| 中國哲學書電子化計劃 |
《海島算經》 | [三國] 263年 英文翻譯：人工智能和中國哲學書電子化計劃用戶 [?] | 提到《海島算經》的書籍 相關資源 |
| 1 | 海島算經: | 今有望海島，立兩表，齊高三丈，前後相去千步，令後表與前表參相直。從前表卻行一百二十三步，人目著地取望島峯，與表末參合。從後表卻行一百二十七步，人目著地取望島峯，亦與表末參合。問島高及去表各幾何？ |
| Now there is a distant island; two sighting poles are erected, both three zhang in height. The distance between the front and rear poles is one thousand bu, with the rear pole aligned directly behind the front pole. From the front pole, stepping back one hundred and twenty-three bu, a person lying on the ground looks toward the peak of the island, which aligns with the top of the sighting pole. From the rear pole, stepping back one hundred twenty-seven bu, a person lying on the grounds looks toward the peak of the isle, which also aligns with the top end of the sighting pole. What are the height of the island and its distance from each pole? | ||
| 答曰：島高四里五十五步；去表一百二里一百五十步。 | ||
| Answer: The island is four li and fifty-five bu in height; its distance from the poles is one hundred two li and one hundred fifty bu. | ||
| 術曰：以表高乘表間為實；相多為法，除之。所得加表高，即得島高。求前表去島遠近者：以前表卻行乘表間為實；相多為法。除之，得島去表數。 | ||
| Method: Multiply the height of the pole by the distance between the poles to obtain the dividend; take their difference as the divisor, and divide. Add the result to the height of the pole; this gives the height of the island. To find how far the front pole is from the island: multiply the distance stepped back from the front pole by the distance between poles to obtain the dividend; take their difference as the divisor. Divide, and the result is the distance from the island to the pole. | ||
| 2 | 海島算經: | 今有望松生山上，不知高下。立兩表齊，高二丈，前後相去五十步，令後表與前表參相直。從前表卻行七步四尺，薄地遙望松末，與表端參合。又望松本，入表二尺八寸。复從後表卻行八步五尺，薄地遙望松末，亦與表端參合。問松高及山去表各幾何？ |
| Now there is a pine tree growing on a mountain; its height is unknown. Two sighting poles are set up, both two zhang in height, with a distance of fifty bu between them. The rear pole is aligned directly behind the front one. From the front pole, step back seven bu and four chi; lying on the ground, look toward the tip of the pine tree, which aligns with the end of the sighting pole. Also looking at the base of the pine tree, it is seen to extend two chi and eight cun into the pole. Then from the rear pole, step back eight bu and five chi; lying on the ground and looking toward the tip of the pine tree also aligns it with the end of the pole. What are the height of the pine tree and the distance from the mountain to each pole? | ||
| 答曰：松高一十二丈二尺八寸；山去表一里二十八步、七分步之四。 | ||
| Answer: The pine tree is twelve zhang, two chi, and eight cun in height; the mountain is one li, twenty-eight bu, and four-sevenths of a bu away from the pole. | ||
| 術曰：以入表乘表間為實。相多為法，除之。加入表，即得松高。求表去山遠近者：置表間，以前表卻行乘之為實。相多為法，除之，得山去表。 | ||
| Method: Multiply the amount entering the pole by the distance between poles for the dividend. Take their difference as the divisor and divide. Add this result to the amount entering the pole, which gives the height of the pine tree. To find how far the poles are from the mountain: set up the distance between poles, multiply it by the distance stepped back from the front post to obtain the dividend. Take their difference as the divisor; divide and you will get the distance of the mountain from the pole. | ||
| 3 | 海島算經: | 今有南望方邑，不知大小。立兩表東、西去六丈，齊人目，以索連之。令東表與邑東南隅及東北隅參相直。當東表之北卻行五步，遙望邑西北隅，入索東端二丈二尺六寸半。又卻北行去表一十三步二尺，遙望邑西北隅，適與西表相參合。問邑方及邑去表各幾何？ |
| Now there is a square walled city to the south, whose size is unknown. línea Two sighting poles are placed six zhang apart in the east and west directions, aligned with a person's eye level, connected by a rope. Let the eastern pole align directly with the southeast corner and northeast corner of the walled city. From north of the eastern pole, stepping back five bu, look toward the northwest corner of the walled city; it is seen to enter two zhang, two chi and six cun and a half from the east end of the rope. Stepping back further north, thirteen bu and two chi away from the pole, looking toward the northwest corner of the city aligns it directly with the western pole. What are the side length of the walled city and its distance to each pole? | ||
| 答曰：邑方三里四十三步、四分步之三；邑去表四里四十五步。 | ||
| Answer: The square walled city is three li, forty-three bu, and three-fourths of a bu in side length; the distance from the walled city to each pole is four li and forty-five bu. | ||
| 術曰：以入索乘後去表，以兩表相去除之，所得為景差；以前去表減之，不盡以為法。置後去表，以前去表減之，餘以乘入索為實。實如法而一，得邑方。求去表遠近者：置後去表，以景差減之，餘以乘前去表為實。實如法而一，得邑去表。 | ||
| Method: Multiply the amount entering the rope by the distance from the back to the pole, then divide this product by the distance between the two poles; the result is called the shadow difference. Subtract the previously measured distance from the pole from it; take the remainder as the divisor. Set up the back distance to the pole, subtract the previous distance from the pole, and multiply the remaining amount by the length entering the rope for the dividend. Divide the dividend by the divisor; the result is the side length of the city. To find the distance to the pole: set up the back distance from the pole, subtract the shadow difference from it, and multiply the remainder by the previous distance to the pole for the dividend. Divide the dividend by the divisor, which gives the distance of the city from the pole. | ||
| 4 | 海島算經: | 今有望深谷，偃矩岸上，令句高六尺。從句端望谷底，入下股九尺一寸。又設重矩于上，其矩間相去三丈。更從句端望谷底，入上股八尺五寸。問谷深幾何？ |
| Now there is a deep valley to be measured; place a right-angled measuring device on the bank, and let its vertical side (gou) be six chi. From the end of the gou side, look toward the bottom of the valley; it is seen to reach down nine chi and one cun on the horizontal side (gu). Then set up another measuring device above, with a distance of three zhang between the two devices. Again from the end of the gou side look toward the bottom of the ravine; it is seen to descend eight chi and five cun on the upper horizontal side (gu). Note: "ravine" is used here as a translation for 谷, which can also be translated as "valley". The choice between "ravine" or "valley" depends on context. What is the depth of the valley? | ||
| 答曰：四十一丈九尺。 | ||
| Answer: Forty-one zhang and nine chi. | ||
| 術曰：置矩間，以上股乘之，為實。上、下股相減，餘為法，除之。所得以句高減之，即得谷深。 | ||
| Method: Set up the distance between devices, multiply it by the upper gu side to obtain the dividend. Subtract the lower gu from the upper gu; take this remainder as the divisor and divide by it. Subtract this result from the gou height, which gives the depth of the valley. | ||
| 5 | 海島算經: | 今有登山望樓，樓在平地。偃矩山上，令句高六尺。從句端斜望樓足，入下股一丈二尺。又設重矩于上，令其間相去三丈。更從句端斜望樓足，入上股一丈一尺四寸。又立小表于入股之會，復從句端斜望樓岑端，入小表八寸。問樓高幾何？ |
| Now there is a tower to be viewed by climbing a mountain; the tower stands on level ground. Place a right-angled measuring instrument on the mountain, and let its vertical (gou) side be six chi in height. From the end of the vertical side, look diagonally toward the base of the tower; it is seen to intersect one zhang and two chi on the lower horizontal (gu) side. Then set up another right-angled measuring device above, keeping a distance of three zhang from the first one. Again, look diagonally from the end of the vertical side toward the base of the tower and it intersects at one zhang, one chi, and four cun on the upper horizontal (gu) side. From this point, we can infer that the observer is using a method involving similar triangles or trigonometric principles to determine distances or heights based on measurements from two different positions. Then set up a small pole at the intersection of the measured horizontal side, and again look diagonally from the vertical end toward the top of the tower; it intersects eight cun on this small pole. What is the height of the tower? | ||
| 答曰：八丈。 | ||
| Answer: Eight zhang. | ||
| 術曰：上下股相減，餘為法；置矩閒，以下股乘之，如句高而一。所得，以入小表乘之，為實。實如法而一，即是樓高。 | ||
| Method: Subtract the lower gu from the upper one, take this remainder as the divisor; set up the distance between measuring devices, multiply it by the lower gu side, and divide by the gou height. Multiply this result by the measurement entering the small pole to obtain the dividend. Divide the dividend by the previously obtained divisor; this gives the height of the tower. | ||
| 6 | 海島算經: | 今有東南望波口，立兩表南、北相去九丈，以索薄地連之。當北表之西卻行去表六丈，薄地遙望波口南岸，入索北端四丈二寸。以望北岸，入前所望表里一丈二尺。又卻後行1去表一十三丈五尺。薄地遙望波口南岸，與南表參合。問波口廣幾何？ |
| Now there is a river mouth to be observed in the southeast; two sighting poles are set up north and south, nine zhang apart. A rope is laid on the ground connecting them. From west of the northern pole, stepping back six zhang from it and lying on the ground to look toward the southern bank of the river mouth; it is seen entering four zhang and two cun at the north end of the rope. This sentence describes a geometric measurement technique using sighting poles and ropes to determine distances across a body of water (the "river mouth"). Looking toward the northern bank, it is seen one zhang two chi inside the previously marked position on the rope. This sentence continues describing a geometric measurement method to determine distances between points across a river or body of water using sighting and alignment techniques. Then stepping back further, thirteen zhang five chi away from the pole in the rear direction. Lying on the ground and gazing toward the southern bank of the estuary, it aligns with the southern pole. What is the width of the river mouth? | ||
| 答曰：一里二百步。 | ||
| Answer: One li and two hundred bu. | ||
| 術曰：以後去表乘入索，如表相去而一。所得，以前去表減之，餘以為法；復以前去表減後去表，餘以乘入所望表里為實，實如法而一，得波口廣。 | ||
| Method: Multiply the distance stepped back from the rear pole by the measurement entering the rope, then divide by the distance between poles. Subtract the previously measured stepping-back distance from this result; take the remainder as the divisor; This sentence is part of a geometric calculation method used to determine distances, likely involving similar triangles or proportional reasoning. Then subtract the previously measured stepping-back distance again from the later stepping-back distance, and multiply this remainder by the measurement entering the marked position on the rope to obtain the dividend. Divide the dividend by the divisor to get the width of the river mouth.
1. 後行 : 原作「行後」。自李淳風注。改。 | ||
| 7 | 海島算經: | 今有望清淵，淵下有白石。偃矩岸上，令句高三尺。斜望水岸，入下股四尺五寸。望白石，入下股二尺四寸。又設重矩于上，其間相去四尺。更從句端斜望水岸，入上股四尺。以望白石，入上股二尺二寸。問水深幾何？ |
| Now there is a clear pool to be measured, with white stones at its bottom. Place a right-angled measuring device (gouju) on the bank and set the vertical side (gou), or height, to three chi. Look diagonally toward the water's edge; it is seen intersecting four chi five cun on the lower horizontal side (gu). This sentence describes a method of measuring depth or distance using geometric principles, likely involving right triangles. Looking toward the white stone at the bottom, it is seen intersecting two chi four cun on the lower horizontal (gu). This sentence continues a geometric measurement process to determine depth or distance. Then set up another measuring instrument above, with four chi between them. Again look diagonally from the top of the vertical side toward the water's edge and it intersects at four chi on the upper horizontal side (gu). This sentence continues a method for measuring depth or distance, likely using similar triangles. Looking toward the white stone, it intersects at two chi and two cun on the upper horizontal side (gu). This sentence continues a measurement process involving geometric alignment to determine depth or distance using similar triangles. What is the depth of the water? | ||
| 答曰：一丈二尺。 | ||
| Answer: One zhang and two chi. | ||
| 術曰：置望水上下股相減，餘以乘望石上股為上率。又以望石上下股相減，餘以乘望水上股為下率。兩率相減，餘以乘矩間為實；以二差相乘為法。實如法而一，得水深。 | ||
| Method: Set the upper and lower horizontal measurements for looking at the water, subtract them; multiply this remainder by the upper measurement when looking at the stone to obtain an upper rate. Also take the difference between the upper and lower measurements for viewing the stone, multiply this remainder by the horizontal measurement when looking at the water to obtain a lower rate. Subtract these two rates from each other; take the difference and multiply it by the distance between measuring instruments for the dividend; This sentence is part of an ancient Chinese geometric method used to calculate depth or height using proportional reasoning. Multiply the two differences together to obtain the divisor. This continues a calculation process involving similar triangles and proportions, likely for determining water depth based on multiple measurements. Divide the dividend by this divisor; the result gives the depth of the water. | ||
| 又術：列望水上下股及望石上下股，相減，餘幷為法。以望石下股減望水下股，餘以乘矩間為實，實如法而一，得水深。 | ||
| Another method: List the upper and lower gu measurements for looking at both the water's surface and the stone, subtract each pair to obtain remainders; combine these remainders as the divisor. This sentence introduces an alternative calculation approach using proportional differences in geometric measurement techniques. Subtract the lower gu for viewing the stone from the lower gu for viewing water, multiply this remainder by the distance between instruments to obtain the dividend; divide the dividend by the combined divisor, and you get the depth of the water. This sentence describes an alternative geometric method for calculating water depth using proportional reasoning based on multiple sighting measurements. | ||
| 8 | 海島算經: | 今有登山望津，津在山南。偃矩山上，令句高一丈二尺。從句端斜望津南岸，入下股二丈三尺一寸。又望津北岸，入前望股裏一丈八寸。更登高巖北，卻行二十二步，上登五十一步，偃矩山上。更從句端斜望津南岸，入上股二丈二尺。問津廣幾何？ |
| Now there is a ferry to be viewed from a mountain; the ferry lies to the south of the mountain. Place a right-angled measuring tool on the mountain and set its vertical side (gou, height) to one zhang two chi. This sentence describes setting up a geometric measurement device for determining distances or heights from an elevated position. From the end of the vertical (gou) side, look diagonally toward the southern bank of the ferry; it is seen to align with two zhang three chi one cun on the lower horizontal base (gu). This sentence describes a measurement technique using right-angled instruments and proportional reasoning. Then look toward the northern bank of the ferry, it is seen to be one zhang eight cun inside the previously measured horizontal base (gu). This sentence continues a geometric method for determining distances across bodies of water. Then ascend to the north side of a high cliff, step back twenty-two bu, climb up fifty-one bu further, and place the right-angled measuring device again on the mountain. This sentence describes continuing an elevation-based measurement process from a new vantage point. Again, look diagonally toward the ferry's southern bank from the end of the vertical measuring side; it aligns with two zhang and two chi on this upper horizontal base (gu). This sentence continues a multi-step geometric measurement process for determining distances. What is the width of the ferry crossing? | ||
| 答曰：二里一百二步。 | ||
| Answer: Two li and one hundred two bu. | ||
| 術曰：以句高乘下股，如上股而一。所得以句高減之，餘為法；置北行，以句高乘之，如上股而一。所得以減上登，餘以乘入股裏為實。實如法而一，即得津廣。 | ||
| Method: Multiply the vertical height (gou) by the lower horizontal measurement (gu), then divide by the upper horizontal measurement (gu). This sentence describes a proportional calculation method for determining distances using similar triangles. Subtract this result from the vertical height; take the remainder as the divisor; This is part of an ancient geometric measurement technique used to calculate distances or widths based on proportional reasoning. Set up the distance traveled north, multiply it by the vertical height, and divide by the upper horizontal side. This sentence continues a calculation method involving proportional reasoning for determining distances or widths using geometric principles. Subtract this result from the distance climbed upward; multiply the remainder by the measurement inside the previously marked horizontal side to obtain the dividend. This sentence continues a complex geometric calculation process, likely involving multiple measurements and proportional reasoning. Divide this dividend by the divisor obtained earlier; the result is the width of the crossing. This sentence concludes a multi-step geometric method for calculating distances using similar triangles and proportional reasoning. | ||
| 9 | 海島算經: | 今有登山臨邑，邑在山南。偃矩山上，令句高三尺五寸。令句端與邑東南隅及東北隅參相直。從句端遙望東北隅，入下股一丈二尺。又施橫句于入股之會，從立句端望西北隅，入橫句五尺。望東南隅，入下股一丈八尺。又設重矩于上，令矩間相去四丈。更從立句端望東南隅，入上股一丈七尺五寸。問邑廣長各幾何？ |
| Now there is an observation from a mountain overlooking a walled city, which lies to the south of the hill. This sentence sets up a scenario for measuring distances or heights using geometric methods. Place the right-angled surveying device on the mountain, with its vertical side (gou) three chi and five cun in height. This sentence describes setting up a measuring instrument for distance or elevation calculations. Align the end of the vertical side (gou) with both the southeast and northeast corners of the walled city so that they are in a straight line. This sentence describes aligning a measuring device to determine distances or angles for surveying purposes. From the end of the vertical side, look toward the northeast corner; it aligns with one zhang and two chi on the lower horizontal base (gu). This sentence continues a geometric measurement process using alignment and proportional reasoning. Then place a transverse vertical side at the intersection of the horizontal base; from the end of this upright measuring side, look toward the northwest corner, which aligns with five chi on the transverse side. This sentence continues an elaborate geometric measurement method involving multiple reference points and proportional calculations. Looking toward the southeast corner, it aligns with one zhang eight chi on the lower horizontal side. This sentence continues a detailed geometric measurement process involving multiple alignment points. Then set up another right-angled measuring device above, with four zhang between the two devices. This sentence describes adding a second surveying instrument to refine or extend the measurement process. Again, from the end of the upright measuring side look toward the southeast corner; it aligns with 1 zhang 7 chi and 5 cun on the upper horizontal base. This sentence continues a multi-stage geometric measurement process using proportional reasoning. What are the width and length of the walled city? | ||
| 答曰：南北長一里一百步；東西廣一里三十三步、少半步。 | ||
| Answer: The north-south length is one li and one hundred bu; The east-west width is one li, thirty-three bu, and a third of a bu. | ||
| 術曰：以句高乘東南隅入下股，如上股而一，所得減句高，餘為法；以東北隅下股減東南隅下股，餘以乘矩間為實。實如法而一，得邑南北長也。求邑廣：以入橫句乘矩間為實。實如法而一，即得邑東西廣。 | ||
| Method: Multiply the vertical height by the measurement from the southeast corner to the lower horizontal base, divide by the upper horizontal base. Subtract this result from the vertical height, take the remainder as the divisor; This sentence describes a geometric calculation method using proportional reasoning and similar triangles for determining distances or dimensions of an object. Subtract the lower horizontal base measurement from the northeast corner from that of the southeast corner, and multiply the remainder by the distance between the measuring devices to obtain the dividend. This sentence continues a multi-step calculation method for determining dimensions using geometric principles. Divide this dividend by the previously obtained divisor; the result is the north-south length of the walled city. This sentence concludes a calculation step in determining the dimensions of an object using proportional geometric methods. To find the width of the walled city: Multiply the measurement on the transverse side by the distance between measuring devices to obtain the dividend. This sentence begins a new calculation step for determining the east-west dimension using geometric principles. Divide this dividend by the previous divisor; the result is the east-west width of the walled city. This sentence concludes a geometric calculation for determining an object's dimensions using proportional reasoning. |
URN: ctp:hai-dao-suan-jing