharistic gifts by the sign of the Cross. The most important significance of the usage of the Pauline Salutation is the open proclamation that the Quddasa is through and through an act of the Holy Trinity.

The Dialogue Prayer

A dialogue type prayer is seen in almost all Eucharistic liturgies irrespective of the tradition. Here, at the commencement of the Quddasa of Mar Addai and Mar Mari too, there is this dialogue prayer. The celebrant reminds the congregation that “their minds must on high.” To this, the congregation reply that “it is to God who is the God of the virtuous and venerable patriarchs that the minds must be raised.” The celebrant then tells that “the Holy Qurbana, the most fitting thanksgiving, is offered to the same God, who is the Lord of all.” The community then affirms that “it is the most righteous and just act of human beings.” The dialogue prayer assures the participation of the congregation in the celebration of the Holy Qurbana.  It also ensures the congregation that they are part and parcel of the celebration, symbolizing the communitarian dimension of the Holy Qurbana.

The deacon recites “peace be with us” and with this, the introductory rites and rubrics of the Quddasa of Mar Addai and Mar Mari come to an end.

Joseph Varghese Kureethara CMI
FrJoseph@ChristUniversity.in

Sunday, October 19, 2014

Square Numbers



Natural numbers are numbers that are used for counting. Among the natural numbers there are some numbers called power numbers. They are nothing but natural numbers which can be written as the product of a number with itself two or more times. Number 1 is multiplied with itself any number of times would give only 1. Therefore, 1 is a trivial power number. Power number is also understood as a number n with the equation xk=n that has the integer solution of for some positive integer k>1.
Number 2 is the number that follows number 1 among the natural numbers. We can easily see that 2 cannot be written as product of identical numbers. Next natural number 3 also has the similar property. Both 2 and 3 are also the first two prime numbers. A prime number is a number that cannot be written as the product two distinct numbers other than 1 and itself.

Power Numbers
But the next natural number 4 is a power number. It is the product of 2 with itself. In connection with the concept of prime numbers, we can say that a power number is a composite number with identical factors. By inspection we can see that among the single digit numbers, only 1, 4, 8 and 9 are power numbers. Power numbers are very rare among the natural numbers. There are only 8 two-digit power numbers and 28 three-digit power numbers.  Among the entire 9000 four digit numbers there are only 84 power numbers. Strange though, the percentage of power numbers below is very meagre for large values of n. For example, there are only .366% of power numbers among all numbers with at most five digits.
Range
Number
%
Range
Number
%
1-9
4
44.44
1-9
4
44.44
10-99
8
8.89
1-99
12
12.12
100-999
28
3.11
1-999
40
4.4
1000-9999
84
.93
1-9999
124
1.28
10000-99999
242
.269
1-99999
366
.366

Square Numbers
Among the single digit power numbers 1, 4 and 9 are called square numbers. A square number is the product of an integer with itself.  The term ‘square’ is attributed to these numbers because of the fact that it is the area of a square with an integer side length.  Number 8 is a cube number because it is the product of 2 three times.

Square Number and the last digit
Let us see the some of the properties of square numbers. First eleven square numbers are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121 ... These square numbers help us in observing a salient feature of square numbers. That is, a square number will never end in 2, 3, 7 or 8. This is a fundamental property of the square numbers.
Among the first 10 square numbers, it can be observed that square of both 1 and 9 end in 1. Similarly squares of 2 and 8 end in 4, squares of 3 and 7 end in 9 and that of 4 and 6 end in 6. It is interesting to note that 1+9=2+8=3+7=4+6=10. Hence, we can conclude that square of a number of the form 10kn, 1n4, ends in 1, 4, 9 and 6 respectively whereas the square of a number that ends in 0 or5 will always end in 0 and5, respectively. It is interesting to note that among these end digits 5 and 6 are not square numbers whereas all others are.

Square Numbers and the last two digits
Now, let us look at the properties of the last two digits of a square number. Though 1, 5, 6 and 9 can be the end digits of square numbers, if they are repeated, then they cannot be the end digits of square numbers. That is to say that only 00 and 44 can be the last two digits of square numbers. Among the first 100 numbers there are exactly 22 numbers that can be the end digits of square numbers. They are 00, 01, 04, 09, 16, 21, 24, 25, 29, 36, 41, 44, 49, 56, 61, 64, 69, 76, 81, 84, 89 and 96.
There are some interesting properties for these numbers. The numbers 00, 16, 44 and 69 in the reverse order also can be the end digits of square numbers. Sum of two single digit numbers can have any value from 0 to 18. But, the sum of the last two digits of square numbers can never be 2, 14, 16 and 18. Hence, similar to that of single digit ends of square numbers, we have a result.
Given below are the types of numbers whose squares have particular end digits. Here k is any nonnegative integer.
n
Last digits of n2

n
Last digits of n2
10k
00

10k
00
10k∓5
25

10k∓5
25
50k∓1
01

50k∓1
01
50k∓2
04

50k∓2
04
50k∓3
09

50k∓3
09
50k∓4
16

50k∓4
16
50k∓11
21

50k∓6
36
50k∓18
24

50k∓7
49
50k∓23
29

50k∓8
64
50k∓6
36

50k∓9
81
50k∓21
41

50k∓11
21
50k∓12
44

50k∓12
44
50k∓7
49

50k∓13
69
50k∓16
56

50k∓14
96
50k∓19
61

50k∓16
56
50k∓8
64

50k∓17
89
50k∓13
69

50k∓18
24
50k∓24
76

50k∓19
61
50k∓9
81

50k∓21
41
50k∓22
84

50k∓22
84
50k∓17
89

50k∓23
29
50k∓14
96

50k∓24
76

Algorithm to find the square
There is an Indian short cut to find the square of a number.  Though the algorithm looks tedious and lengthy, we can write the square in a line, if we master the algorithm.
Let dkdk-1...d4d3d2d1 be a k-digit number.
Step 1: Take the square of the last digit i.e., d1. Last digit of it is the last digit of the square.  Keep the carryover, if any.
Step 2: Multiply d2 with d1 and take twice of it and add the carryover from the previous step. The last digit of this number is the last but one digit of the square. Keep the carryover, if any.
Step 3: Take the square of d2. Multiply d3 with d1 and take twice of it. Add these two sums with the carryover from Step 3. The last digit of this number is the e last but two digit of the square. Keep the carryover, if any.
Step 4: Multiply d4 with d1, d3 with d2 and take twice of both and add the carryover from the previous step. The last digit of this number is the last but three digit of the square. Keep the carryover, if any.
Continuing in this manner, in 2k-1 steps, we get the required square.

Triangular Numbers and Square Numbers
Square numbers have a fascinating link to triangular numbers. A triangular number is the sum of first n consecutive natural numbers. They are 1, 3, 6, 10, 15 ...  It is striking to note that any square number is the sum of two consecutive triangular numbers i.e., 1=1+0, 4=3+1,  9=6+3, 16=10+6, ... This leads us to conclude that twice the sum of first n natural numbers added to n+1 is a square number. i.e., if Tn is the nth triangular number, then 2Tn+ (n+1) and 2Tn-n are square numbers.

Parity of Numbers and Square Numbers
Alas! We have another result. Difference of two consecutive square numbers is always an odd number.  Therefore, we can have the sequence of odd numbers as the difference of square numbers viz., 1-0,  4-1, 9-4, 16-9, 25-16, ... In general, the nth odd number, say, On=n2-(n-1)2.
Similarly, even numbers also can be found from square numbers. The parity of square numbers is same as the parity of natural numbers. Hence, the difference between every pair of alternate square numbers is as an even number. Moreover, they are all multiples of 4. Halving them we get the sequence of even numbers. Quartering them we get the natural numbers. That is to say that for any natural number n, (n+1)2-(n-1)2=22n.  Therefore, n= [(n+1)2-(n-1)2]/22.

Prime Numbers and Square Numbers
For any given number, we know that the largest non-trivial factor it can have is its square root. Hence, in finding the factors of a number one needs to check up to its square root only. Are square numbers some way helpful to know about the distribution of prime numbers? It is easy to see that for any given n, the number of prime numbers less than n is much larger than the number of square numbers. We have seen earlier square numbers alternates parity. This gives us some clue. Take only the even square numbers. Its immediate number is likely to be a prime number.  That is, the sequence, an= (2n)2+1, where n does not end in 1, 4, 6 or 9, has plenty of prime numbers.  Some of the initial members of the sequence are 17, 37, 101, 197, 257, 401, 577, 677, 901, 1157, 1297, 1601,  ... Some remembrances of the sequences of Mersenne Primes and Fermat Primes.
For training in mathematics research, the collection of square numbers is a good area. Following questions are worth searchable for beginners in number theory research. Is the number of k-digit powers numbers always an even number? Is there a better connection between square numbers and prime numbers? What are the properties of the last k-digits of power (square) numbers? Various proofs for such properties would really be a hard test. How far the properties of power numbers help us in solving the integer factorization problem? To sum up, one can escalate one’s mathematical powers by doing research on power numbers and can become a square personality mathematically.