The Project Gutenberg EBook of Hai Dao Suan Jing, by Hui Liu This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org Title: Hai Dao Suan Jing Author: Hui Liu Release Date: October 20, 2008 [EBook #26979] Language: Chinese Character set encoding: UTF-8 *** START OF THIS PROJECT GUTENBERG EBOOK HAI DAO SUAN JING *** Produced by Yvonne 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？ 答曰：島高四里五十五步；去表一百二里一百五十步。 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。 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？ 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。 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。 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？ 答曰：邑方三里四十三步、四分步之三；邑去表四里四十五步。 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 。 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？ 答曰：四十一丈九尺。 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。 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？ 答曰：八丈。 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。 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？ 答曰：一里二百步。 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。 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？ 答曰：一丈二尺。 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。 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。 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？ 答曰：二里一百二步。 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 。 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？ 答曰：南北長一里一百步；東西廣一里三十三步、少半步。 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。 End of the Project Gutenberg EBook of Hai Dao Suan Jing, by Hui Liu *** END OF THIS PROJECT GUTENBERG EBOOK HAI DAO SUAN JING *** ***** This file should be named 26979-0.txt or 26979-0.zip ***** This and all associated files of various formats will be found in: https://www.gutenberg.org/2/6/9/7/26979/ Produced by Yvonne Updated editions will replace the previous one--the old editions will be renamed. Creating the works from public domain print editions means that no one owns a United States copyright in these works, so the Foundation (and you!) can copy and distribute it in the United States without permission and without paying copyright royalties. Special rules, set forth in the General Terms of Use part of this license, apply to copying and distributing Project Gutenberg-tm electronic works to protect the PROJECT GUTENBERG-tm concept and trademark. Project Gutenberg is a registered trademark, and may not be used if you charge for the eBooks, unless you receive specific permission. If you do not charge anything for copies of this eBook, complying with the rules is very easy. You may use this eBook for nearly any purpose such as creation of derivative works, reports, performances and research. They may be modified and printed and given away--you may do practically ANYTHING with public domain eBooks. Redistribution is subject to the trademark license, especially commercial redistribution. *** START: FULL LICENSE *** THE FULL PROJECT GUTENBERG LICENSE PLEASE READ THIS BEFORE YOU DISTRIBUTE OR USE THIS WORK To protect the Project Gutenberg-tm mission of promoting the free distribution of electronic works, by using or distributing this work (or any other work associated in any way with the phrase "Project Gutenberg"), you agree to comply with all the terms of the Full Project Gutenberg-tm License (available with this file or online at https://gutenberg.org/license). Section 1. General Terms of Use and Redistributing Project Gutenberg-tm electronic works 1.A. By reading or using any part of this Project Gutenberg-tm electronic work, you indicate that you have read, understand, agree to and accept all the terms of this license and intellectual property (trademark/copyright) agreement. If you do not agree to abide by all the terms of this agreement, you must cease using and return or destroy all copies of Project Gutenberg-tm electronic works in your possession. If you paid a fee for obtaining a copy of or access to a Project Gutenberg-tm electronic work and you do not agree to be bound by the terms of this agreement, you may obtain a refund from the person or entity to whom you paid the fee as set forth in paragraph 1.E.8. 1.B. "Project Gutenberg" is a registered trademark. It may only be used on or associated in any way with an electronic work by people who agree to be bound by the terms of this agreement. There are a few things that you can do with most Project Gutenberg-tm electronic works even without complying with the full terms of this agreement. See paragraph 1.C below. There are a lot of things you can do with Project Gutenberg-tm electronic works if you follow the terms of this agreement and help preserve free future access to Project Gutenberg-tm electronic works. See paragraph 1.E below. 1.C. The Project Gutenberg Literary Archive Foundation ("the Foundation" or PGLAF), owns a compilation copyright in the collection of Project Gutenberg-tm electronic works. Nearly all the individual works in the collection are in the public domain in the United States. If an individual work is in the public domain in the United States and you are located in the United States, we do not claim a right to prevent you from copying, distributing, performing, displaying or creating derivative works based on the work as long as all references to Project Gutenberg are removed. Of course, we hope that you will support the Project Gutenberg-tm mission of promoting free access to electronic works by freely sharing Project Gutenberg-tm works in compliance with the terms of this agreement for keeping the Project Gutenberg-tm name associated with the work. You can easily comply with the terms of this agreement by keeping this work in the same format with its attached full Project Gutenberg-tm License when you share it without charge with others. 1.D. The copyright laws of the place where you are located also govern what you can do with this work. Copyright laws in most countries are in a constant state of change. If you are outside the United States, check the laws of your country in addition to the terms of this agreement before downloading, copying, displaying, performing, distributing or creating derivative works based on this work or any other Project Gutenberg-tm work. The Foundation makes no representations concerning the copyright status of any work in any country outside the United States. 1.E. Unless you have removed all references to Project Gutenberg: 1.E.1. The following sentence, with active links to, or other immediate access to, the full Project Gutenberg-tm License must appear prominently whenever any copy of a Project Gutenberg-tm work (any work on which the phrase "Project Gutenberg" appears, or with which the phrase "Project Gutenberg" is associated) is accessed, displayed, performed, viewed, copied or distributed: This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org 1.E.2. If an individual Project Gutenberg-tm electronic work is derived from the public domain (does not contain a notice indicating that it is posted with permission of the copyright holder), the work can be copied and distributed to anyone in the United States without paying any fees or charges. If you are redistributing or providing access to a work with the phrase "Project Gutenberg" associated with or appearing on the work, you must comply either with the requirements of paragraphs 1.E.1 through 1.E.7 or obtain permission for the use of the work and the Project Gutenberg-tm trademark as set forth in paragraphs 1.E.8 or 1.E.9. 1.E.3. If an individual Project Gutenberg-tm electronic work is posted with the permission of the copyright holder, your use and distribution must comply with both paragraphs 1.E.1 through 1.E.7 and any additional terms imposed by the copyright holder. Additional terms will be linked to the Project Gutenberg-tm License for all works posted with the permission of the copyright holder found at the beginning of this work. 1.E.4. Do not unlink or detach or remove the full Project Gutenberg-tm License terms from this work, or any files containing a part of this work or any other work associated with Project Gutenberg-tm. 1.E.5. Do not copy, display, perform, distribute or redistribute this electronic work, or any part of this electronic work, without prominently displaying the sentence set forth in paragraph 1.E.1 with active links or immediate access to the full terms of the Project Gutenberg-tm License. 1.E.6. You may convert to and distribute this work in any binary, compressed, marked up, nonproprietary or proprietary form, including any word processing or hypertext form. However, if you provide access to or distribute copies of a Project Gutenberg-tm work in a format other than "Plain Vanilla ASCII" or other format used in the official version posted on the official Project Gutenberg-tm web site (www.gutenberg.org), you must, at no additional cost, fee or expense to the user, provide a copy, a means of exporting a copy, or a means of obtaining a copy upon request, of the work in its original "Plain Vanilla ASCII" or other form. Any alternate format must include the full Project Gutenberg-tm License as specified in paragraph 1.E.1. 1.E.7. Do not charge a fee for access to, viewing, displaying, performing, copying or distributing any Project Gutenberg-tm works unless you comply with paragraph 1.E.8 or 1.E.9. 1.E.8. You may charge a reasonable fee for copies of or providing access to or distributing Project Gutenberg-tm electronic works provided that - You pay a royalty fee of 20% of the gross profits you derive from the use of Project Gutenberg-tm works calculated using the method you already use to calculate your applicable taxes. The fee is owed to the owner of the Project Gutenberg-tm trademark, but he has agreed to donate royalties under this paragraph to the Project Gutenberg Literary Archive Foundation. Royalty payments must be paid within 60 days following each date on which you prepare (or are legally required to prepare) your periodic tax returns. Royalty payments should be clearly marked as such and sent to the Project Gutenberg Literary Archive Foundation at the address specified in Section 4, "Information about donations to the Project Gutenberg Literary Archive Foundation." - You provide a full refund of any money paid by a user who notifies you in writing (or by e-mail) within 30 days of receipt that s/he does not agree to the terms of the full Project Gutenberg-tm License. You must require such a user to return or destroy all copies of the works possessed in a physical medium and discontinue all use of and all access to other copies of Project Gutenberg-tm works. - You provide, in accordance with paragraph 1.F.3, a full refund of any money paid for a work or a replacement copy, if a defect in the electronic work is discovered and reported to you within 90 days of receipt of the work. - You comply with all other terms of this agreement for free distribution of Project Gutenberg-tm works. 1.E.9. If you wish to charge a fee or distribute a Project Gutenberg-tm electronic work or group of works on different terms than are set forth in this agreement, you must obtain permission in writing from both the Project Gutenberg Literary Archive Foundation and Michael Hart, the owner of the Project Gutenberg-tm trademark. Contact the Foundation as set forth in Section 3 below. 1.F. 1.F.1. Project Gutenberg volunteers and employees expend considerable effort to identify, do copyright research on, transcribe and proofread public domain works in creating the Project Gutenberg-tm collection. Despite these efforts, Project Gutenberg-tm electronic works, and the medium on which they may be stored, may contain "Defects," such as, but not limited to, incomplete, inaccurate or corrupt data, transcription errors, a copyright or other intellectual property infringement, a defective or damaged disk or other medium, a computer virus, or computer codes that damage or cannot be read by your equipment. 1.F.2. LIMITED WARRANTY, DISCLAIMER OF DAMAGES - Except for the "Right of Replacement or Refund" described in paragraph 1.F.3, the Project Gutenberg Literary Archive Foundation, the owner of the Project Gutenberg-tm trademark, and any other party distributing a Project Gutenberg-tm electronic work under this agreement, disclaim all liability to you for damages, costs and expenses, including legal fees. YOU AGREE THAT YOU HAVE NO REMEDIES FOR NEGLIGENCE, STRICT LIABILITY, BREACH OF WARRANTY OR BREACH OF CONTRACT EXCEPT THOSE PROVIDED IN PARAGRAPH F3. YOU AGREE THAT THE FOUNDATION, THE TRADEMARK OWNER, AND ANY DISTRIBUTOR UNDER THIS AGREEMENT WILL NOT BE LIABLE TO YOU FOR ACTUAL, DIRECT, INDIRECT, CONSEQUENTIAL, PUNITIVE OR INCIDENTAL DAMAGES EVEN IF YOU GIVE NOTICE OF THE POSSIBILITY OF SUCH DAMAGE. 1.F.3. LIMITED RIGHT OF REPLACEMENT OR REFUND - If you discover a defect in this electronic work within 90 days of receiving it, you can receive a refund of the money (if any) you paid for it by sending a written explanation to the person you received the work from. If you received the work on a physical medium, you must return the medium with your written explanation. The person or entity that provided you with the defective work may elect to provide a replacement copy in lieu of a refund. If you received the work electronically, the person or entity providing it to you may choose to give you a second opportunity to receive the work electronically in lieu of a refund. If the second copy is also defective, you may demand a refund in writing without further opportunities to fix the problem. 1.F.4. Except for the limited right of replacement or refund set forth in paragraph 1.F.3, this work is provided to you 'AS-IS' WITH NO OTHER WARRANTIES OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO WARRANTIES OF MERCHANTIBILITY OR FITNESS FOR ANY PURPOSE. 1.F.5. Some states do not allow disclaimers of certain implied warranties or the exclusion or limitation of certain types of damages. If any disclaimer or limitation set forth in this agreement violates the law of the state applicable to this agreement, the agreement shall be interpreted to make the maximum disclaimer or limitation permitted by the applicable state law. The invalidity or unenforceability of any provision of this agreement shall not void the remaining provisions. 1.F.6. INDEMNITY - You agree to indemnify and hold the Foundation, the trademark owner, any agent or employee of the Foundation, anyone providing copies of Project Gutenberg-tm electronic works in accordance with this agreement, and any volunteers associated with the production, promotion and distribution of Project Gutenberg-tm electronic works, harmless from all liability, costs and expenses, including legal fees, that arise directly or indirectly from any of the following which you do or cause to occur: (a) distribution of this or any Project Gutenberg-tm work, (b) alteration, modification, or additions or deletions to any Project Gutenberg-tm work, and (c) any Defect you cause. Section 2. Information about the Mission of Project Gutenberg-tm Project Gutenberg-tm is synonymous with the free distribution of electronic works in formats readable by the widest variety of computers including obsolete, old, middle-aged and new computers. It exists because of the efforts of hundreds of volunteers and donations from people in all walks of life. Volunteers and financial support to provide volunteers with the assistance they need, is critical to reaching Project Gutenberg-tm's goals and ensuring that the Project Gutenberg-tm collection will remain freely available for generations to come. In 2001, the Project Gutenberg Literary Archive Foundation was created to provide a secure and permanent future for Project Gutenberg-tm and future generations. To learn more about the Project Gutenberg Literary Archive Foundation and how your efforts and donations can help, see Sections 3 and 4 and the Foundation web page at https://www.pglaf.org. Section 3. Information about the Project Gutenberg Literary Archive Foundation The Project Gutenberg Literary Archive Foundation is a non profit 501(c)(3) educational corporation organized under the laws of the state of Mississippi and granted tax exempt status by the Internal Revenue Service. The Foundation's EIN or federal tax identification number is 64-6221541. Its 501(c)(3) letter is posted at https://pglaf.org/fundraising. Contributions to the Project Gutenberg Literary Archive Foundation are tax deductible to the full extent permitted by U.S. federal laws and your state's laws. The Foundation's principal office is located at 4557 Melan Dr. S. Fairbanks, AK, 99712., but its volunteers and employees are scattered throughout numerous locations. Its business office is located at 809 North 1500 West, Salt Lake City, UT 84116, (801) 596-1887, email business@pglaf.org. Email contact links and up to date contact information can be found at the Foundation's web site and official page at https://pglaf.org For additional contact information: Dr. Gregory B. Newby Chief Executive and Director gbnewby@pglaf.org Section 4. Information about Donations to the Project Gutenberg Literary Archive Foundation Project Gutenberg-tm depends upon and cannot survive without wide spread public support and donations to carry out its mission of increasing the number of public domain and licensed works that can be freely distributed in machine readable form accessible by the widest array of equipment including outdated equipment. Many small donations ($1 to $5,000) are particularly important to maintaining tax exempt status with the IRS. The Foundation is committed to complying with the laws regulating charities and charitable donations in all 50 states of the United States. Compliance requirements are not uniform and it takes a considerable effort, much paperwork and many fees to meet and keep up with these requirements. We do not solicit donations in locations where we have not received written confirmation of compliance. To SEND DONATIONS or determine the status of compliance for any particular state visit https://pglaf.org While we cannot and do not solicit contributions from states where we have not met the solicitation requirements, we know of no prohibition against accepting unsolicited donations from donors in such states who approach us with offers to donate. International donations are gratefully accepted, but we cannot make any statements concerning tax treatment of donations received from outside the United States. U.S. laws alone swamp our small staff. Please check the Project Gutenberg Web pages for current donation methods and addresses. Donations are accepted in a number of other ways including including checks, online payments and credit card donations. To donate, please visit: https://pglaf.org/donate Section 5. General Information About Project Gutenberg-tm electronic works. Professor Michael S. Hart was the originator of the Project Gutenberg-tm concept of a library of electronic works that could be freely shared with anyone. For thirty years, he produced and distributed Project Gutenberg-tm eBooks with only a loose network of volunteer support. Project Gutenberg-tm eBooks are often created from several printed editions, all of which are confirmed as Public Domain in the U.S. unless a copyright notice is included. Thus, we do not necessarily keep eBooks in compliance with any particular paper edition. Most people start at our Web site which has the main PG search facility: https://www.gutenberg.org This Web site includes information about Project Gutenberg-tm, including how to make donations to the Project Gutenberg Literary Archive Foundation, how to help produce our new eBooks, and how to subscribe to our email newsletter to hear about new eBooks.