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SOLVING SANGAKU : A TRADITIONAL SOLUTION TO A NINETEENTH CENTURY JAPANESE TEMPLE PROBLEM Rosalie Joan Hosking Abstract. This paper demonstrates how a nineteenth century Japanese votive temple problem known as a sangaku from Okayama prefecture can be solved using traditional mathematical methods of the Japanese Edo (1603-1868 CE). We compare a modern solution to a sangaku problem from Sacred Geometry: Japanese Temple Problems of Tony Rothman and Hidetoshi Fukagawa with a traditional solution of Ōhara Toshiaki (?-1828). Our investigation into the solution of Ōhara provides an example of traditional Edo period mathematics using the tenzan jutsu symbolic manipulation method, as well as producing new insights regarding the contextual nature of the rules of this technique.
1. Introduction During the Japanese Edo period (1603-1868 CE), a unique practice of creating votive tablets with mathematical content emerged. These tablets - known as sangaku 算額 - contained a variety of geometrical problems and were dedicated to Shinto shrines and Buddhist temples throughout the country. This tradition remained largely hidden to historians until the publishing of Japanese Temple Problems: San Gaku by David Pedoe and Fukagawa Rothman in 1989 CE. This work provided a collection of problems taken directly from various sangaku, along with a selection of solutions using modern mathematical methods and terminology. This was followed up with the more extensive Sacred Mathematics - Japanese Temple Geometry by Tony Rothman and Fukagawa Hidetoshi in 2008 CE, which provided further historical and contextual information regarding the sangaku tradition. However, the work of Pedoe, Rothman and Hidetoshi has been criticised for its strictly modern approach to these problems. One critic, Peter J Lu, writes that overall their work "achieves only limited success in showcasing sangaku as exemplars of a uniquely Japanese style of mathematics, because that style is never elucidated" [10]. This paper aims to directly respond to such concerns by demonstrating how a nineteenth century Japanese sangaku problem solved by Rothman and Hidetoshi with modern methods from the Katayamahiko shrine 2010 Mathematics Subject Classification. A01A27. Key words and phrases. sangaku, wasan, tenzan jutsu, boshō hō, Sanpō Tenzan Shinan, Katayamahiko, Ōhara Toshiaki. 1
in Okayama prefecture can be solved using traditional methods of the Japanese Edo period. The traditional solution used is taken from the work of Ōhara Toshiaki 大 原利明 (?-1828), a mathematician from Umeda village in the Adachi ward of modern day Tokyo [6]. In 1810 CE, Ōhara produced the Sanpō Tenzan Shinan 算法点竄指南 Instruction in Tenzan Mathematics, a three volume work which provided solutions to a number of geometrical problems in mathematics using the traditional manipulation technique known as tenzan jutsu 点竄術. The third volume of this work refers to Ōhara as the master of the Ōhara school of mathematics, indicating that he was a mathematical instructor. The method provided by Ōhara is transcribed and translated in section 4.2. Our investigation into the style of calculation used by Ōhara also provides insight into the contextual nature of calculation in the Edo period, as the use of certain methods appears to have been influenced by the mathematical school one belonged to. 2. The Sangaku Tradition Sangaku first appeared in the Edo period, which started when Ieyasu Tokugawa became the military leader in 1603 CE. The new Tokugawa feudal regime instigated many changes, including the national seclusion policy - sakoku which was enforced in 1639 CE. This policy saw scientific works on mathematics, astronomy, and geography become included on lists of banned books [18]. Texts on mathematics reduced in volume due to the ban, prompting the Japanese to begin producing their own works on the subject. This led to the development of a uniquely Japanese mathematical tradition known as wasan. Out of this tradition sprung sangaku, where san 算 means calculation and gaku 額 plaque. These tablets were made from solid wood, and came in a variety of sizes. An examination of 250 sangaku listed on the Sangaku website of Hiroshi Kotera [9] indicates these tablets had an average width of 153 cm and average height of 69.4 cm. While there are few mentions of the specific type of wood used for sangaku, the Kanchobetsu Kampo Shuroku refers to a sangaku made from hinoki, indicating this tree was used for their canvas [7]. Sangaku are believed to have been influenced by a pre-existing tradition of hanging tablets in Shino shrines known as ema 絵馬 – where e 絵 means picture or painting and ma 馬 a horse [17]. In ancient times, shrines would ritually sacrifice live horses. However, due to the expense, parishioners began to instead sacrifice paintings of horses on wooden boards as a substitute [8]. During the Muromachi (1333 – 1573 CE) period, larger ema began to appear which depicted famous battles, people, or scenes. To house these larger and more artistic tablets, shrines and temples created special areas within their grounds known as emaden or emadō 絵馬堂 which functioned as informal public art galleries. In the literature, historians such as Shimodaira have described sangaku as "mathematical ema" [19] due to their similarities in canvas and
location. For example, sangaku are commonly found either in the emaden of shrines, on the eaves of the shrine or temple halls, or situated attached to a smaller shrine located within the larger complex. Ema were also often made from hinoki [23], and could be over one yard in length and width [15]. While sangaku may have borrowed their size and location from ema, their content was vastly different. Sangaku presented mathematical problems which examined the geometry of certain configurations of figures. For example, one might be given a large circle containing five smaller circles of varying sizes and asked to find the value of one of these smaller circles given the diameter of another. These problems ranged in difficulty, and their subject matter included results similar to the Malfatti circles (see [2]), Casey's theorem [3], and Soddy's hexlet theorem [22] which appeared on sangaku prior to being known in Europe [17]. These problems were elaborately decorated using pigment paints. Sangaku usually presented their problems in three different sections: (1) Problem: Introductory section describing the diagram and indicating which figure from the diagram the observer needs to find the value of. (2) Answer: Section giving either the numerical value of the sought figure or advising to refer to the next section. (3) Formula: Section which provided a formula for obtaining the solution. While sangaku usually provided a formula for finding the solution, they did not provide any working, leaving the observer to find the solution on their own. With regard to numbers and labelling, sangaku employed the traditional Japanese base ten counting system. The numbers from one to ten in the Japanese style are listed in the right-hand column of Table 1. Kanji
Kanji Japanese
甲 乙 丙 丁 戊 己 庚 辛 壬 癸
kō 1st 一 ichi otsu 2nd 二 ni hei 3rd 三 san tei 4th 四 shi/yon bo 5th 五 go ki 6th 六 roku kō 7th 七 shichi/nana shin 8th 八 hachi jin 9th 九 kyū ki 10th 十 jyū Table 1. Common Labels Used in Wasan
Reading one two three four five six seven eight nine ten
In this system the largest value is placed before the smaller. For instance, 15 is written jyu go 十五, literally 'ten five'. A value such as 20 is expressed ni jyu 二十 - 'two ten'. For values in the hundreds and thousands, hyaku 百 and sen 千 are used in a similar manner.
To label figures and represent unknowns, characters from the Chinese calender such as kō 甲, otsu 乙, hei 丙, and tei 丁 (shown in the left-hand column of Table 1) would often be used. These characters were treated as ordinal numbers similar to a, b, c, d, etc or first, second, third, etc [4]. 3. Tenzan Jutsu One of the major acheivements of Edo period was the development of a symbolic manipulation technique known first as bōsho hō 傍書法 and later as tenzan jutsu. Sometimes referred to as the method of side-writing [12], it was developed to solve equations which had multiple unknowns. The bōsho hō is accredited to the mathematician Seki Takakazu 関孝和 (? - 1708). The system used vertical lines and kanji characters to represent unknowns and express operations. As the Edo period progressed, the method was later renamed tenzan jutsu 点竄術 in the work of the mathematician Matsunaga Yoshinsuke 松永良弼 (1693-1744), who was a second generation pupil of the mathematial school founded by Seki [21]. In tenzan jutsu, vertical lines were used to represent numerical values. For example, | 甲 (where as mentioned 甲 kō is a character used to represent the first in a series of figures or unknowns. See Table 1) could be used to express 1 甲, || 甲 to express 2 甲, and so on. Negation was shown by a line being crossed through a vertical line. To represent fractions, the numerator would be represented by the kanji character to the right of the line and the denominator by a character on the left, such as 乙 | 甲 for 甲 ÷ 乙. In general, the blank space separating values indicates addition. For example, in Figure 1, the space between | 甲 and 2 | 乙 indicates the calculation | 甲 + 2 | 乙. In our transcriptions, we translate 甲 as a, 乙 as b, 丙 as c, 丁 as d, 戊 as e, and so on. Figure 1 displays how tenzan jutsu could express the calculation a + 2b − c. 甲
乙 2 b
Ec E E
(+) a
b 2
Figure 1. Left: Original tenzan jutsu calculation. Middle: Transcription. Right: Translation.
3.1. Rules of Tenzan Jutsu In the Sanpo Tenzan Shinan, Ōhara includes a set of rules instructing the reader how to manipulate the symbols and lines of the tenzan jutsu system to solve a variety of geometrical problems. We have examined and translated the introduction section of the Sanpo Tenzan Shinan to produce the following summary of the rules of tenzan jutsu.
Figure 2. Original text from the Sanpo Tenzan Shinan explaining the 'convert' rule 自乘 Self Multiplication The 'self multiplication rule' 自乘 is a means of squaring. For example, applying the rule to 2a produces 4a2 . When there are two terms, squaring and expansion of the square occurs. For example, applying the rule to a + b produces a2 + 2ab + b2 . 括之 Put Together The 'put together' rule 括之 is used to combine elements in various ways. Firstly it can be used to combine terms listed separately with different lines in the tenzan system to bring them under one line. For example, say we have the following addition. The rule is also used for combining like terms in the operation equivalent to factorisation. 解之 Splitting
The 'splitting' rule has the opposite function of the put together rule. It allows combined terms to be split into individual terms. For instance, if we have a(b + c), the rule changes this into (a · b) + (a · c). Splitting is also the term used when substitution occurs. When a variable has been defined by a set of terms, the use of the splitting rule indicates that variable is to be decomposed into the terms that make it up. For example, say we have y + c = x and know that x = a + b. The splitting rule indicates we can substitute the value of a + b for x and change this expression into y + c = a + b. 遍省過乗 Eliminate Surplus Factors When we have an equation such as 3x(a + b + c) − 6ab = 0, where all terms have a common factor of 3, this rule indicates we are able to eliminate multiplication by three to produce x(a + b + c) − 2ab = 0. 同加異減 Add Same Subtract Different This rule states that when we have two terms with similar signs, we can add them, but for different signs we subtract. For example, when there are two similar terms with positive signs - such as +2ab + 2ab - they are combined to form 4ab. This also applies for two negative values, such that −2ab − 2ab become −4ab. However when there are similar terms which have different signs - such as +a2 − a2 - we subtract. For example, say we have (a + b)2 − (a − b)2 which produces a2 + 2ab + b2 − a2 + 2ab − b2 . These can be grouped into similar terms to form (+a2 − a2 ), (+2ab + 2ab) and (+b2 − b2 ). 變換 Conversion and 開平方商變換 Square Root Converstion The 'conversion' rule was used to convert a value into an equivalent value of a different form. In the Sanpo Tenzan Shinan the following example, referring to a right-angle triangle, is given: short side · long side = hypotenuse · altitude a2 − b2 = (a + b)(a − b) The 'square root conversion' rule is used to change a square root into an expanded and equivalent form For √ example,√the rule can be √ √ √using binomials. √ used to convert √ a into a( a + x)( a − x). As ( a + x)( a − x) =√1, this is equivalent to a. This rule is used when another binomial of either a − x √ or a + x appears in the other terms of the equation. By having the binomial as common to all terms, it can be eliminated from the equation. The following conversions are presented to the reader: √ √ √ √ √ √ 1 = ( 2 − 1)( 2 + 1) = ( 2 − 1)2 ( 2 + 1)2 = ( 3 − 2)( 3 + 2) √ √ = ( 5 − 2)( 5 + 2) √ √ √ √ 2 = ( 2 − 1)2 ( 2 + 2)2 = ( 3 − 1)( 3 + 1) √ √ √ √ √ 2 = ( 2 − 1)( 2 + 2) = ( 2 − 2)2 ( 2 + 1)2 √ √ 4 = ( 5 − 1)( 5 + 1)
The additional conversions are also given, but they are stated to have a positive value when the values are negative: √ √ √ √ 2 = ( 2 − 2)( 2 + 2) = ( 3 − 2)( 3 + 1)2 √ √ √ 2 = ( 2 − 2)( 2 + 1) 乗除括之 Multiplication and Division Together and 加減括之 Addition and Subtraction Together With the 'multiplication and division together' rule, when there is one fraction in an equation the rest of the equation is put over a common denominator. a2 a2 2ab b2 It converts an expression such as + 2a + b into + + . b b b b The rule of 'addition and subtraction together' allows for a negative term to be created in an equation with only positive terms. For instance, say we have a2 + ab + b2 . This then has a negative term added by converting ab into 2ab − ab to give a2 + 2ab − ab + b2 . This then becomes (a + b)2 − ab. Another example of this is a2 + 2ab + b2 , which can be converted into a2 + 4ab − 2ab + b2 and then (a − b)2 + 4ab. 4. Katayamahiko Sangaku
Figure 3. Sangaku at Katayamahiko Shrine. (Image by author).
The Katayamahiko sangaku was dedicated in October of 1873 CE to the Katayamahiko shrine 片山日子神社 located in the small town of Osafune in Okayama prefecture. It contains sixteen different problems, all of which are beautifully painted and follow the common problem-answer-technique format. In Sacred Mathematics - Japanese Temple Geometry, fifteen of the sixteen problems are described with modern mathematics. The problem we examine is the fourteenth problem, located sixth from the left on the second row on the tablet (see Figure 3). A close up of the diagram is shown in Figure 4. This problem has been chosen because as will be shown, there are interesting features in the traditional solution which differ to modern approaches and provide unique insight into the wasan tradition.
Figure 4. The Katayamahiko problem diagram. (Image by author). The problem text has been translated and transcribed below: Transcription
今有如圖方内隔斜容甲乙円各 As in the diagram, there is a square segmented 二個只云甲円径一寸問乙円径 by diagonal lines which contains two circles kō 甲 幾何 and otsu 乙. Say the diameter of circle kō 甲 is 1 sun. Problem - what is the diameter of circle otsu 乙? 答云 乙円径五分八厘五毛
Answer: The diameter of circle otsu 乙 is 5 bu 8 rin 5 mo
術日 置二個開平方以域二個 Technique: Put 2 ko and take the square root. 余乘甲円径得乙円径合問 Subtract 2 ko. Multiply the remainder by the diameter of circle kō 甲. Obtain the diameter of circle otsu 乙 as required.
4.1. Modern Solution A modern translation of the Katayamahiko problem can be found on page 100 of Sacred Mathematics - Japanese Temple Geometry. Rothman and Hidetoshi describe the problem as such: [T]wo circles of radius r are inscribed in a square and touch each other at the centre. Each of two smaller circles with radius t touches two sides of the square as well as the common tangent between the two larger circles. Find t in terms of r. [17]
Figure 5. Diagram similar to that provided by Rothman and Fukagawa in Sacred Mathematics - Japanese Temple Geometry to illustrate their solution to the problem. They provide a diagram similar to that shown in Figure 5 and the following solution: [W]e see that, √ on the one hand, the length of the central diagonal is 2r+2 2r. On the √ other hand, it is also equal to 2P T +2r. However √ P T = t(1 + 2). Equality the two expressions gives: √ 2 t= √ r = (2 − 2)r = 0.585786r. [17] 2+1 This solution finds the length of the diagonal of the square in Figure 5 in two different ways. The first is in terms of the larger circles (labelled r in Figure 5) and uses the diagonal which runs from the bottom left-hand corner of Figure 5 to the top right-hand corner. The second is in terms of both the smaller circles (labelled t) and larger from the diagonal which runs from the top left-hand corner of the diagram to the bottom right-hand corner. These values - which
are equivalent - are then rearranged and used to find the value of the smaller circles in terms of the larger.
Figure 6. Original text from the Sanpo Tenzan Shinan showing the calculation using tenzan jutsu. 4.2. Traditional Solution of Ōhara Toshiaki As noted, sangaku, while providing a formula, did not provide any working, making it difficult for modern historians to ascertain how they were solved. To overcome this issue and locate traditional methods which could be applied to the Katayamahiko problem, we investigated a number of texts created by wasan practitioners. This research uncovered the Sanpo Tenzan Shinan of Ōhara Toshiaki, which provided instructions on how to solve problems traditionally using the tenzan jutsu method. Additionally, this text also provided a solution to a problem almost identical to that found on the Katayamahiko tablet. In this section, we detail the solution provided by Ōhara. We present a transcription of the original text below in the left column, a translation using modern symbols in the middle column, and translation in modern notation in
the right. An image of the original text can be seen in Figure 6. The problem reads:
今有如圖方内隔斜容甲 As in the diagram, there is a square segmented 乙圓各二個只云甲圓径 by diagonal lines which contains two circles kō 甲 一寸問乙圓径幾何 and otsu 乙. Say the diameter of circle kō 甲 is 1 sun. Problem - what is the diameter of circle otsu 乙? 答云 乙圓径五分八厘 Answer: The diameter of circle otsu 乙 is 5 bu 8 五毛 rin 5 mo 術日 置二個開平方以 Technique: Put 2 ko. Take the square root. 域二個余乘甲圓径得 Subtract 2 ko. Multiply the remainder by the 乙圓径合問 diameter of circle kō 甲. Obtain the diameter of circle otsu 乙 as required.
Given Instructions:
Figure 7. Left: Diagram from the Sanpo Tenzan Shinan. Right: Transcription.
[Let 甲 be represented as a, 乙 as b, and 方斜 (the diagonal) as x.] 解日置乙乘方斜率加甲乙和
Solution: Put 乙 [b] and multiply by the diagonal square ratio.1. Add 甲 [a] plus 乙 [b]
乙二 商
b 2 √
甲 乙 和
a b +
√ 2·b
(+) a + b = x
寄左 ⃝ 又置甲乘方斜率加甲 Move left. Furthermore, put 甲 [a] and multiply by the square diagonal ratio and add 甲 [a].
甲二 商
a 2 √
√ 2·a
(+) a = x
與寄左相消求矩合 And move left and cancel to find equation.
乙二 商
b 2 √
√ 2·b
1The diagonal square ratio refers to
√ 2.
甲 乙 和
a b +
E E a 2 √ E E E
E E a E E E 0
(+) a + b
√ − 2·a
−a = 0
異域括之 Different terms put together. 二乙 商 加 一
2 b √
E E 2 a E√ E E 0
√ b( 2 + 1)
+ 1
−a ·
√ 2=0
變換 Convert. 二乙 商 加 一
2 b √ + 1
√ b( 2 + 1)
2 2 a √ √ − 1
√ √ √ −a( 2(( 2 − 1)( 2 + 1)))
遍省過乗得 Using elimination of higher power terms many times obtain.
2 2 a E √ √ √ √ E −a( 2( 2 − 1)) = 0 E E− E1 0
解之 Splitting.2
E E a E E E
- 2a
√ √ 2( 2−1)) becomes
2We are to separate/expand the terms with a common factor. So −a(
√ −2a + a 2.
二甲 商 合矩定
2 a √
√ (+) a 2 = 0
如定例求得乙径歸除式 As per the typical technique for obtaining the diameter of 乙 [b] using the division formula.3
二甲 商 得乙径式
E E 2a E 甲 E a √ E E E E E E Can find b formula
−2a +
√ 2a = b
故絶答術則如左 Therefore absolute answer technique [the answer is] shown left. 5. Solution Comparison It will be observed that both the solution of Fukagawa and Rothman and that of Ōhara rely on finding two different ways of interpreting the diagonal (line x in Figure 6) and then using this information to find the size of the small circles (b) in terms of the large (a). In Sacred Mathematics - Japanese Temple Geometry, where 2r from Figure 5 is represented as a and 2t as b, Rothman and Fukagawa give √ √ a + 2a = b( 2 + 1) + a (1) √ 2 (2) b= √ ·a 2+1 √ However, to get to the end result of (2 − 2)a additional steps are required. The denominator must be rationalized, which can be done by multiplying the √ denominator and numerator by 2 − 1 as follows √ √ 2( 2 − 1) √ (3) ·a b= √ ( 2 + 1)( 2 − 1) 3What this division formula refers to is not at this point entirely clear.
(4) (5)
√ √ 2( 2 − 1) b= ·a √1 b = (2 − 2)a
Though similar, in Ōhara's solution the creation of the fraction in√ (2) is completely √ √ avoided by the use of the 'convert' rule which multiplies 2a by ( 2 + 1)( 2 + 1) to give √ √ (6) b( 2 + 1) − 2a = 0 √ √ √ √ b( 2 + 1) − 2a( 2 + 1)( 2 − 1) = 0 (7) √ √ (8) b − 2a( 2 − 1) = 0 √ While Ōhara incorrectly√rearranges this to give −2a + 2a = b, the correct rearranging produces 2a − 2a = b which in turn can be simplified as the same √ (2 − 2)a = b we find in the solution of Rothman and Fukagawa. An examination of the problems of the three volumes of the Sanpo Tenzan Shinan √ shows that √ the 'convert' rule can be used whenever a binomial of the form a − x or a + x appears - unless it is in the denominator or part of the final answer. In the Sanpo Tenshoho Shinan 算法天生法指南 of Aida Yasuaki 会田安明 (1747–1817), written in 1810 CE, a similar set of rules for tenzan jutsu is given to the reader. However, while the lists are very similar, the 'convert' rule of the Sanpo Tenzan Shinan is absent from Aida's instructions. This indicates that tenzan jutsu did not have a standard set of rules. As mentioned, Ōhara was the master of the Ōhara school of mathematics, and Aida himself founded a school of mathematics called the Saijyo after falling out with the master of the Seki Takakazu school, Fujita Sadasuke 藤田定資 (1734 - 1807) [20]. As each of these masters presented slightly different rules for calculation with tenzan jutsu, it seems that the calculation style used by practitioners was to some degree dependent upon the school they were part of and the texts they had access to. 6. Conclusions While the modern and traditional methods have the same approach to the solution, the calculation steps are√different as Ōhara uses the 'convert' rule to 2 avoid creating the fraction b = √ · a used by Rothman and Hidetoshi. 2+1 By examining and comparing these modern and traditional solutions to the Katayamahiko sangaku problem, it can be seen that while the modern method provides a correct answer, it does not provide any insight into the original calculation methods of the Japanese and the way in which they would have approached the problem. By investigating an original method, we uncover something interesting about the wasan tradition as a whole, as we discover that calculation methods could differ from school to school, meaning tenzan jutsu not being standardised.
In conclusion, careful attention must be given to finding and translating original solutions to sangaku in the literature in order to accurately represent and understand the methods used in the Edo period. Acknowledgement We would like to thank Tomohiro Uchiyama, Jia-Ming Ying, Clemency Montelle, John Hannah, and Hidetoshi Fukagawa for their help and input with translations and calculations. Sincerest gratitude is also expressed for the Katayamahiko shrine in Osafune for allowing their sangaku to be visited and photographed. Lastly the author thanks both the Waseda University Library and Hiroshi Kotera who provided online access to the original mathematical texts used in this research. References [1] Y. Aida, Sanpo Tenshoho Shinan, http://www.wasan.jp/archive/tenseihosinan1.pdf, 1810. 会田安明, 算法天生法指南, 1810. [2] O. Bottema, The Malfatti problem, Forum Geometricorum 1 (2001), 43-50. [3] J. Casey, A Sequel to Euclid, Hodges, Figgis, and Co., Dublin, 1881. [4] N. Dershowitz and E. M. Reingold, Calendrical Calculations, Cambridge University Press, New York, 2008. [5] H. Fukagawa and D. Pedoe, Japanese Temples Geometry Problems: San Gaku, Charles Babbage Research Foundation, Winnipeg, 1989. [6] A. Hirayama, Gakujutsu o Chūshin to Shita Wasan Shijō no Hitobito, Tokyo: Fuji Tanki Daigaku Shuppanbu, 1965. 平山諦, 学術を中心とした和算史上の人々, 富士短期大学出版 部, 1965. [7] Kanchobetsu Kampo Shuroku, Gihōdō, 1994. 防衛庁防衛年鑑刊行会, 防衛年鑑, 防衛年鑑 刊行会, 1994. [8] E. Kiritani, Vanishing Japan: Traditions, Crafts and Culture, Tuttle Publishing, Singapore, 1995. [9] H. Kotera, Japanese Temple Geometry Problem Sangaku, http://www.wasan.jp. [10] P. J. Lu, The Blossoming of Japanese Mathematics, Nature 454 (2008). [11] M. Morimoto and T. Ogawa, Mathematical Treatise on Technique of Linkage: An Annotated English Translation of Takebe Katahiro's Tetsujutsu Sankei, SCIAMVS 13 (2012), 157-286. [12] M. Morimoto, The Suanxue Qimen and Its Influence on Japanese Mathematics, Seki, Founder of Modern Mathematics in Japan. A Commemoration on His Tercentenary, E. Knobloch, H. Komatsu, and D. Liu (eds), Springer Proceedings in Mathematics and Statistics 39 (2013), 119-132. [13] T. Ōhara, Sanpo Tenzan Shinan, 1810. Waseda University Kotenseki Sogo Database. 大 原利明, 算法点竄指南, 1810. 早稲田大古典籍総合データベース. [14] I. Reader, Letters to the Gods: The Form and Meaning of Ema, Japanese Journal of Religious Studies 18(1) (1991), 23-50. [15] J. Robertson, Ema-gined Community: Votive Tablets (ema) and Strategic Ambivalence in Wartime Japan, Asian Ethnology 67(1) (2008), 43-77. [16] T. Rothman, Japanese Temple Geometry, Scientific American 278(5) (1997), 84-91. [17] T. Rothman and H. Fukagawa, Sacred Mathematics - Japanese Temple Geometry, Princeton University Press, Princeton, 2008. [18] S. Shio, Prohibition of Import of Certain Chinese Books and the Policy of the Edo Government, Journal of the American Oriental Society 57(3) (1937), 290-303.
[19] K. Shimodaira, Mathematics of the Japanese: Wasan, Kawade Shobō Shinsha, Tokyo, 1972. 下平和夫, 日本人の数学: 和算, 河出書房新社, 1972. [20] K. Shimodaira, Aida Yasuaki, Complete Dictionary of Scientific Biography, ed. Charles Coulsont Gillispie and Frederic Lawrence Holmes and Noretta Koertge and Thomson Gale, Charles Scribner's Sons, Detroit, 2008. [21] D. E. Smith and Y. Mikami, A History of Japanese Mathematics, Cosimo, Inc: New York, 2000. [22] F. Soddy, The bowl of integers and the hexlet, Nature 139 (1927), 77–79. [23] J. Winter, East Asian Paintings: Materials, Structures and Deterioration Mechanisms, Archetype Publications, London, 2008. Rosalie Hosking School of Mathematics and Statistics University of Canterbury Christchurch, New Zealand E-mail address:
[email protected]
J. Hist. Math. SOLVING SANGAKU : A TRADITIONAL SOLUTION TO A NINETEENTH CENTURY JAPANESE TEMPLE PROBLEM Rosalie Joan Hosking Abstract. This paper dem...
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Law Institute Journal
01 Apr 2016
Practice Direction No 1 of 2016: Correcting penalties imposed contrary to law Background. Section 104B of the Sentencing Act 1991 allows a court to reopen a proceeding to correct a sentence imposed contrary to law or which...
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From 6 December 2007 law practices whose matters are governed by the Legal Profession Act 2004 cannot use the penalty interest rate for their accounts...
For the financial year commencing 1 July 2015, the value of a penalty unit is $151.67. The value of a fee unit is $13.60 (Government Gazette SG86, 17 April 2015).
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Mandie and others v Memert Nominees Pty Ltd [2016] VSCA 4 (unreported, 5 February 2016, S APCI 2015 0097, Kyrou, Ferguson and McLeish JJA)...
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In CGU Insurance Limited v Blakely [2016] HCA 2 (11 February 2016) the High Court upheld a decision to join a third party insurer to determine the insurer's liability to indemnify a defendant. Liquidators of Akron Roads Pty Ltd...
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Magistrates Megan Aumair and Tim Gattuso were welcomed to the Magistrates' Court of Victoria on 8 February 2016. Among the speakers was LIV president Steven Sapountsis. These are edited versions of his speeches...
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01 Apr 2016
As part of the Supreme Court's 175th anniversary celebrations this month, Victoria's first solicitor will be honoured. Mystery surrounds the 1837 arrival of Victoria's first solicitor William Meek...
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The LIV is delighted to welcome the appointment of Andrea Tsalamandris as a judge of the County Court of Victoria.
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Federal Agricultural Chemicals Security Credit
Extracted from PDF file 2018-federal-form-8931.pdf, last modified October 2008
Agricultural Chemicals Security Credit8931 Form (Rev. December 2010) Department of the Treasury Internal Revenue Service Name shown on return Agricultural Chemicals Security Credit OMB No. 1545-2122 See instructions. Attach to your tax return. ▶ ▶ Attachment Sequence No. 162 Identifying number Facility Enter on the applicable line below the qualified agricultural chemicals security costs described (see instructions) . . . . 1 a Employee security training and background checks . . . . (a) e f 1b Tagging, locking tank valves, and chemical additives to prevent theft or to render chemicals unfit for illegal use . . . . . . d Perimeter protection of agricultural chemicals . . . . . . 1c 1d . Installation of security lighting, cameras, recording equipment, and intrusion detection sensors . . . . . . . . . . . 1e Implementation of measures to increase computer or computer network security . . . . . . . . . . . . . . . . 1f g Conducting a security vulnerability assessment . . . . . . 1g h . . . . . . 1h 2 Implementing a site security plan . . . . . Total qualified agricultural chemicals security costs. Add the amounts in columns (a), (b), and (c) on lines 1a through 1h . . . 2 3 Multiply the amounts in columns (a), (b), and (c) on line 2 by 30% 3 4 Maximum credit per facility. Subtract the total of the credits claimed for the facility in the 5 prior tax years from $100,000 . . 4 Agricultural chemicals security credit. Enter the smaller of line 3 or line 4 for each facility. For additional facilities, see instructions . 5 5 (c) 1a . b Limitation and prevention of access to controls of agricultural chemicals stored . . . . . . . . . . . . . . . . c (b) 6 Add the amounts for all facilities on line 5 . . . . . . . . . . . . . . 6 7 Credit from partnerships, S corporations, estates, and trusts . . . . . . . . . . . . . . 7 8 Add lines 6 and 7. Enter the result, but not more than $2,000,000. Estates and trusts, go to line 9; partnerships and S corporations, report this amount on Schedule K; all others, report this amount on the appropriate line of Form 3800 (e.g., line 1v of the 2010 Form 3800) . . . . . . . . . . 8 Amount allocated to beneficiaries of the estate or trust (see instructions) . 9 9 10 . . . . . . . . . . . . . . . Estates and trusts, subtract line 9 from line 8. Report the credit on the appropriate line of Form 3800 (e.g., line 1v of the 2010 Form 3800) . . . . . . . . . . . . . . . . . . . . . . For Paperwork Reduction Act Notice, see page 2. Cat. No. 37745A 10 Form 8931 (Rev. 12-2010) Page 2 Form 8931 (Rev. 12-2010) General Instructions Section references are to the Internal Revenue Code unless otherwise noted. Purpose of Form Use Form 8931 to claim the tax credit for qualified agricultural chemicals security costs paid or incurred by eligible agricultural businesses. All the costs must be paid or incurred to protect specified agricultural chemicals at a facility. The credit is part of the general business credit. Your deduction for otherwise deductible agricultural chemicals security costs is reduced by the amount of the agricultural chemicals security credit. The amount of the credit for any taxpayer for any tax year cannot exceed $2 million. If you have more than one facility, the costs are reported in columns (a), (b), and (c) as needed. Taxpayers (other than partnerships, S corporations, estates, and trusts) whose only source of this credit is from those pass-through entities are not required to complete or file this form. Instead, report this credit directly on Form 3800, General Business Credit. Who Can Claim the Credit An eligible agricultural business that paid or incurred qualified agricultural chemicals security costs after May 22, 2008, and before January 1, 2013, may claim the credit. This includes an eligible individual, corporation, partnership, S corporation, estate, or trust. Definitions Eligible Agricultural Business An eligible agricultural business is one that: • Sells agricultural products, including specified agricultural chemicals, at retail predominantly to farmers and ranchers; or • Manufactures, formulates, distributes, or aerially applies specified agricultural chemicals. Specified Agricultural Chemicals A specified agricultural chemical is either of the following: 1. Fertilizer commonly used in agricultural operations that is listed under any of the following. a. Section 302(a)(2) of the Emergency Planning and Community Right-to-Know Act of 1986. b. Section 101 of part 172 of title 49, Code of Federal Regulations. c. Part 126, 127, or 154 of title 33, Code of Federal Regulations. 2. Any pesticide (as defined in section 2(u) of the Federal Insecticide, Fungicide, and Rodenticide Act), including all active and inert ingredients, that are customarily used on crops grown for food, feed, or fiber. Member of a Controlled Group or Business Under Common Control For purposes of figuring the credit, all members of a controlled group of corporations (as defined in section 41(f) (5)) and all members of a group of businesses under common control (as defined in Regulations section 1.52-1 (b) through (g)), are treated as a single taxpayer. As a member, figure your credit using only your proportionate share of the $100,000 facility limit and the $2 million annual limit on the group's credit on lines 4 and 5. Attach a statement showing how your share of the amounts on these lines was figured, and write "See attached" next to the entry spaces for lines 4 and 5. Specific Instructions Line 1 Enter only costs (as described on each line) to the extent they were paid or incurred during the tax year after May 22, 2008, and before January 1, 2013, to protect specified agricultural chemicals. Line 5 If you have more than three facilities, complete and attach as many Forms 8931 as you need. Also, attach an additional Form 8931 showing the total for all the facilities and enter "Total for all facilities" at the top of the form. Line 6 Enter the total credit for all facilities. Line 9 Allocate the credit on line 8 between the estate or trust and the beneficiaries in the same proportion as income was allocated and enter the beneficiaries' share on line 9. Paperwork Reduction Act Notice. We ask for the information on this form to carry out the Internal Revenue laws of the United States. You are required to give us the information. We need it to ensure that you are complying with these laws and to allow us to figure and collect the right amount of tax. You are not required to provide the information requested on a form that is subject to the Paperwork Reduction Act unless the form displays a valid OMB control number. Books or records relating to a form or its instructions must be retained as long as their contents may become material in the administration of any Internal Revenue law. Generally, tax returns and return information are confidential, as required by section 6103. The time needed to complete and file this form will vary depending on individual circumstances. The estimated burden for individual taxpayers filing this form is approved under OMB control number 1545-0074 and is included in the estimates shown in the instructions for their individual income tax return. The estimated burden for all other taxpayers who file this form is shown below. Recordkeeping . . . . . . . . . . 4 hrs., 32 min. Learning about the law or the form . . . . . . . . . . . . 35 min. Preparing and sending the form to the IRS . . . . . . . . . . . 42 min. If you have comments concerning the accuracy of these time estimates or suggestions for making this form simpler, we would be happy to hear from you. See the instructions for the tax return with which this form is filed.
Form 8931 (Rev December 2010)
More about the Federal Form 8931 Other Tax Credit TY 2018
We last updated the Agricultural Chemicals Security Credit in February 2019, so this is the latest version of Form 8931, fully updated for tax year 2018. You can download or print current or past-year PDFs of Form 8931 directly from TaxFormFinder. You can print other Federal tax forms here.
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The Internal Revenue Service usually releases income tax forms for the current tax year between October and January, although changes to some forms can come even later. We last updated Federal Form 8931 from the Internal Revenue Service in February 2019.
Form 8931 is a Federal Other form. States often have dozens of even hundreds of various tax credits, which, unlike deductions, provide a dollar-for-dollar reduction of tax liability. Some common tax credits apply to many taxpayers, while others only apply to extremely specific situations. In most cases, you will have to provide evidence to show that you are eligible for the tax credit, and calculate the amount of the credit to which you are entitled.
Historical Past-Year Versions of Federal Form 8931
We have a total of six past-year versions of Form 8931 in the TaxFormFinder archives, including for the previous tax year. Download past year versions of this tax form as PDFs here:
Form 8931 (Rev December 2010)
Form 8931 (Rev December 2010)
Form 8931 (Rev December 2010)
Form 8931 (Rev December 2010)
Form 8931 (Rev December 2010)
Form 8931 (Rev December 2010)
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Riverdale's got some family business to deal with tonight.
Now that Hiram (Mark Consuelos) and Hermione (Marisol Nichols) are out of prison and Veronica (Camila Mendes) has met her shady half-sister, the Lodge family is about to have a sit down to figure out what to do next.
Hiram, of all people, declares that it's time for no more secrets and no more lies, but yes to more rum. Apparently, according to his secret new daughter, his passion is rum! Who knew?
Hiram also reveals that Hermosa (his secret daughter's name) is sticking around, and maybe she could work at Le Bonne Nuit alongside Veronica?
We'll give you one guess as to how Veronica feels about that idea. Watch the clip above!


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What We Do at Cat Eye Girls!
Buy Old Things: Funky Junk, Vintage Elements, Industrial Kits, Whimsical Kabootles
Cat Eye Girls are generalists -- which makes our buying criteria simple: what we buy should be easily recognizable as retro, vintage, antique, primitive or industrial and manufactured prior to the 1950s.
The Cat Eye Girls are interested in everything from vintage clothing to antique pottery to 1930s quilts to depression era glassware to vintage outdoor furniture to concrete garden art to primitive toys.
As buyers we are fair and transparent. If you have a box, closet, garage or house full of things that you no longer love or no longer feel compelled to store, please call us. We are at your service!
Teresa Mitchell (Agent 99)
We routinely buy:
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- Cat eye glass frames from the 1950s and 60s (with or without jewels).
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- Antique horse tack and cowboy habiliments (parade bridles, spurs, chaps).
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- Depression era glassware.
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- Kitchenware (Fireking, Pyrex, Bauer, green and red handled utensils, cast iron).
- Furniture (retro, antique, funky junk with peeling paint, vintage).
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- Textiles (1930s quilts, 1950s tablecloths, tatting and fine doilies).
- Americana (48 star flags, posters, banners).
- Baskets (Oak, splint, Indian).
- Rusty junk.


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Iq intelligent quattro
Intelligent mechatronic automation & remote control systems
for hydraulic-operated vehicles, to overcome limitations of
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The participants are willing to develop a totally new automation and control system. Development of new mechatronic components will be included. The limitations of hydraulic components and systems will be eliminated, and at the same time, the advantages of hydraulics (the speed of movements and power-size ratio) can be additionally improved. Main parts of the system will be: 1. Intelligent hydraulic actuators. 2. Intelligent modular digital control system with serial intermachine data bus.
Project ID:
Start date:
Project Duration:
Project costs:
2 100 000.00€
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12 Theories About 'Game Of Thrones' Season 8 Now That The Night King Has Been Defeated
Spoilers ahead for Game of Thrones Season 8, Episode 3. During "The Long Night," Arya stabbed the Night King with her Valyrian steel dagger and the army of the dead is seemingly done with. With three episodes left in the final season, this leaves the show wide open for possibilities. Based on some Game Of Thrones Season 8 theories, fans should brace themselves for more death. And a whole lot more of Cersei.
The Battle of Winterfell had a number of casualties, but the side of the living won in the end due to a girl who does have a name, Arya Stark. While no one would say this battle that took up 82 minutes of screentime was necessarily easy, it's kind of a shock that the Night King was actually killed by the end of just the third episode this season. The White Walkers have been a threat on Game of Thrones since the very first scene of the series, but now that the Night King and his army of wights, White Walkers, and giants (among other creatures) has been seemingly defeated, what's left for the remaining characters to do?
If you're feeling lost about what will happen next, you're not alone. Here are 12 theories about the end of Game of Thrones that show there's still plenty of great TV moments in store.
Arya Will Kill Cersei
Melisandre and Arya's reunion during the Battle of Winterfell recalled their Season 3 meeting where the Red Woman noted the brown, blue, and green eyes that Arya would shut forever. While there is some debate over some character's eye colors, Arya has already killed someone with brown eyes (Meryn Trant) and someone with blue eyes (the Night King). So to truly fulfill Melisandre's prophecy, Arya just might need to kill the green-eyed Cersei and cross that name off her death list. But, as some fans have pointed out, Littlefinger had green eyes too, so this death might not be hers to take. Which while that might be a bummer for some fans, don't forget she just killed the Night King, so she got perhaps the most epic kill of the entire series.
Jaime Will Kill Cersei
Instead of Arya killing Cersei, many fans want the valonqar prophecy from the books to happen. That prophecy dictates that her younger brother will kill her and while Tyrion was the frontrunner for a while, Jaime being the one to do it feels even more right to many fans.
Sansa & Tyrion Will Return To King's Landing
Sansa's allegiance is to the North, but would she be willing to venture down South again to face off with Cersei one last time? With her former husband by her side, she might be more emboldened. Especially since fans were feeling the love between Sansa and Tyrion when they thought they would die down in the crypts together.
Daenerys Will Need Yara's Reinforcements
Theon died a hero and a good man when he attacked the Night King for Bran. But his sister Yara sat the Battle of Winterfell out after heading back to the Iron Islands. Since many members of Daenerys' army have died (including almost all of the Dothraki), the Mother of Dragons might need to recruit Yara's Ironborn forces before facing Cersei and the Golden Company. And Yara finding out how Theon died is bound to be emotional.
Arya & Gendry Will Continue Their Romance
In the preview for the fourth episode, it sure looks like Arya is kissing Gendry again. So their night-before-battle sex scene may not be a one-time thing. Based on Arya winning the whole damn battle, Gendry should be really feeling the love for his lady. And since he's a Baratheon, some fans want Gendry and Arya to rule Westeros.
Euron & Cersei Will Marry
Euron is on one knee in front of Cersei in the preview for Episode 4, so will they throw together a quick wedding? If so, Game of Thrones fans know that weddings always end in some major bloodshed.
Bronn Will Kill Jaime & Tyrion
Yes, it was only in the Season 8 premiere, but it feels like so long ago that Cersei sent Qyburn to task Bronn with killing her brothers. Yet, Bronn won't kill his two buddies just for money, right? Right?!
Cersei Will Kill Brienne
Grey Worm & Missandei Will Leave Daenerys
The North Will Fight For Daenerys
Daenerys took her eye off the Iron Throne prize and lost most of her army in the fight against the Night King. She protected the North (and the rest of Westeros), so even though the North wants their independence, they should fight for her in the battle for the Iron Throne ... especially cause she's going to need all the help she can get.
Daenerys Will Be Pregnant With Jon's Baby
Jon and Daenerys aren't on the best terms after his Targaryen confession, but they might have to maintain their (ahem) very close nephew-aunt relationship if Dany is pregnant with his baby. But also, a three-fourths Targaryen and one-fourth Stark baby would certainly help out with the whole succession question.
The Night King Will Rise Again
If you really thought the Night King fell too easily, then you may be into this terrifying theory that he will come back after Daenerys, Jon, and company beats Cersei. Game of Thrones is known to be cruel, so don't rule it out.
Some of these theories seem more legit than others, but perhaps take a lesson from what happened during "The Long Night" and don't anticipate anything fans come up with to come true. Because Game of Thrones enjoys subverting viewers' expectations.


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I am extremely proud of my children who have grown up to be outstanding, respectful, resilient and successful adults. I think that Paul and I must have done something right in raising them.
When Paul and I came together, we became an instant family and I became step mum to two boys who were reasonably young at the time. After our marriage we started our own family and had another son and then two years later, our long awaited daughter.
Our little family was complete and happy no matter what was thrown at us, we were an awesome dynamic unit.
As the years have gone by, I am very happy that all the kids have come together and become close over the years, backing and supporting each other no matter what. They know that they can call upon each other if they need anything and they often do.
As a single mum now, I am fortunate that both my children are adults although they both still live at home.
During Paul's illness all of the kids were amazing and supportive especially Sarah our daughter who was my rock when we were at our lowest and darkest moments.
I am blessed that I have an open and honest relationship with all the kids and we have some great conversations and they can ask me anything. We are still a close and tight family unit as family was extremely important to Paul and that still remains intact.
For me I am one lucky mum who has four beautiful kids and one step grandson. Paul's legacy continues in all the kids; his love of photography through Matthew, his passion for cars and racing through Garry, his love of wheeling and dealing through Brendan, and his passion for mentoring and teaching through Sarah. And most of all his positive outlook and passion for life through me.
I am truly blessed and proud to be called mum.


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|zhe jiang sheng nong ye ke xue yuan zhe jiang da xue
|zhe jiang nong ye ke xue bian ji bu
|The publication, Zhejiang Academy of Agricultural Sciences and Zhejiang University jointly sponsored a comprehensive agricultural science and technology publications, founded in 1960, is the founder of an earlier agricultural science and technology publications, but also the comprehensive Chinese Journal of Agricultural Science Core. The main new results reported agricultural research and new products at home and abroad the latest developments and practical information technology and agriculture, including grain, fruit, vegetable, microbial, animal husbandry, cotton, sericulture, stains linen and new pesticides, fertilizers and other improved varieties or flat products introduced, high-quality, high yield, high yield (including aquaculture), and agricultural products processing and utilization technologies. Publications home and abroad over the authority of Papers Abstracts gathers a large number of institutions included in the country (except Hong Kong, Macao and Taiwan outside) were issued. (Translated by machine.)


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Thanks, Jade, for the wonderful conversation about my life experiences from Military, Red Cross, Business and Instructing at Mount Royal University.
I am so happy to have joined you and to have the opportunity to share my story with your Facebook Live audience.
Originally broadcast on Facebook Live on 14 Aug 2019


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February 1, 2019
Albums – Network Removals
This is a complete update of network removals for the Albums category.
Removed From Troubles: Problem resolved (NOT open for application):
Goodbye Lullaby (Lavigne, Avril); Under My Skin (Lavigne, Avril); Songs About Jane (Maroon 5)
Characters: TV – F/U/C/O/T
This is a complete update of finished, updated, closed, overdue and troubles for the Characters: TV category.
Finished Characters: TV
Jeremiah, Joey (Degrassi Series); Ryan, Caitlin (Degrassi Series)
Updated Characters: TV
No update forms have been received since the last update.
Closed Characters: TV (Open for application)
Wilkins III, Richard aka The Mayor (Buffy the Vampire Slayer); [+] All characters (Firefly); Tudor, Princess Mary (Tudors, The); Michonne (Walking Dead, The)
Overdue Characters: TV
No fanlistings are overdue.
Overdue: Temporary Layouts
No overdue layouts since the last update.
Report Forms
No report forms have been received since the last update.
Troubled Characters: TV (Not open for application)
The following fanlistings are scheduled for removal on February 8, 2019. If your fanlisting is listed below, please fix the problem and reply to the troubles email!
Thanks to Jess and Laura for helping me check the category!
Dead Link: 404/403
Bristow, Sydney (Alias); Gibbs, Leroy Jethro (NCIS)
Inactivity: 2 months + (0 pending)
Winters, Lana (American Horror Story); [+] All Characters (Big Bang Theory, The); Calendar, Jenny (Buffy the Vampire Slayer); Hart, Maya (Girl Meets World); [+] All Characters (True Blood)
Inactivity: Pending Members Within Past 2 Months Not Added/Updated
Bradbury, Charlie (Supernatural)
Navigation: Menu Not Visible
[+] Bad Female Characters (Angel); [+] Bad Female Characters (Buffy the Vampire Slayer); Fleming, Sharona (Monk); Monk, Adrian (Monk)
Navigation: Script Errors
Keller, Jennifer (Stargate: Atlantis); Mitchell, Cameron (Stargate SG-1); Tuvok (Star Trek: Voyager)


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Teeth Cleaning Voucher Clickfunnels Dentist
Even though Clickfunnels could possibly be the most popular revenue funnel software ever, you frequently hear people speaking about earning a lot of money as a Clickfunnels affiliate. But can you? Teeth Cleaning Voucher Clickfunnels Dentist
The answer is absolutely. However, not everybody will.
It all depends on what your skillset is, and how you go about promoting it.
>Try Clickfunnels Risk Free for 14 Days<<
Clickfunnels Affiliate Promotion Approaches
In the following article, I'll lay out a few of the basic ways individuals promote Clickfunnels as an affiliate. I will discuss what works best for me, also have tried a few ways .
First, let's lay out some of the most frequent methods that are promotional.
Please be aware, this is not each the methods. But, being a part of many Clickfunnels Facebook classes, these are the methods I most commonly see mentioned.
- Facebook Advertising
- Google Adwords
- Bing Ads
- Email Marketing
- Share Funnels
- Solo Advertising
Below, we will explore each method slightly more in depth. Hopefully this might help you choose which method best contrasts with you.
Facebook Advertising
One of the most usual methods of boosting Clickfunnels as an affiliate, would be to use Facebook Ads. Matter of fact, Clickfunnels teaches you to do this in their Affiliate Bootcamp. Teeth Cleaning Voucher Clickfunnels Dentist
The majority of people who ordinarily advertise via Facebook, use one of Russell Brunson's books as the direct provide. Either Dotcom Secrets, or Expert Secrets.
Now, is that this manner effective?
For me it was not. I found that because this is educated in the bootcamp, it's somewhat saturated. Imagine tens of thousands of people, all running the exact same advertisement, for the identical product.
Basically, that is what it is. Now, some people offer a complimentary lead magnet, to obtain an email address; and at the follow up try and get the customer to take the free + transport book offer.
This proves to be a much more effective strategy, if you are wishing to go the Facebook advertisement route.
Google Adwords
Next, is another paid traffic origin; Google Adwords.
Now, a lot of people have made a whole lot of money as a Clickfunnels affiliate, with Adwords. The very popular and well-known Clickfunnels affiliate, is Spencer Mecham.
That is with good reason, seeing as he has broken 1 million bucks as an affiliate.
The other hand to Google Adwords, is how competitive it is. Especially since the release of Spencer's course Affiliate Secrets 2.0.
Because of the popularity of the course, many individuals have dove headfirst to the Adwords method, forcing the price of a single Clickfunnels trial through the roof.
I have seen people paying upwards of $400 for a single click! Not a demo; a click.
Unsure about you, but that's a bit too rich for my blood.
Bing Advertising
For the reason of Adwords being high, some people have turned to advertisements on Bing.
Although Bing is not quite as popular as Google, keep in mind it's still responsible for approximately 33 percent of all web traffic.
Recently in case you've got an Android phone, you may have noticed when clicking a link in your own email, the URL opens in Bing rather than Google.
Because of this, I would anticipate Bing visitors to increase. So, it may be a excellent prospect for Bing Advertising. In reality, I know of a few men and women that are extremely successful together.
Email Marketing
This technique could work in conjunction with paid traffic, or as a stand alone entity.
First off, you need to have an email list constructed first. You want the listing to be constructed with a viewer interested in earning money online (also known as the MMO niche).
Another viewer that may do nicely as a Clickfunnels Affiliate, is if you've got business contacts in your list. Especially those in a market such as Health or Real Estate.
From that point you would want to promote Clickfunnels on your email followups. Instead that is by supplying a great deal of free value, or a direct magnet. Or, even a talk funnel, which we will discuss next.
Since it's stated that it takes someone around 7 times of viewing something average, before they purchase; the goal here would be to keep showing them the value of Clickfunnels.
The end result will be that a certain percentage of people may end up taking the free trial of Clickfunnels. Once they do, provide them guidance, or free tools etc to help them along. Teeth Cleaning Voucher Clickfunnels Dentist
The end goal is obviously you want them to become a paying subscriber, in which you would create a recurring commission for every month they remain.
Share Funnels
This is a really effective method for getting Clickfunnels trials. The drawback to it, is you have to be a Clickfunnels manhood, and be adequate at building a funnel.
What you may do, is drive visitors to a landing page, where you literally give away these funnels for free.
The purpose here, is for individuals not to be Clickfunnels members already, so they'll have to sign up with your connection, then import the funnel in their dashboard.
Not only does this work, it was the very first strategy that really worked on me a year or so ago. This is a win-win in most cases.
The brand new subscriber instantly has a funnel, which can potentially allow them to start collecting trials instantly. Also at precisely the exact same time, it lands the Clickfunnels consumer who provided the funnel, fresh trials. From that point, you'd send some useful followup emails, with guidance on the best way to land the traffic, and how to turn them into paying members.
Most likely, you'll also be getting emails from the funnel builder. Because just like you are working to convert trials along with your funnel, they're attempting to convert you.
It is not quite as simple as it sounds. Because in order for this to work, you have to have traffic of course. This may be accomplished by either paid visitors, email advertising or SEO; which we'll talk about now.
For those of you that do not know, SEO is my bread and butter. Sure, it takes longer to start making progress. But, it requires little to no money to get started.
The goal behind SEO, would be to compose high quality and beneficial posts (such as this one), which will rank in the search engines of Google, Bing, Yahoo etc..
Once it takes some time to rank, after it does a fantastic article can bring in traffic for many years to come; using minimal maintenance.
This is my favorite method when starting out, because I did not want to sink a good deal of money into paid traffic. The disadvantage concerning compensated traffic, is that although you can get traffic quick, you have to constantly keep sinking money into ads, or the traffic stops.
You don't need to worry about this with SEO.
Although I have not made a million bucks using Clickfunnels as an affiliate, all my earnings are profit.
Here is a screenshot of my dashboard in the time of this writing.
I began getting Clickfunnels traffic around April of this past year. So, about 9 weeks ago.
As you can see, I have earned almost 15k in Clickfunnels Affiliate commissions.
The good thing about SEO, is the way that it starts to chemical once your website (s) begin to rank.
If you're considering how I personally learned SEO, you can take a look at the class I took here. Teeth Cleaning Voucher Clickfunnels Dentist
Solo Ads
Last on my list, is solo advertisements.
Solo ads is among the most popular strategies to building an email list.
You most likely won't receive any trials, possibly a wander 1 or two from running a solo advertising. But, you should get a nice number of optins for your listing.
From that point, you would use the email advertising strategy, to try and convert your new list subscribers to Clickfunnels trials. With effective email marketing, this may be a great way to earn commissions in the long term.
The good news about solo advertisements, is they are a list in which you will permanently own.
You do not possess Google traffic. Or, Facebook traffic.
You do possess your email list.
If Facebook shut down tomorrow, I know a lot of people , who would be SOL.
Another plus to solo ads, is that as soon as you build your list, you can promote many different affiliate supplies. It is not for Clickfunnels.
Affiliate Bootcamp
Clickfunnels has its Clickfunnels Affiliate training, called the Affiliate Bootcamp. You can register here for free. The motto for this, is teaching you"How to Retire in 100 Days As a Clickfunnels Affiliate". Though I find that somewhat extreme, it is extremely informative, and can be quite helpful if you choose to undergo it. In order to perform all the coaching, you will at least need to have the 14 day trial of Clickfunnels, which can be found here.
If after 2 weeks of trialing, you don't feel like paying it, Clickfunnels is simple to cancel. I've done it myself in the past.
It's done via the dashboard, therefore there is no need to have to speak to anybody about it, if you decide it's not for you.
So, as you can see – it can be very rewarding as a Clickfunnels affiliate. Like anything, there are various men and women who neglect. But in the event that you simply take action, and tweak things as you proceed, you'll be successful. Teeth Cleaning Voucher Clickfunnels Dentist
My advice for you would be to select 1 method which resonates with you, and master it. I see a lot of individuals who stay stuck, because they split their attention in too many directions.
This does not help you climb. It holds you back.
>>Try Clickfunnels Risk Free for 14 Days<<


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Hi There! Is this Your First Time?
Did you know if you Register you have access to free search tools including the ability to save listings and property searches? Did you know that you can bypass the search altogether and have listings sent directly to your email address? Check out our how-to page for more info.
- Price$2,200
- Sq. Ft.500
2168 Westchester, Bronx
GREAT LOCATION!!! for an office space, located in the Castle Hill area of the Bronx. There is Heavy-foot and car traffic. This office space has lots of opportunity, and it is a great location to get a good return on your investments.
Essential Information
- Price$2,200
- Bathrooms0.00
- Square Footage500
- Acres0.00
- TypeCommercial
- StatusActive
- HeatingRadiator
- CoolingWindow Units
Listing Details
- Listing OfficeCommunity Realty Alliance Corp
Community Information
- Address2168 Westchester
- AreaBronx
- CityBronx
- CountyBronx
- StateNY
- Zip Code10462
Additional Information
- ZoningC1-2
The data relating to real estate for sale or lease on this web site comes in part from HGMLS. Real estate listings held by brokerage firms other than Monroe Realty Center. Inc are marked with the HGMLS logo or an abbreviated logo and detailed information about them includes the name of the listing broker.
Information deemed reliable but not guaranteed.
© 2019 Hudson Gateway Multiple Listing Service, Inc. All rights reserved.
Listing information last updated on December 7th, 2019 at 5:28pm EST.


text:
You are being redirected to the story the piece of content is found in so you can read it in context. Please click the following link if you are not automatically redirected within a couple seconds:
Brazil: Federal Supreme Court maintains rules for indigenous areas
Author: Letícia Casado e Rubens Valente, Folha de Sao Paulo/UOL (Brazil), Published on: 17 August 2017
"Federal Supreme Court Maintains Rules for Indigenous Areas", 17 August 2017
In a unanimous decision on Wednesday the 16th, the Federal Supreme Court (STF) ruled against the government of Mato Grosso which was seeking indemnification for the expropriation of land for the Xingu national park and for reservations occupied by Nhambiquaras and Parecis Indians...[T]he justices found that the Indians had already been occupying the area, which belongs to the Federal Government, and that the State has no standing to seek redress. The court analyzed two cases by the state government questioning the possession of the areas by Indians. Both were rejected. Mato Grosso had been demanding indemnification from the Federal Government and FUNAI (National Indian Foundation) arguing that the areas had been improperly included in the limits of the Xingu park, without any compensation paid by the Federal Government. It also argued that the Indians weren't living in the region when the demarcation was effected...For...[FUNAI]..., the lands are "immemorially indigenous". The State argued that the areas have belonged to it since the 1891 Constitution, but the Federal Government, which is the other party in the case, argued that the government of Mato Grosso was unable to prove ownership of the territory.


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China rubbishes Dalai Lama's remarks
China has rubbished the Dalai Lama's accusation that Beijing itself could be behind the recent violence in Tibet.world Updated: Apr 01, 2008 11:11 IST
China has rubbished the Dalai Lama's accusation that Beijing itself could be behind the recent violence in Tibet, saying it showed his "guilty conscience" and charged the Tibetan spiritual leader with "trying to pass the buck using rumours and cheating".
"The Dalai Lama's citing of hearsay rumours cannot change the truth of the incident. It only shows the sense of guilty conscience in his deep heart so that he is trying to pass the buck using rumours and cheating," Chinese Foreign Ministry spokesperson Jiang Yu said.
Countering China's persistent attacks accusing him of having "masterminded" the violence in Tibet, the Dalai Lama had remarked in New Delhi on Saturday, "We have heard a few hundred Chinese received monks' dress."
"They (soldiers) dressed like monks. So, for a lay person, they look like monks. But the swords they had were not Tibetan, they were Chinese swords," the Nobel lareaute living in exile in India had said, seeking to turn the tables on Beijing.
Jiang, quoted by official Xinhua news agency, said: "the truth of the violent crimes in Lhasa is there for all to see and proved by ironclad evidences that cannot be denied."
The most vicious monks-led anti-government protests in two decades in Lhasa had burst into violence on March 14, leaving at least 20 people dead in Tibet and elsewhere.


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c(3)G is a gene in Drosophila melanogaster defined by two independently isolated mutants on the third chromosome. When homozygous in females, the mutants—c(3)G17 or c(3)G68—result in the elimination of meiotic crossing over and a great increase in nondisjunction at the first meiotic division. The gametic frequency of X-, second-, or third-chromosome nondisjunction is approximately .3 in c(3)G17, and .4 in c(3)G68; for the fourth chromosome, the frequency is .2 in c(3)G17 and .3 in c(3)G68. These values are at least two hundred fold greater than for spontaneous nondisjunction, though not high enough to indicate that chromosomes are distributed at random to the first meiotic division poles. Chromosomes loss is inferred from an excess of nullo-exceptional over diplo-exceptional ova. Loss is more frequent in c(3)G68. If c(3)G females mate at low temperature, crossing over is still absent, but nondisjunction is decreased. c(3)G17 is more temperature sensitive than c(3)G68.—Nonhomologous chromosomes tend to undergo nondisjunction in the same meiotic cells in c(3)G. Moreover, there is substantial nonhomologous pairing involving the larger chromosomes of the genome, inferred from the tendency for nonhomologs to disjoin from each other. Nonhomologous segregation is not observed between chromosome 4 and any other chromosome. c(3)G68 exhibits more nonhomologous segregation than does c(3)G17, and, for either allele, the degree of nonhomologous segregation is directly proportional to the similarity in length of the two nonhomologs being considered. The degree of nonhomologous segregation is increased at low temperature.—Heterozygosity for inversions tends to increase c(3)G-mediated nondisjunction, and to alter the patterns of nonhomologous segregations. The effects are observed even if the inversion does not disrupt centromeric heterochromatin, and even though the inversions do not change the lengths of the chromosomes involved. In XXY females, c(3)G17 shows more separation of the two X's from the Y chromo-some than does c(3)G68. Fourth-chromosome nondisjunction is increased by the presence of a Y chromosome in both kinds of mutant females. But in XXY;c(3)G females which are also heterozygous for an X inversion, frequencies of fourth-chromosome nondisjunction are little different from those in XX; c(3)G females, while the degrees of XX-from-Y disjunction are increased.—The chromosome behavior of the two alleles of c(3)G is readily rationalized by a model which assumes that c(3)G+ controls a stage of meiosis prior to synapsis and crossing over. If exchange is directly disrupted in c(3)G homozygotes, disjunctional consequences should be the same in c(3)G17 and c(3)G68. They are not. If, however, c(3)G+ controls a precondition to crossing over—such as the association of homologous and nonhomologous chromosomes—then the two alleles could each abolish crossing over, but lead to different amounts and patterns of nondisjunction.
- Received September 3, 1971.
- Revision received March 7, 1972.
- Copyright © 1972 by the Genetics Society of America


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GTC505 Engine Ignition Analyzer
เหมาะสำหรับตรวจสอบระบบจุดระเบิด เช่น :-
- Coil Near Plug
- Distributor / Coil
- Waste Spark
- Magneto
สะดวกรวดเร็ว ช่วยประหยัดเวลาในการค้นหาปัญหา
ใช้งานง่าย รู้ผลทันที่ ผลวิเคราะห์แม่นยำ เชื่อถือได้
ราคาพิเศษฃ่วงแนะนำ - กรุณาโทรสอบถาม
ผู้ผลิตสินค้า ประเทศ แคนาดา
Engine Ignition Analyzer
The GTC505 Engine Ignition Analyzer is a handheld instrument designed from the ground up for the diagnosis
and performance measurement of all types of spark ignition systems, from conventional to the newest Coil on Plug
in use.
Capable of displaying and measuring the following in real time: secondary ignition waveform, primary coil current
ramp time, dwell time, spark burn time and RPM. It also provides two innovative and unique display formats used to
compare measurements between cylinders, and to look at data for hundreds of spark events over time in order to
detect intermittent problems, misfires and other performance issues which are very difficult to detect using
conventional scopes.
This hand held device is powered by a rechargeable lithium battery, which provides up to 8 hours of battery life
and uses a fast response 3.5" color TFT display with a bright backlight and wide viewing angle.
- Display waveforms in real time
- Detect and display dwell angle (in high voltage sparkplug-wire systems), and ramp time (for COP systems)
- Graph burn time, spark plug voltage, dwell/ramp time, and RPM for each spark, allowing for detection of misfires
- Compare readings between cylinders with a "candlestick" graph that displays maximum, minimum, and average readings for each cylinder
- Display everything on a 3.5" color TFT LCD screen
- Rechargeable Li-Ion battery provide 6+ hours of working time
- Pass the rigorous "MIL-STD-810" drop test
Useful for troubleshooting:
- Non-starts
- Misfires
- Intermittent problems
- Primary and secondary coil circuits faults
- Fouled or damaged spark plugs
- Damaged spark plug wires
Quick powerful and easy to use
- Plug the appropiate sensor pick-up into the flexible probe
- Select the number of cycles of the engine to be measured
- Select the measurement type and display format
- Place the pick-up on top of the ignition module, or over the spark plug wire
- The GTC505 will automatically detect and adjust all parameters to provide accurate measurements and clear graphs
Technical Specifications
- Display: Color 3.5" TFT (320x240 pixels)
- Ignition system compatibility: magneto, conventional distributor, waste spark (DIS), coil near plug, coil on plug, etc.
- Engine cycle: 2, 4 and waste spark (DIS)
- Probe: 12" (30 cm) long with interchangeable COP and SPW pick-up
- Internal power Source: 1500 mA/h rechargeable LiFePO4 battery
- Battery life: Approximately 6 hours of operation with a fully charged battery
- Power Saving: Auto power off
- Physical dimensions : 6.5"(H) x 4.9"(W) x 1.4"(D)or 165 mm x 124 mm x 36 mm (without probe)
- Weight: 14.8 oz or 420 g (without probe)
- Range: 120 to 19,999 RPM
- Resolution: 1 RPM
- Accuracy: ±0.5% + 1 LSD
Spark (peak) voltage
- Range: 0 to 50 kV
- Resolution: 0.1 kV
- Repetitivity: ±0.2 kV
Spark burn (firing) time
- Range: 0 to 10.0 mS
- Resolution: 0.1 mS
- Accuracy: ±0.1 mS + 1 LSD
Ignition primary current ramp time (COP mode)
- Range: 0 to 10.0 mS
- Resolution: 0.1 mS
- Accuracy: ±0.1 mS + 1 LSD
Dwell time (spark plug wire)
- range: 0 to 10.0 mS
- Resolution: 0.1 mS
- Accuracy: ±0.1 mS + 1 LSD
Electrical Specifications
- Input MicroUSB connector: 5.0 volt, 500 mA maximum
For use in
- Automobiles
- Trucks
- Boats
- Lawnmowers
- Chainsaw
- Power generators
- Marine engines
- Motorcycles, etc.
- Flexible probe
- Coil on plug (COP) pickup
- Spark plug wire (SPW) pickup
- USB micro cable and AC charger adapter
- Protective rubber holster
- Hard padded case for carrying and storage
- User's Manual
- 1 year parts and labor warranty
ดาวน์โหลด เอกสารที่ด้านล่างครับ