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Contents
1. Cover
2. Title Page
3. Copyright Page
4. Dedication
5. About the Author
6. Contents at a Glance
7. Contents
8. Acknowledgments
9. Introduction
10. Part I Introduction and Installation
1. Chapter 1 Introduction to Linux, Distributions, and FOSS
1. Linux: The Operating System
2. Open Source Software and GNU: Overview
1. The GNU Public License
2. Upstream and Downstream
3. The Advantages of Open Source Software
4. Understanding the Differences Between Windows
and Linux
1. Single Users vs. Multiple Users vs.
Network Users
2. The Monolithic Kernel and the Micro-
Kernel
3. Separation of the GUI and the Kernel
4. My Network Places
5. The Registry vs. Text Files
6. Domains and Active Directory
5. Summary

2. Chapter 2 Installing a Linux Server
1. Hardware and Environmental Considerations
2. Server Design
1. Uptime
3. Methods of Installation
4. Installing Fedora
1. Project Prerequisites
5. The Installation
6. Installation Summary
1. Localization Section
2. Software Section
3. System Section
4. User Settings Section
5. The Installation
7. Installing Ubuntu Server
1. Start the Installation
2. Configure the Network
3. Configure Proxy
4. Configure Ubuntu Archive Mirror
5. File System Setup
6. Profile Setup
7. SSH Setup
8. Featured Server Snaps
9. Install Complete
8. Summary
3. Chapter 3 Deploying Linux Servers in the Cloud
1. Behind the Cloud

2. Obtaining and Spinning Up New Virtual Linux
Servers
1. Free-to-Run Virtual Linux Servers
2. Commercial Cloud Service Providers
3. Summary
11. Part II Single-Host Administration
1. Chapter 4 The Command Line
1. An Introduction to Bash
1. Job Control
2. Environment Variables
3. Pipes
4. Redirection
2. Command-Line Shortcuts
1. Filename Expansion
2. Environment Variables as Parameters
3. Multiple Commands
4. Backticks
3. Documentation Tools
1. The man Command
2. The texinfo System
4. Files (Types, Ownership, and Permissions)
1. Normal Files
2. Directories
3. Hard Links
4. Symbolic Links
5. Block Devices
6. Character Devices
7. Listing Files: ls
8. Change Ownership: chown

9. Change Group: chgrp
10. Change Mode: chmod
5. File Management and Manipulation
1. Copy Files: cp
2. Move Files: mv
3. Link Files: ln
4. Find a File: find
5. File Compression: gzip
6. File Compression: bzip2
7. File Compression: xz
8. Create a Directory: mkdir
9. Remove Files or Directories: rm
10. Show Present Working Directory: pwd
11. Tape Archive: tar
12. Concatenate Files: cat
13. Display a File One Screen at a Time: more
or less
14. Show the Directory Location of a File:
which
15. Locate a Command: whereis
6. Editors
1. vi
2. emacs
3. pico
4. sed
7. Miscellaneous Tools
1. Disk Utilization: du
2. Disk Free: df
3. List Processes: ps
4. Show an Interactive List of Processes: top
5. Send a Signal to a Process: kill
6. Show System Information: uname
7. Who Is Logged In: who
8. A Variation on who: w
9. Switch User: su

8. Putting It All Together (Moving a User and Its
Home Directory)
9. Summary
2. Chapter 5 Managing Software
1. The Red Hat Package Manager
2. Managing Software Using RPM
1. Querying for Information the RPM Way
(Getting to Know One Another)
2. Installing Software with RPM (Moving in
Together)
3. Uninstalling Software with RPM (Ending
the Relationship)
4. Other Things RPM Can Do
3. Yum
4. DNF
5. GUI RPM Package Managers
1. Fedora or Ubuntu
2. openSUSE and SLE
6. The Debian Package Management System
1. APT
7. Software Management in Ubuntu
1. Querying for Information
2. Installing Software in Ubuntu
3. Removing Software in Ubuntu
8. Compile and Install GNU Software
1. Getting and Unpacking the Source
Package
2. Looking for Documentation
3. Configuring the Package
4. Compiling the Package

5. Installing the Package
6. Testing the Software
7. Cleanup
9. Common Problems when Building from Source
Code
1. Problems with Libraries
2. Missing Configure Script
3. Broken Source Code
10. Summary
3. Chapter 6 Managing Users and Groups
1. What Exactly Constitutes a User?
2. Where User Information Is Kept
1. The /etc/passwd File
2. The /etc/shadow File
3. The /etc/group File
3. User Management Tools
1. Command-Line User Management
2. GUI User Managers
4. Users and Access Permissions
1. Understanding SetUID and SetGID
Programs
2. Sticky Bit
5. Pluggable Authentication Modules
1. How PAM Works
2. PAM’s Files and Their Locations
3. Configuring PAM
4. A Sample PAM Configuration File
5. The “Other” File
6. D’oh! I Can’t Log In!
7. Debugging PAM

6. A Grand Tour
1. Creating Users with useradd
2. Creating Groups with groupadd
3. Modifying User Attributes with usermod
4. Modifying Group Attributes with
groupmod
5. Deleting Users and Groups with userdel
and groupdel
7. Summary
4. Chapter 7 Booting and Shutting Down
1. Boot Loaders
1. GRUB Legacy
2. GRUB 2
3. Bootstrapping
2. The init Process
3. Systemd Scripts
1. Writing Your Own rc Script
4. Enabling and Disabling Services
1. Enabling a Service
2. Disabling a Service
5. Odds and Ends of Booting and Shutting Down
1. fsck!
2. Booting into Single-User (“Recovery”)
Mode
6. Summary
5. Chapter 8 File Systems

1. The Makeup of File Systems
1. i-Nodes
2. Blocks
3. Superblocks
4. ext4
5. Btrfs
6. XFS
2. Managing File Systems
1. Mounting and Unmounting Local Disks
2. Using fsck
3. Adding a New Disk
1. Overview of Partitions
2. Traditional Disk and Partition Naming
Conventions
4. Volume Management
1. Creating Partitions and Logical Volumes
5. Creating File Systems
6. Summary
6. Chapter 9 Core System Services
1. systemd
1. systemd’s Role
2. How systemd Works
2. xinetd
1. The /etc/xinetd.conf File
2. Examples: A Simple (echo) Service Entry
3. The Logging Daemon

1. rsyslogd
2. systemd-journald (journald)
4. The cron Program
1. The crontab File
2. Editing the crontab File
5. Summary
7. Chapter 10 The Linux Kernel
1. What Exactly Is a Kernel?
2. Finding the Kernel Source Code
1. Getting the Correct Kernel Version
2. Unpacking the Kernel Source Code
3. Building the Kernel
1. Preparing to Configure the Kernel
2. Kernel Configuration
3. Compiling the Kernel
4. Installing the Kernel
5. Booting the Kernel
6. The Author Lied! It Didn’t Work!
4. Patching the Kernel
1. Downloading and Applying Patches
2. If the Patch Worked
3. If the Patch Didn’t Work
5. Summary
8. Chapter 11 Knobs and Dials: API (Virtual) File Systems
1. What’s Inside the /proc Directory?
1. Tweaking Files Inside of /proc

2. Some Useful /proc Entries
1. Enumerated /proc Entries
3. Common proc Settings and Reports
1. SYN Flood Protection
2. Issues on High-Volume Servers
4. SysFS
5. cgroupfs
6. tmpfs
1. tmpfs Example
7. Summary
12. Part III Networking and Security
1. Chapter 12 TCP/IP for System Administrators
1. The Layers
1. Packets
2. TCP/IP Model and the OSI Model
2. Headers
1. Ethernet
2. IP (IPv4)
3. TCP
4. UDP
3. A Complete TCP Connection
1. Opening a Connection
2. Transferring Data
3. Closing the Connection
4. How ARP Works

1. The ARP Header: ARP Works with Other
Protocols, Too!
5. Bringing IP Networks Together
1. Hosts and Networks
2. Subnetting
3. Netmasks
4. Static Routing
5. Dynamic Routing with RIP
6. tcpdump Bits and Bobs
1. Reading and Writing Dumpfiles
2. Capturing More or Less per Packet
3. Performance Impact
4. Don’t Capture Your Own Network Traffic
5. Troubleshooting Slow Name Resolution
(DNS) Issues
7. IPv6
1. IPv6 Address Format
2. IPv6 Address Types
3. IPv6 Backward Compatibility
8. Summary
2. Chapter 13 Network Configuration
1. Modules and Network Interfaces
1. Network Device Configuration Utilities
(ip, ifconfig, and nmcli)
2. Sample Usage: ifconfig, ip, and nmcli
3. Setting Up NICs at Boot Time
2. Managing Routes
1. Sample Usage: Route Configuration
2. Displaying Routes

3. A Simple Linux Router
1. Routing with Static Routes
4. VPCs, Subnets, IPs, and Route Configuration
(AWS Cloud Example)
1. VPCs and Subnets (AWS)
2. Internet Gateways and Routing (AWS)
3. Security Groups (AWS)
4. Launch a Linux Server in Its Own Subnet
(AWS)
5. Hostname Configuration
6. Summary
3. Chapter 14 Linux Firewall (Netfilter)
1. How Netfilter Works
1. A NAT Primer
2. Chains
2. Installing Netfilter
1. Enabling Netfilter in the Kernel
3. Configuring Netfilter
1. Saving Your Netfilter Configuration
2. The iptables Command
3. firewalld
4. Cookbook Solutions
1. Simple NAT: iptables
2. Simple NAT: nftables
3. Simple Firewall: iptables
5. Summary

4. Chapter 15 Local Security
1. Common Sources of Risk
1. SetUID Programs
2. Unnecessary Processes
2. Picking the Right Runlevel
3. Nonhuman User Accounts
4. Limited Resources
5. Mitigating Risk
1. chroot
2. SELinux
3. AppArmor
6. Monitoring Your System
1. Logging
2. Using ps and netstat
3. Watch That Space (Using df)
4. Automated Monitoring
5. Staying in the Loop (Mailing Lists)
7. Summary
5. Chapter 16 Network Security
1. TCP/IP and Network Security
1. The Importance of Port Numbers
2. Tracking Services
1. Using the netstat Command
2. Security Implications of netstat’s Output
3. Binding to an Interface
4. Shutting Down Services

1. Shutting Down xinetd and inetd Services
2. Shutting Down Non-xinetd Services
5. Monitoring Your System
1. Making the Best Use of syslog
2. Monitoring Bandwidth with MRTG
6. Handling Attacks
1. Trust Nothing (and No One)
2. Change Your Passwords
3. Pull the Plug
7. Network Security Tools
1. nmap
2. Snort
3. Nessus and OpenVAS
4. Wireshark/tcpdump
8. Summary
13. Part IV Internet Services
1. Chapter 17 Domain Name System (DNS)
1. The Hosts File
2. How DNS Works
1. Domain and Host Naming Conventions
2. The Root Domain
3. Subdomain
4. The in-addr.arpa Domain
5. Types of Servers
3. Installing a DNS Server
1. Understanding the BIND Configuration
File
2. The Specifics

4. Configuring a DNS Server
1. Defining a Primary Zone in the
named.conf File
2. Defining a Secondary Zone in the
named.conf File
3. Defining a Caching Zone in the
named.conf File
5. DNS Records Types
1. SOA: Start of Authority
2. NS: Name Server
3. A and AAAA: Address Records
4. PTR: Pointer Record
5. MX: Mail Exchanger
6. CNAME: Canonical Name
7. RP and TXT: The Documentation Entries
6. Setting Up BIND Database Files
1. DNS Server Setup Walkthrough
7. The DNS Toolbox
1. host
2. dig
3. resolvectl
4. nslookup
5. whois
6. nsupdate
7. The rndc Tool
8. Configuring DNS Clients
1. The Resolver
2. Configuring the Client (Traditional)
9. Summary

2. Chapter 18 File Transfer Protocol (FTP)
1. The Mechanics of FTP
1. Client/Server Interactions
2. Obtaining and Installing vsftpd
1. Configuring vsftpd
2. Starting and Testing the FTP Server
3. Customizing the FTP Server
1. Setting Up an Anonymous-Only FTP
Server
2. Setting Up an FTP Server with Virtual
Users
4. Summary
3. Chapter 19 Apache Web Server
1. Understanding HTTP
1. Headers
2. Ports
3. Process Ownership and Security
2. Installing the Apache HTTP Server
1. Apache Modules
3. Starting Up and Shutting Down Apache
1. Starting Apache at Boot Time
4. Testing Your Installation
5. Configuring Apache

1. Creating a Simple Root-Level Page
2. Apache Configuration Files
3. Common Configuration Options
6. Troubleshooting Apache
7. Summary
4. Chapter 20 Simple Mail Transfer Protocol (SMTP)
1. Understanding SMTP
1. Rudimentary SMTP Details
2. Security Implications
3. E-mail Components
2. Installing the Postfix Server
1. Installing Postfix via DNF in Fedora,
CentOS, or RHEL
2. Installing Postfix via APT in Ubuntu
3. Configuring the Postfix Server
1. The main.cf File
2. Checking Your Configuration
4. Running the Server
1. Checking the Mail Queue
2. Flushing the Mail Queue
3. The newaliases Command
4. Making Sure Everything Works
5. Summary
5. Chapter 21 Post Office Protocol and Internet Mail Access
Protocol (POP and IMAP)
1. POP3 and IMAP Protocol Basics
2. Dovecot (IMAP and POP3 Server)
3. Installing Dovecot

1. Dovecot Configuration Files and Options
2. Configuring Dovecot
3. Running Dovecot
4. Checking Basic POP3 Functionality
5. Checking Basic IMAP Functionality
4. Other Issues with Mail Services
1. SSL/TLS Security
2. Availability
3. Log Files
5. Summary
6. Chapter 22 Voice over Internet Protocol (VoIP)
1. VoIP Overview
1. VoIP Server
2. Analog Telephone Adapter (ATA)
3. IP Phones
4. VoIP Protocols
2. VoIP Implementations
3. Asterisk
1. How Asterisk Works
4. Asterisk Installation
1. Starting and Stopping Asterisk
5. Understanding Asterisk Configuration Files and
Structure
1. The Dialplan: extensions.conf
2. Modules: modules.conf
6. Asterisk Network, Port, and Firewall Requirements
1. Configuring the Local Firewall for Asterisk

7. Configuring the PBX
1. Local Extensions
2. Outside Connection (VoIP Trunking)
3. Trunking Using Twilio Elastic SIP Trunks
8. Asterisk Maintenance and Troubleshooting
1. Asterisk CLI Commands
2. Helpful CLI Commands
3. Common Issues with VoIP
9. Summary
7. Chapter 23 Secure Shell (SSH)
1. Understanding Public Key Cryptography
1. Key Characteristics
2. SSH Backstory (Versions)
1. OpenSSH and OpenBSD
2. Alternative Vendors for SSH Clients
3. Installing OpenSSH on RPM-Based
Systems
4. Installing OpenSSH via APT in Ubuntu
3. Server Startup and Shutdown
4. SSHD Configuration File
5. Using OpenSSH
1. Secure Shell (ssh) Client Program
2. Secure Copy (scp) Program
3. Secure FTP (sftp) Program
6. Files Used by the OpenSSH Client
7. Summary
14. Part V Intranet Services

1. Chapter 24 Network File System (NFS)
1. The Mechanics of NFS
1. Versions of NFS
2. Security Considerations for NFS
3. Mount and Access a Partition
2. Enabling NFS in Fedora, RHEL, and CentOS
3. Enabling NFS in Ubuntu and Debian
4. The Components of NFS
1. Kernel Support for NFS
5. Configuring an NFS Server
1. The /etc/exports Configuration File
6. Configuring NFS Clients
1. The mount Command
2. Soft vs. Hard Mounts
3. Cross-Mounting Disks
4. The Importance of the intr Option
5. Performance Tuning
7. Troubleshooting Client-Side NFS Issues
1. Stale File Handles
2. Permission Denied
8. Sample NFS Client and NFS Server Configuration
9. Common Uses for NFS
10. Summary
2. Chapter 25 Samba
1. The Mechanics of SMB

1. Usernames and Passwords
2. Encrypted Passwords
3. Samba Daemons
4. Installing Samba via RPM
5. Installing Samba via APT
2. Samba Administration
1. Starting and Stopping Samba
3. Creating a Share
1. Using smbclient
4. Mounting Remote Samba Shares
5. Samba Users
1. Creating Samba Users
2. Allowing Null Passwords
3. Changing Passwords with smbpasswd
6. Using Samba to Authenticate Against a Windows
Server
1. winbindd Daemon
7. Troubleshooting Samba
8. Summary
3. Chapter 26 Distributed File Systems (DFS)
1. DFS Overview
2. DFS Implementations
1. GlusterFS
3. Summary
4. Chapter 27 Lightweight Directory Access Protocol (LDAP)
1. LDAP Basics

1. LDAP Directory
2. Client/Server Model
3. Uses of LDAP
4. LDAP Terminology
2. OpenLDAP
1. Server-Side Daemons
2. OpenLDAP Utilities
3. Installing OpenLDAP
4. Configuring OpenLDAP
5. Configuring slapd
6. Starting and Stopping slapd
3. Configuring OpenLDAP Clients
1. Creating Directory Entries
4. Searching, Querying, and Modifying the Directory
5. Using OpenLDAP for User Authentication
1. Configuring the Server
2. Configuring the Client
6. Summary
5. Chapter 28 Printing
1. Printing Terminologies
2. The CUPS System
1. Running CUPS
2. Installing CUPS
3. Configuring CUPS
3. Adding Printers
1. Local Printers and Remote Printers
2. Using the Web Interface to Add a Printer
3. Using Command-Line Tools to Add a
Printer

4. Routine CUPS Administration
1. Setting the Default Printer
2. Enabling, Disabling, and Deleting Printers
3. Accepting and Rejecting Print Jobs
4. Managing Printing Privileges
5. Managing Printers via the Web Interface
5. Using Client-Side Printing Tools
1. lpr
2. lpq
3. lprm
6. Summary
6. Chapter 29 Dynamic Host Configuration Protocol (DHCP)
1. The Mechanics of DHCP
2. The DHCP Server
1. Installing DHCP Software via RPM
2. Installing DHCP Software via APT in
Ubuntu
3. Configuring the DHCP Server
4. A Sample dhcpd.conf File
3. The DHCP Client Daemon
4. Summary
7. Chapter 30 Virtualization
1. Why Virtualize?
1. Virtualization Concepts
2. Virtualization Implementations
1. Hyper-V
2. Kernel-Based Virtual Machine (KVM)

3. QEMU
4. VirtualBox
5. VMware
6. Xen
3. KVM
1. KVM Example
2. Managing KVM Virtual Machines
4. Setting Up KVM in Ubuntu/Debian
5. Containers
1. Containers vs. Virtual Machines
2. Docker
6. Summary
8. Chapter 31 Backups
1. Evaluating Your Backup Needs
1. Amount of Data
2. Backup Hardware and Backup Medium
3. Network Throughput
4. Speed and Ease of Data Recovery
5. Data Deduplication
6. Tape Management
2. Command-Line Backup Tools
1. dump and restore
2. tar
3. rsync
3. Miscellaneous Backup Solutions
4. Summary
15. Part VI Appendixes
1. A Creating a Linux Installer on Flash/USB Devices
1. Overview

1. Native Solutions
2. Distro-Specific Solutions
3. Universal Solutions
2. B Demo Virtual Machine and Container
1. Basic Host System Requirements
2. Installing the Virtualization Applications and
Utilities
3. Download and Prep the Demo VM Image File
4. Import the Demo VM Image and Create a New VM
Instance
1. Managing the Demo Virtual Machine
5. Connecting to the Demo VM
1. Virtual Network Computing (VNC)
2. Virtual Serial TTY Console
3. Connecting via SSH
4. Cockpit Application
5. Just Use It!
6. Demo Containers (Docker, podman, buildah, and
kubectl)
7. Feedback
16. Index
Guide
1. Cover
2. Title Page
3. Linux Administration: A Beginner’s Guide, Eighth Edition
Page List
1. i

2. ii
3. iii
4. iv
5. v
6. vi
7. vii
8. viii
9. ix
10. x
11. xi
12. xii
13. xiii
14. xiv
15. xv
16. xvi
17. xvii
18. xviii
19. xix
20. xx
21. xxi
22. xxii
23. xxiii
24. xxiv
25. xxv
26. xxvi
27. xxvii
28. xxviii
29. xxix
30. xxx
31. 1
32. 2
33. 3
34. 4
35. 5
36. 6
37. 7
38. 8
39. 9
40. 10
41. 11
42. 12

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If the parts of
any
ponderant
press the
beams of the
scale every
where
equally, all
the parts cut
off, reckoned
from centre
let there be applied the ponderant O, having to the ponderant C the
proportion of A B to A N. I say, the ponderants in B and N will be
equally poised. For the proportion of the moment of the ponderant
O, in the point N, to the moment of the ponderant C in the point B,
is by the 5th article, compounded of the proportions of the weight O
to the weight C, and of the distance from the centre of the scale A N
or A L to the distance from the centre of the scale A B. But seeing
we have supposed, that the distance A B to the distance A N is in
reciprocal proportion of the weight O to the weight C, the proportion
of the moment of the ponderant O, in the point N, to the moment of
the ponderant C, in the point B, will be compounded of the
proportions of A B to A N, and of A N to A B. Wherefore, setting in
order A B, A N, A B, the moment of O to the moment of C will be as
the first to the last, that is, as A B to A B. Their moments therefore
are equal; and consequently the plane which passes through A will
(by the fifth definition) be a plane of equiponderation. Wherefore
they will be equally poised; as was to be proved.
Now the converse of this is manifest. For if there be
equiponderation and the proportion of the weights and distances be
not reciprocal, then both the weights will always have the same
moments, although one of them have more weight added to it or its
distance changed.
Coroll. When ponderants are of the same species, and their
moments be equal; their magnitudes and distances from the centre
of the scale will be reciprocally proportional. For in homogeneous
bodies, it is as weight to weight, so magnitude to magnitude.
8. If to the whole length of the beam there be
applied a parallelogram, or a parallelopipedum, or a
prisma, or a cylinder, or the superficies of a cylinder,
or of a sphere, or of any portion of a sphere or
prisma; the parts of any of them cut off with planes
parallel to the base will have their moments in the
same proportion with the parts of a triangle, which
has its vertex in the centre of the scale, and for one
of its sides the beam itself, which parts are cut off
by planes parallel to the base.

of the scale,
will have
their
moments in
the same
proportion
with that of
the parts of a
triangle cut
off from the
vertex by
strait lines
parallel to
the base.
First, let the rectangled parallelogram A B C D (in
figure 4) be applied to the whole length of the beam
A B; and producing C B howsoever to E, let the
triangle A B E be described. Let now any part of the
parallelogram, as A F, be cut off by the plane F G,
parallel to the base C B; and let F G be produced to
A E in the point H. I say, the moment of the whole A
B C D to the moment of its part A F, is as the
triangle A B E to the triangle A G H, that is, in
proportion duplicate to that of the distances from
the centre of the scale.
For, the parallelogram A B C D being divided into
equal parts, infinite in number, by strait lines drawn parallel to the
base; and supposing the moment of the strait line C B to be B E, the
moment of the strait line F G will (by the 7th article) be G H; and the
moments of all the strait lines of that parallelogram will be so many
strait lines in the triangle A B E drawn parallel to the base B E; all
which parallels together taken are the moment of the whole
parallelogram A B C D; and the same parallels do also constitute the
superficies of the triangle A B E. Wherefore the moment of the
parallelogram A B C D is the triangle A B E; and for the same reason,
the moment of the parallelogram A F is the triangle A G H; and
therefore the moment of the whole parallelogram to the moment of
a parallelogram which is part of the same, is as the triangle A B E to
the triangle A G H, or in proportion duplicate to that of the beams to
which they are applied. And what is here demonstrated in the case
of a parallelogram may be understood to serve for that of a cylinder,
and of a prisma, and their superficies; as also for the superficies of a
sphere, of an hemisphere, or any portion of a sphere. For the parts
of the superficies of a sphere have the same proportion with that of
the parts of the axis cut off by the same parallels, by which the parts
of the superficies are cut off, as Archimedes has demonstrated; and
therefore when the parts of any of these figures are equal and at
equal distances from the centre of the scale, their moments also are
equal, in the same manner as they are in parallelograms.

The diameter
of
equiponderat
ion of figures
which are
deficient
according to
commensura
ble
proportions
of their
altitudes and
bases,
divides the
axis, so that
the part
taken next
the vertex is
to the other
part as the
complete
figure to the
deficient
figure.
Secondly, let the parallelogram A K I B not be rectangled; the
strait line I B will nevertheless press the point B perpendicularly in
the strait line B E; and the strait line L G will press the point G
perpendicularly in the strait line G H; and all the rest of the strait
lines which are parallel to I B will do the like. Whatsoever therefore
the moment be which is assigned to the strait line I B, as here, for
example, it is supposed to be B E, if A E be drawn, the moment of
the whole parallelogram A I will be the triangle A B E; and the
moment of the part A L will be the triangle A G H. Wherefore the
moment of any ponderant, which has its sides equally applied to the
beam, whether they be applied perpendicularly or obliquely, will be
always to the moment of a part of the same in such proportion as
the whole triangle has to a part of the same cut off by a plane which
is parallel to the base.
9. The centre of equiponderation of any figure,
which is deficient according to commensurable
proportions of the altitude and base diminished, and
whose complete figure is either a parallelogram or a
cylinder, or a parallelopipedum, divides the axis, so,
that the part next the vertex, to the other part, is as
the complete figure to the deficient figure.
For let C I A P E (in fig. 5) be a deficient figure,
whose axis is A B, and whose complete figure is C D
F E; and let the axis A B be so divided in Z, that A Z
be to Z B as C D F E is to C I A P E. I say, the centre
of equiponderation of the figure C I A P E will be in
the point Z.
First, that the centre of equiponderation of the
figure C I A P E is somewhere in the axis A B is
manifest of itself; and therefore A B is a diameter of
equiponderation. Let A E be drawn, and let B E be
put for the moment of the strait line C E; the
triangle A B E will therefore (by the third article) be
the moment of the complete figure C D F E. Let the
axis A B be equally divided in L, and let G L H be
drawn parallel and equal to the strait line C E, cutting the crooked

line C I A P E in I and P, and the strait lines A C and A E in K and M.
Moreover, let Z O be drawn parallel to the same C E; and let it be, as
L G to L I, so L M to another, L N; and let the same be done in all
the rest of the strait lines possible, parallel to the base; and through
all the points N, let the line A N E be drawn; the three-sided figure A
N E B will therefore be the moment of the figure C I A P E. Now the
triangle A B E is (by the 9th article of chapter XVII) to the three-
sided figure A N E B, as A B C D + A I C B is to A I C B twice taken,
that is, as C D F E + C I A P E is to C I A P E twice taken. But as C I
A P E is to C D F E, that is, as the weight of the deficient figure is to
the weight of the complete figure, so is C I A P E twice taken to C D
F E twice taken. Wherefore, setting in order C D F E + C I A P E. 2 C
I A P E. 2 C D F E; the proportion of C D F E + C I A P E to C D F E
twice taken will be compounded of the proportion of C D F E + C I A
P E to C I A P E twice taken, that is, of the proportion of the triangle
A B E to the three-sided figure A N E B, that is, of the moment of the
complete figure to the moment of the deficient figure, and of the
proportion of C I A P E twice taken to C D F E twice taken, that is, to
the proportion reciprocally taken of the weight of the deficient figure
to the weight of the complete figure.
Again, seeing by supposition A Z. Z B:: C D F E. C I A P E are
proportionals; A B. A Z:: C D F E + C I A P E. C D F E will also, by
compounding, be proportionals. And seeing A L is the half of A B, A
L. A Z:: C D F E + C I A P E. 2 C D F E will also be proportionals. But
the proportion of C D F E + C I A P E to 2 C D F E is compounded,
as was but now shown, of the proportions of moment to moment,
&c., and therefore the proportion of A L to A Z is compounded of the
proportion of the moment of the complete figure C D F E to the
moment of the deficient figure C I A P E, and of the proportion of
the weight of the deficient figure C I A P E to the weight of the
complete figure C D F E; but the proportion of A L to A Z is
compounded of the proportions of A L to B Z and of B Z to A Z. Now
the proportion of B Z to A Z is the proportion of the weights
reciprocally taken, that is to say, of the weight C I A P E to the
weight C D F E. Therefore the remaining proportion of A L to B Z,
that is, of L B to B Z, is the proportion of the moment of the weight

C D F E to the moment of the weight C I A P E. But the proportion of
A L to B Z is compounded of the proportions of A L to A Z and of A Z
to Z B; of which proportions that of A Z to Z B is the proportion of
the weight C D F E to the weight C I A P E. Wherefore (by art. 5 of
this chapter) the remaining proportion of A L to A Z is the proportion
of the distances of the points Z and L from the centre of the scale,
which is A. And, therefore, (by art. 6) the weight C I A P E shall
hang from O in the strait line O Z. So that O Z is one diameter of
equiponderation of the weight C I A P E. But the strait line A B is the
other diameter of equiponderation of the same weight C I A P E.
Wherefore (by the 7th definition) the point Z is the centre of the
same equiponderation; which point, by construction, divides the axis
so, that the part A Z, which is the part next the vertex, is to the
other part Z B, as the complete figure C D F E is to the deficient
figure C I A P E; which is that which was to be demonstrated.
Coroll. I. The centre of equiponderation of any of those plane
three-sided figures, which are compared with their complete figures
in the table of art. 3, chap. XVII, is to be found in the same table, by
taking the denominator of the fraction for the part of the axis cut off
next the vertex, and the numerator for the other part next the base.
For example, if it be required to find the centre of equiponderation
of the second three-sided figure of four means, there is in the
concourse of the second column with the row of three-sided figures
of four means this fraction 5
⁄7, which signifies that that figure is to
its parallelogram or complete figure as 5
⁄7 to unity, that is, as 5
⁄7 to
7
⁄7, or as 5 to 7; and, therefore the centre of equiponderation of that
figure divides the axis, so that the part next the vertex is to the
other part as 7 to 5.
Coroll. II. The centre of equiponderation of any of the solids of
those figures, which are contained in the table of art. 7 of the same
chap. XVII, is exhibited in the same table. For example, if the centre
of equiponderation of a cone be sought for, the cone will be found to
be 1
⁄3 of its cylinder; and, therefore, the centre of its
equiponderation will so divide the axis, that the part next the vertex
to the other part will be as 3 to 1. Also the solid of a three-sided
figure of one mean, that is, a parabolical solid, seeing it is 2
⁄4, that is

The diameter
of
equiponderat
ion of the
complement
of the half of
any of the
said deficient
figures,
divides that
line which is
drawn
through the
vertex
parallel to
the base, so
that the part
next the
vertex is to
the other
part as the
complete
figure to the
complement.
½ of its cylinder, will have its centre of equiponderation in that
point, which divides the axis, so that the part towards the vertex be
double to the part towards the base.
10. The diameter of equiponderation of the
complement of the half of any of those figures
which are contained in the table of art. 3, chap. XVII,
divides that line which is drawn through the vertex
parallel and equal to the base, so that the part next
the vertex will be to the other part, as the complete
figure to the complement.
For let A I C B (in the same fig. 5) be the half of a
parabola, or of any other of those three-sided
figures which are in the table of art. 3, chap. XVII,
whose axis is A B, and base B C, having A D drawn
from the vertex, equal and parallel to the base B C,
and whose complete figure is the parallelogram A B
C D. Let I Q be drawn at any distance from the side
C D, but parallel to it; and let A D be the altitude of
the complement A I C D, and Q I a line ordinately
applied in it. Wherefore the altitude A L in the
deficient figure A I C B is equal to Q I the line
ordinately applied in its complement; and contrarily,
L I the line ordinately applied in the figure A I C B is
equal to the altitude A Q in its complement; and so
in all the rest of the ordinate lines and altitudes the mutation is such,
that that line, which is ordinately applied in the figure, is the altitude
of its complement. And, therefore, the proportion of the altitudes
decreasing to that of the ordinate lines decreasing, being
multiplicate according to any number in the deficient figure, is
submultiplicate according to the same number in its complement.
For example, if A I C B be a parabola, seeing the proportion of A B
to A L is duplicate to that of B C to L I, the proportion of AD to A Q
in the complement A I C D, which is the same with that of B C to L I,
will be subduplicate to that of C D to Q I, which is the same with
that of A B to A L; and consequently, in a parabola, the complement
will be to the parallelogram as 1 to 3; in a three-sided figure of two

The centre of
equiponderat
ion of the
half of any of
the deficient
figures in the
first row of
the table of
art. 3,
chapter xvii,
may be found
out by the
numbers of
the second
row.
means, as 1 to 4; in a three-sided figure of three means, as 1 to 5,
&c. But all the ordinate lines together in A I C D are its moment; and
all the ordinate lines in A I C B are its moment. Wherefore the
moments of the complements of the halves of deficient figures in the
table of art. 3 of chap. XVII, being compared, are as the deficient
figures themselves; and, therefore, the diameter of equiponderation
will divide the strait line A D in such proportion, that the part next
the vertex be to the other part, as the complete figure A B C D is to
the complement A I C D.
Coroll. The diameter of equiponderation of these halves may be
found by the table of art. 3 of chap. XVII, in this manner. Let there be
propounded any deficient figure, namely, the second three-sided
figure of two means. This figure is to the complete figure as 3
⁄5 to 1,
that is 3 to 5. Wherefore the complement to the same complete
figure is as 2 to 5; and, therefore, the diameter of equiponderation
of this complement will cut the strait line drawn from the vertex
parallel to the base, so that the part next the vertex will be to the
other part as 5 to 2. And, in like manner, any other of the said three-
sided figures being propounded, if the numerator of its fraction
found out in the table be taken from the denominator, the strait line
drawn from the vertex is to be divided, so that the part next the
vertex be to the other part, as the denominator is to the remainder
which that subtraction leaves.
11. The centre of equiponderation of the half of
any of those crooked-lined figures, which are in the
first row of the table of art. 3 of chap. XVII, is in that
strait line which, being parallel to the axis, divides
the base according to the numbers of the fraction
next below it in the second row, so that the
numerator be answerable to that part which is
towards the axis.
For example, let the first figure of three means be
taken, whose half is A B C D (in fig. 6), and let the
rectangle A B E D be completed. The complement
therefore will be B C D E. And seeing A B E D is to
the figure A B C D (by the table) as 5 to 4, the same

A B E D will be to the complement B C D E as 5 to 1. Wherefore, if F
G be drawn parallel to the base D A, cutting the axis so that A G be
to G B as 4 to 5, the centre of equiponderation of the figure A B C D
will, by the precedent article, be somewhere in the same F G. Again,
seeing, by the same article, the complete figure A B E D, is to the
complement B C D E as 5 to 1, therefore if B E and A D be divided in
I and H as 5 to 1 the centre of equiponderation of the complement B
C D E will be somewhere in the strait line which connects H and I.
Let now the strait line L K be drawn through M the centre of the
complete figure, parallel to the base; and the strait line N O through
the same centre M, perpendicular to it; and let the strait lines L K
and F G cut the strait line H I in P and Q. Let P R be taken quadruple
to P Q; and let R M be drawn and produced to F G in S. R M
therefore will be to M S as 4 to 1, that is, as the figure A B C D to its
complement B C D E. Wherefore, seeing M is the centre of the
complete figure A B E D, and the distances of R and S from the
centre M be in proportion reciprocal to that of the weight of the
complement B C D E to the weight of the figure A B C D, R and S will
either be the centres of equiponderation of their own figures, or
those centres will be in some other points of the diameters of
equiponderation H I and F G. But this last is impossible. For no other
strait line can be drawn through the point M terminating in the strait
lines H I and F G, and retaining the proportion of M R to M S, that is,
of the figure A B C D to its complement B C D E. The centre,
therefore, of equiponderation of the figure A B C D is in the point S.
Now, seeing P M hath the same proportion to Q S which R P hath to
R Q, Q S will be 5 of those parts of which P M is four, that is, of
which I N is four. But I N or P M is 2 of those parts of which E B or F
G is 6; and, therefore, if it be as 4 to 5, so 2 to a fourth, that fourth
will be 2½. Wherefore Q S is 2½ of those parts of which F G is 6.
But F Q is 1; and, therefore, F S is 3½. Wherefore the remaining
part G S is 2½. So that F G is so divided in S, that the part towards
the axis is in proportion to the other part, as 2½ to 3½, that is as 5
to 7; which answereth to the fraction 5
⁄7 in the second row, next
under the fraction 4
⁄5 in the first row. Wherefore drawing S T parallel
to the axis, the base will be divided in like manner.

The centre of
equiponderat
ion of the
half of any of
the figures of
the second
row of the
same table
may be found
out by the
numbers of
the fourth
row.
By this method it is manifest, that the base of a semiparabola will
be divided into 3 and 5; and the base of the first three-sided figure
of two means, into 4 and 6; and of the first three-sided figure of
four means, into 6 and 8. The fractions, therefore, of the second row
denote the proportions, into which the bases of the figures of the
first row are divided by the diameters of equiponderation. But the
first row begins one place higher than the second row.
any of the figures in the second row of the same
table of art. 3, chap. XVII, is in a strait line parallel to
the axis, and dividing the base according to the
numbers of the fraction in the fourth row, two
places lower, so as that the numerator be
answerable to that part which is next the axis.
Let the half of the second three-sided figure of
two means be taken; and let it be A B C D (in fig.
7); whose complement is B C D E, and the rectangle
completed A B E D. Let this rectangle be divided by
the two strait lines L K and N O, cutting one another
in the centre M at right angles; and because A B E D is to A B C D as
5 to 3, let A B be divided in G, so that A G to B G be as 3 to 5; and
let F G be drawn parallel to the base. Also because A B E D is (by
art. 9) to B C D E as 5 to 2, let B E be divided in the point I, so that
B I be to I E as 5 to 2; and let I H be drawn parallel to the axis,
cutting L K and F G in P and Q. Let now P R be so taken, that it be
to P Q as 3 to 2, and let R M be drawn and produced to F G in S.
Seeing, therefore, R P is to P Q, that is, R M to M S, as A B C D is to
its complement B C D E, and the centres of equiponderation of A B C
D and B C D E are in the strait lines F G and H I, and the centre of
equiponderation of them both together in the point M; R will be the
centre of the complement B C D E, and S the centre of the figure A
B C D. And seeing P M, that is I N, is to Q S, as R P is to R Q; and I
N or P M is 3 of those parts, of which B E, that is F G, is 14;
therefore Q S is 5 of the same parts; and E I, that is F G, 4; and F S,
9; and G S, 5. Wherefore the strait line S T being drawn parallel to
the axis, will divide the base A D into 5 and 9. But the fraction 5
⁄9 is

The centre of
equiponderat
ion of the
half of any of
the figures in
the same
table being
known, the
centre of the
excess of the
same figure
above a
triangle of
the same
altitude and
base is also
known.
found in the fourth row of the table, two places below the fraction ⅗
in the second row.
By the same method, if in the same second row there be taken
the second three-sided figure of three means, the centre of
equiponderation of the half of it will be found to be in a strait line
parallel to the axis, dividing the base according to the numbers of
the fraction 6
⁄10, two places below in the fourth row. And the same
way serves for all the rest of the figures in the second row. In like
manner, the centre of equiponderation of the third three-sided figure
of three means will be found to be in a strait line parallel to the axis,
dividing the base, so that the part next the axis be to the other part
as 7 to 13, &c.
Coroll. The centres of equiponderation of the halves of the said
figures are known, seeing they are in the intersection of the strait
lines S T and F G, which are both known.
any of the figures, which (in the table of art. 3,
chap. XVII) are compared with their parallelograms,
being known; the centre of equiponderation of the
excess of the same figure above its triangle is also
known.
For example, let the semiparabola A B C D (in fig.
8) be taken, whose axis is A B; whose complete
figure is A B E D; and whose excess above its
triangle is B C D B. Its centre of equiponderation
may be found out in this manner. Let F G be drawn
parallel to the base, so that A F be a third part of
the axis; and let H I be drawn parallel to the axis, so
that A H be a third part of the base. This being
done, the centre of equiponderation of the triangle
A B D will be I. Again, let K L be drawn parallel to the base, so that
A K be to A B as 2 to 5; and M N parallel to the axis, so that A M be
to A D as 3 to 8; and let M N terminate in the strait line K L. The
centre, therefore, of equiponderation of the parabola A B C D is N;
and therefore we have the centres of equiponderation of the
semiparabola A B C D, and of its part the triangle A B D. That we

The centre of
equiponderat
may now find the centre of equiponderation of the remaining part B
C D B, let I N be drawn and produced to O, so that N O be triple to I
N; and O will be the centre sought for. For seeing the weight of A B
D to the weight of B C D B is in proportion reciprocal to that of the
strait line N O to the strait line I N; and N is the centre of the whole,
and I the centre of the triangle A B D; O will be the centre of the
remaining part, namely, of the figure B D C B; which was to be
found.
Coroll. The centre of equiponderation of the figure B D C B is in
the concourse of two strait lines, whereof one is parallel to the base,
and divides the axis, so that the part next the base be ⅗ or 9
⁄15 of
the whole axis; the other is parallel to the axis, and so divides the
base, that the part towards the axis be ½, or 12
⁄24 of the whole
base. For drawing O P parallel to the base, it will be as I N to N O,
so F K to K P, that is, so 1 to 3, or 5 to 15. But A F is 5
⁄15, or ⅓ of
the whole A B; and A K is 6
⁄15, or ⅖; and F K 1
⁄15; and KP 3
⁄15; and
therefore A P is 9
⁄15 of the axis A B. Also A H is ⅓, or 8
⁄24; and A M
⅜, or 9
⁄24 of the whole base; and therefore O Q being drawn parallel
to the axis, M Q, which is triple to H M, will be 3
⁄24. Wherefore A Q is
12
⁄24, or ½ of the base A D.
The excesses of the rest of the three-sided figures in the first row
of the table of art. 3, chap. XVII, have their centres of
equiponderation in two strait lines, which divide the axis and base
according to those fractions, which add 4 to the numerators of the
fractions of a parabola 9
⁄15 and 12
⁄24; and 6 to the denominators, in
this manner:—
In a parabola, the axis 9⁄15, the base 12⁄24.
In the first three-sided figure, the axis 13⁄21, the base 16⁄30.
In the second three-sided figure, the axis 17⁄27, the base 20⁄36, &c.
And by the same method, any man, if it be worth the pains, may
find out the centres of equiponderation of the excesses above their
triangles of the rest of the figures in the second and third row, &c.
14. The centre of equiponderation of the sector of
a sphere, that is, of a figure compounded of a right

ion of a solid
sector is in
the axis so
divided, that
the part next
the vertex be
to the whole
axis, wanting
half the axis
of the portion
of the sphere,
as 3 to 4.
cone, whose vertex is the centre of the sphere, and
the portion of the sphere whose base is the same
with that of the cone, divides the strait line which is
made of the axis of the cone and half the axis of the
portion together taken, so that the part next the
vertex be triple to the other part, or to the whole
strait line as 3 to 4.
For let A B C (in fig. 9) be the sector of a sphere,
whose vertex is the centre of the sphere A; whose
axis is A D; and the circle upon B C is the common
base of the portion of the sphere and of the cone
whose vertex is A; the axis of which portion is E D, and the half
thereof F D; and the axis of the cone, A E. Lastly, let A G be 3
⁄4 of
the strait line A F. I say, G is the centre of equiponderation of the
sector A B C.
Let the strait line F H be drawn of any length, making right angles
with A F at F; and drawing the strait line A H, let the triangle A F H
be made. Then upon the same centre A let any arch I K be drawn,
cutting A D in L; and its chord, cutting A D in M; and dividing M L
equally in N, let N O be drawn parallel to the strait line F H, and
meeting with the strait line A H in O.
Seeing now B D C is the spherical superficies of the portion cut off
with a plane passing through B C, and cutting the axis at right
angles; and seeing F H divides E D, the axis of the portion, into two
equal parts in F; the centre of equiponderation of the superficies B D
C will be in F (by art. 8); and for the same reason the centre of
equiponderation of the superficies I L K, K being in the strait line A
C, will be in N. And in like manner, if there were drawn, between the
centre of the sphere A and the outermost spherical superficies of the
sector, arches infinite in number, the centres of equiponderation of
the spherical superficies, in which those arches are, would be found
to be in that part of the axis, which is intercepted between the
superficies itself and a plane passing along by the chord of the arch,
and cutting the axis in the middle at right angles.
Let it now be supposed that the moment of the outermost
spherical superficies B D C is F H. Seeing therefore the superficies B

D C is to the superficies I L K in proportion duplicate to that of the
arch B D C to the arch I L K, that is, of B E to I M, that is, of F H to
N O; let it be as F H to N O, so N O to another N P; and again, as N
O to N P, so N P to another N Q; and let this be done in all the strait
lines parallel to the base F H that can possibly be drawn between the
base and the vertex of the triangle A F H. If then through all the
points Q there be drawn the crooked line A Q H, the figure A F H Q
A will be the complement of the first three-sided figure of two
means; and the same will also be the moment of all the spherical
superficies, of which the solid sector A B C D is compounded; and by
consequent, the moment of the sector itself. Let now F H be
understood to be the semidiameter of the base of a right cone,
whose side is A H, and axis A F Wherefore, seeing the bases of the
cones, which pass through F and N and the rest of the points of the
axis, are in proportion duplicate to that of the strait lines F H and N
O, &c., the moment of all the bases together, that is, of the whole
cone, will be the figure itself A F H Q A; and therefore the centre of
equiponderation of the cone A F H is the same with that of the solid
sector. Wherefore, seeing A G is ¾ of the axis A F, the centre of
equiponderation of the cone A F H is in G; and therefore the centre
of the solid sector is in G also, and divides the part A F of the axis so
that A G is triple to G F; that is, A G is to A F as 3 to 4; which was to
be demonstrated.
Note, that when the sector is a hemisphere, the axis of the cone
vanisheth into that point which is the centre of the sphere; and
therefore it addeth nothing to half the axis of the portion.
Wherefore, if in the axis of the hemisphere there be taken from the
centre ¾ of half the axis, that is, ⅜ of the semidiameter of the
sphere, there will be the centre of equiponderation of the
hemisphere.

Vol. 1. Lat. & Eng.
C.XXIII.
Fig. 1-9
Fig
1.
Fig
2.
Fig
3.
Fig
4.
Fig
5.
Fig
6.
Fig
7.
Fig
8.
Fig
9.

Definitions.
CHAPTER XXIV.
OF REFRACTION AND REFLECTION.
1. Definitions.—2. In perpendicular motion there is no refraction.—3. Things
thrown out of a thinner into a thicker medium are so refracted that the angle
refracted is greater than the angle of inclination.—4. Endeavour, which from
one point tendeth every way, will be so refracted, as that the sine of the angle
refracted will be to the sine of the angle of inclination, as the density of the
first medium is to the density of the second medium, reciprocally taken.—5.
The sine of the refracted angle in one inclination is to the sine of the refracted
angle in another inclination, as the sine of the angle of that inclination is to
the sine of the angle of this inclination.—6. If two lines of incidence, having
equal inclination, be the one in a thinner, the other in a thicker medium, the
sine of the angle of inclination will be a mean proportional between the two
sines of the refracted angles.—7. If the angle of inclination be semirect, and
the line of inclination be in the thicker medium, and the proportion of their
densities be the same with that of the diagonal to the side of a square, and
the separating superficies be plane, the refracted line will be in the separating
superficies.—8. If a body be carried in a strait line upon another body, and do
not penetrate the same, but be reflected from it, the angle of reflection will
be equal to the angle of incidence.—9. The same happens in the generation
of motion in the line of incidence.
DEFINITIONS.
I. Refraction is the breaking of that strait line, in
which a body is moved or its action would proceed
in one and the same medium, into two strait lines, by reason of the
different natures of the two mediums.
II. The former of these is called the line of incidence; the latter the
refracted line. III. The point of refraction is the common point of the
line of incidence, and of the refracted line.

In
perpendicular
motion there
is no
refraction.
IV. The refracting superficies, which also is the separating
superficies of the two mediums, is that in which is the point of
refraction.
V. The angle refracted is that, which the refracted line makes in the
point of refraction with that line, which from the same point is drawn
perpendicular to the separating superficies in a different medium.
VI. The angle of refraction is that which the refracted line makes
with the line of incidence produced.
VII. The angle of inclination is that which the line of incidence
makes with that line, which from the point of refraction is drawn
perpendicular to the separating superficies.
VIII. The angle of incidence is the complement to a right angle of
the angle of inclination.
And so, (in fig. 1) the refraction is made in A B F. The refracted
line is B F. The line of incidence is A B. The point of incidence and of
refraction is B. The refracting or separating superficies is D B E. The
line of incidence produced directly is A B C. The perpendicular to the
separating superficies is B H. The angle of refraction is C B F. The
angle refracted is H B F. The angle of inclination is A B G or H B C.
The angle of incidence is A B D.
IX. Moreover the thinner medium is understood to be that in which
there is less resistance to motion, or to the generation of motion;
and the thicker that wherein there is greater resistance.
X. And that medium in which there is equal resistance everywhere,
is a homogeneous medium. All other mediums are heterogeneous.
If a body pass, or there be generation of motion
from one medium to another of different density, in
a line perpendicular to the separating superficies,
there will be no refraction.
For seeing on every side of the perpendicular all
things in the mediums are supposed to be like and equal, if the
motion itself be supposed to be perpendicular, the inclinations also
will be equal, or rather none at all; and therefore there can be no
cause from which refraction may be inferred to be on one side of the
perpendicular, which will not conclude the same refraction to be on

Things
thrown out of
a thinner into
a thicker
medium are
so refracted
that the
angle
refracted is
greater than
the angle of
inclination.
the other side. Which being so, refraction on one side will destroy
refraction on the other side; and consequently either the refracted
line will be everywhere, which is absurd, or there will be no refracted
line at all; which was to be demonstrated.
Coroll. It is manifest from hence, that the cause of refraction
consisteth only in the obliquity of the line of incidence, whether the
incident body penetrate both the mediums, or without penetrating,
propagate motion by pressure only.
3. If a body, without any change of situation of its
internal parts, as a stone, be moved obliquely out of
the thinner medium, and proceed penetrating the
thicker medium, and the thicker medium be such, as
that its internal parts being moved restore
themselves to their former situation; the angle
refracted will be greater than the angle of
inclination.
For let D B E (in the same first figure) be the
separating superficies of two mediums; and let a
body, as a stone thrown, be understood to be
moved as is supposed in the strait line A B C; and let A B be in the
thinner medium, as in the air; and B C in the thicker, as in the water.
I say the stone, which being thrown, is moved in the line A B, will
not proceed in the line B C, but in some other line, namely, that,
with which the perpendicular B H makes the refracted angle H B F
greater than the angle of inclination H B C.
For seeing the stone coming from A, and falling upon B, makes
that which is at B proceed towards H, and that the like is done in all
the strait lines which are parallel to B H; and seeing the parts moved
restore themselves by contrary motion in the same line; there will be
contrary motion generated in H B, and in all the strait lines which are
parallel to it. Wherefore, the motion of the stone will be made by the
concourse of the motions in A G, that is, in D B, and in G B, that is,
in B H, and lastly, in H B, that is, by the concourse of three motions.
But by the concourse of the motions in A G and B H, the stone will
be carried to C; and therefore by adding the motion in H B, it will be

Endeavour,
which from
one point
tendeth every
way, will be
so refracted,
as that the
sine of the
angle
refracted will
be to the sine
of the angle
of inclination,
as the
carried higher in some other line, as in B F, and make the angle H B
F greater than the angle H B C.
And from hence may be derived the cause, why bodies which are
thrown in a very oblique line, if either they be any thing flat, or be
thrown with great force, will, when they fall upon the water, be cast
up again from the water into the air.
For let A B (in fig. 2) be the superficies of the water; into which,
from the point C, let a stone be thrown in the strait line C A, making
with the line B A produced a very little angle C A D; and producing B
A indefinitely to D, let C D be drawn perpendicular to it, and A E
parallel to C D. The stone therefore will be moved in C A by the
concourse of two motions in C D and D A, whose velocities are as
the lines themselves C D and D A. And from the motion in C D and
all its parallels downwards, as soon as the stone falls upon A, there
will be reaction upwards, because the water restores itself to its
former situation. If now the stone be thrown with sufficient obliquity,
that is, if the strait line C D be short enough, that is, if the
endeavour of the stone downwards be less than the reaction of the
water upwards, that is, less than the endeavour it hath from its own
gravity (for that may be), the stone will by reason of the excess of
the endeavour which the water hath to restore itself, above that
which the stone hath downwards, be raised again above the
superficies A B, and be carried higher, being reflected in a line which
goes higher, as the line A G.
4. If from a point, whatsoever the medium be,
endeavour be propagated every way into all the
parts of that medium; and to the same endeavour
there be obliquely opposed another medium of a
different nature, that is, either thinner or thicker;
that endeavour will be so refracted, that the sine of
the angle refracted, to the sine of the angle of
inclination, will be as the density of the first medium
to the density of the second medium, reciprocally
taken.
First, let a body be in the thinner medium in A
(fig. 3), and let it be understood to have endeavour

density of the
first medium
is to the
density of the
second
medium,
reciprocally
taken.
every way, and consequently, that its endeavour
proceed in the lines A B and A b; to which let B b
the superficies of the thicker medium be obliquely
opposed in B and b, so that A B and A b be equal;
and let the strait line B b be produced both ways.
From the points B and b, let the perpendiculars B C
and b c be drawn; and upon the centres B and b,
and at the equal distances B A and b A, let the circles A C and A c be
described, cutting B C and b c in C and c, and the same C B and c b
produced in D and d, as also A B and A b produced in E and e. Then
from the point A to the strait lines B C and b c let the perpendiculars
A F and A f be drawn. A F therefore will be the sine of the angle of
inclination of the strait line A B, and A f the sine of the angle of
inclination of the strait line A h, which two inclinations are by
construction made equal. I say, as the density of the medium in
which are B C and b c is to the density of the medium in which are B
D and b d, so is the sine of the angle refracted, to the sine of the
angle of inclination.
Let the strait line F G be drawn parallel to the strait line A B,
meeting with the strait line b B produced in G.
Seeing therefore A F and B G are also parallels, they will be equal;
and consequently, the endeavour in A F is propagated in the same
time, in which the endeavour in B G would be propagated if the
medium were of the same density. But because B G is in a thicker
medium, that is, in a medium which resists the endeavour more than
the medium in which A F is, the endeavour will be propagated less in
B G than in A F, according to the proportion which the density of the
medium, in which A F is, hath to the density of the medium in which
B G is. Let therefore the density of the medium, in which B G is, be
to the density of the medium, in which A F is, as B G is to B H; and
let the measure of the time be the radius of the circle. Let H I be
drawn parallel to B D, meeting with the circumference in I; and from
the point I let I K be drawn perpendicular to B D; which being done,
B H and I K will be equal; and I K will be to A F, as the density of the
medium in which is A F is to the density of the medium in which is I
K. Seeing therefore in the time A B, which is the radius of the circle,

the endeavour is propagated in A F in the thinner medium, it will be
propagated in the same time, that is, in the time B I in the thicker
medium from K to I. Therefore, B I is the refracted line of the line of
incidence A B; and I K is the sine of the angle refracted; and A F the
sine of the angle of inclination. Wherefore, seeing I K is to A F, as
the density of the medium in which is A F to the density of the
medium in which is I K; it will be as the density of the medium in
which is A F or B C to the density of the medium in which is I K or B
D, so the sine of the angle refracted to the sine of the angle of
inclination. And by the same reason it may be shown, that as the
density of the thinner medium is to the density of the thicker
medium, so will K I the sine of the angle refracted be to A F the sine
of the angle of inclination.
Secondly, let the body, which endeavoureth every way, be in the
thicker medium at I. If, therefore, both the mediums were of the
same density, the endeavour of the body in I B would tend directly
to L; and the sine of the angle of inclination L M would be equal to I
K or B H. But because the density of the medium, in which is I K, to
the density of the medium, in which is L M, is as B H to B G, that is,
to A F, the endeavour will be propagated further in the medium in
which L M is, than in the medium in which I K is, in the proportion of
density to density, that is, of M L to A F. Wherefore, B A being
drawn, the angle refracted will be C B A, and its sine A F. But L M is
the sine of the angle of inclination; and therefore again, as the
density of one medium is to the density of the different medium, so
reciprocally is the sine of the angle refracted to the sine of the angle
of inclination; which was to be demonstrated.
In this demonstration, I have made the separating superficies B b
plane by construction. But though it were concave or convex, the
theorem would nevertheless be true. For the refraction being made
in the point B of the plane separating superficies, if a crooked line,
as P Q, be drawn, touching the separating line in the point B; neither
the refracted line B I, nor the perpendicular B D, will be altered; and
the refracted angle K B I, as also its sine K I, will be still the same
they were.

The sine of
the refracted
angle in one
inclination is
to the sine of
the refracted
angle in
another
inclination,
as the sine of
the angle of
that
inclination is
to the sine of
the angle of
this
inclination.
If two lines
of incidence,
having equal
inclination,
be one in a
thinner the
other in a
thicker
medium, the
sine of the
angle of
inclination
will be a
mean
proportional
between the
two sines of
the refracted
angles.
5. The sine of the angle refracted in one
inclination is to the sine of the angle refracted in
another inclination, as the sine of the angle of that
inclination to the sine of the angle of this inclination.
For seeing the sine of the refracted angle is to the
sine of the angle of inclination, whatsoever that
inclination be, as the density of one medium to the
density of the other medium; the proportion of the
sine of the refracted angle, to the sine of the angle
of inclination, will be compounded of the
proportions of density to density, and of the sine of
the angle of one inclination to the sine of the angle
of the other inclination. But the proportions of the
densities in the same homogeneous body are
supposed to be the same. Wherefore refracted
angles in different inclinations are as the sines of the angles of those
inclinations; which was to be demonstrated.
6. If two lines of incidence, having equal
inclination, be the one in a thinner, the other in a
thicker medium, the sine of the angle of their
inclination will be a mean proportional between the
two sines of their angles refracted.
For let the strait line A B (in fig. 3) have its
inclination in the thinner medium, and be refracted
in the thicker medium in B I; and let E B have as
much inclination in the thicker medium, and be
refracted in the thinner medium in B S; and let R S,
the sine of the angle refracted, be drawn. I say, the
strait lines R S, A F, and I K are in continual
proportion. For it is, as the density of the thicker
medium to the density of the thinner medium, so R
S to A F. But it is also as the density of the same
thicker medium to that of the same thinner medium,
so A F to I K. Wherefore R S. A F :: A F. I K are
proportionals; that is, R S, A F, and I K are in continual proportion,
and A F is the mean proportional; which was to be proved.

If the angle
of inclination
be semirect,
and the line
of inclination
be in the
thicker
medium, and
the
proportion of
their
densities be
the same
with that of
the diagonal
to the side of
a square, and
the
separating
superficies be
plain, the
refracted line
will be in the
separating
superficies.
7. If the angle of inclination be semirect, and the
line of inclination be in the thicker medium, and the
proportion of the densities be as that of a diagonal
to the side of its square, and the separating
superficies be plain, the refracted line will be in that
separating superficies.
For in the circle A C (fig. 4) let the angle of
inclination A B C be an angle of 45 degrees. Let C B
be produced to the circumference in D; and let C E,
the sine of the angle E B C, be drawn, to which let B
F be taken equal in the separating line B G. B C E F
will therefore be a parallelogram, and F E and B C,
that is F E and B G equal. Let A G be drawn, namely
the diagonal of the square whose side is B G, and it
will be, as A G to E F so B G to B F; and so, by
supposition, the density of the medium, in which C
is, to the density of the medium in which D is; and
so also the sine of the angle refracted to the sine of
the angle of inclination. Drawing therefore F D, and
from D the line D H perpendicular to A B produced,
D H will be the sine of the angle of inclination. And
seeing the sine of the angle refracted is to the sine
of the angle of inclination, as the density of the medium, in which is
C, is to the density of the medium in which is D, that is, by
supposition, as A G is to F E, that is as B G is to D H; and seeing D H
is the sine of the angle of inclination, B G will therefore be the sine
of the angle refracted. Wherefore B G will be the refracted line, and
lye in the plain separating superficies; which was to be
demonstrated.
Coroll. It is therefore manifest, that when the inclination is greater
than 45 degrees, as also when it is less, provided the density be
greater, it may happen that the refraction will not enter the thinner
medium at all.

If a body be
carried in a
strait line
upon another
body, and do
not penetrate
it, but be
reflected
from it, the
angle of
reflection will
be equal to
the angle of
incidence.
8. If a body fall in a strait line upon another body,
and do not penetrate it, but be reflected from it, the
angle of reflection will be equal to the angle of
incidence.
Let there be a body at A (in fig. 5), which falling
with strait motion in the line A C upon another body
at C, passeth no further, but is reflected; and let the
angle of incidence be any angle, as A C D. Let the
strait line C E be drawn, making with D C produced
the angle E C F equal to the angle A C D; and let A
D be drawn perpendicular to the strait line D F. Also
in the same strait line D F let C G be taken equal to
C D; and let the perpendicular G E be raised, cutting C E in E. This
being done, the triangles A C D and E C G will be equal and like. Let
C H be drawn equal and parallel to the strait line A D; and let H C be
produced indefinitely to I. Lastly let E A be drawn, which will pass
through H, and be parallel and equal to G D. I say the motion from A
to C, in the strait line of incidence A C, will be reflected in the strait
line C E.
For the motion from A to C is made by two coefficient or
concurrent motions, the one in A H parallel to D G, the other in A D
perpendicular to the same D G; of which two motions that in A H
works nothing upon the body A after it has been moved as far as C,
because, by supposition, it doth not pass the strait line D G; whereas
the endeavour in A D, that is in H C, worketh further towards I. But
seeing it doth only press and not penetrate, there will be reaction in
H, which causeth motion from C towards H; and in the meantime
the motion in H E remains the same it was in A H; and therefore the
body will now be moved by the concourse of two motions in C H and
H E, which are equal to the two motions it had formerly in A H and H
C. Wherefore it will be carried on in C E. The angle therefore of
reflection will be E C G, equal, by construction, to the angle A C D;
which was to be demonstrated.
Now when the body is considered but as a point, it is all one
whether the superficies or line in which the reflection is made be

The same
happens in
the
generation of
motion in the
line of
incidence.
strait or crooked; for the point of incidence and reflection C is as well
in the crooked line which toucheth D G in C, as in D G itself.
9. But if we suppose that not a body be moved,
but some endeavour only be propagated from A to
C, the demonstration will nevertheless be the same.
For all endeavour is motion; and when it hath
reached the solid body in C, it presseth it, and
endeavoureth further in C I. Wherefore the reaction
will proceed in C H; and the endeavour in C H
concurring with the endeavour in H E, will generate the endeavour in
C E, in the same manner as in the repercussion of bodies moved.
If therefore endeavour be propagated from any point to the
concave superficies of a spherical body, the reflected line with the
circumference of a great circle in the same sphere will make an
angle equal to the angle of incidence.
For if endeavour be propagated from A (in fig. 6) to the
circumference in B, and the centre of the sphere be C, and the line C
B be drawn, as also the tangent D B E; and lastly if the angle F B D
be made equal to the angle A B E, the reflection will be made in the
line B F, as hath been newly shown. Wherefore the angles, which the
strait lines A B and F B make with the circumference, will also be
equal. But it is here to be noted, that if C B be produced howsoever
to G, the endeavour in the line G B C will proceed only from the
perpendicular reaction in G B; and that therefore there will be no
other endeavour in the point B towards the parts which are within
the sphere, besides that which tends towards the centre.
And here I put an end to the third part of this discourse; in which
I have considered motion and magnitude by themselves in the
abstract. The fourth and last part, concerning the phenomena of
nature, that is to say, concerning the motions and magnitudes of the
bodies which are parts of the world, real and existent, is that which
follows.

Vol. 1. Lat. & Eng.
C. XXIV.
Fig. 1-6
Fig 1. Fig 2. Fig 3. Fig 4. Fig 5. Fig 6.

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