Let me start with the first part of the excerpt — Page 55, Lesson 44

I want to walk you through how the Flight Management System handles one of its core navigation tasks: converting the distance the aircraft has travelled east or west into a change of longitude. This is a critical step in automatically updating your present position. Let me start with the first part of the excerpt. It presents four options describing how a quantity called "departure" is used to find "d'long" — that's shorthand for difference of longitude, measured in minutes. The correct relationship is that departure is multiplied by the secant of the present latitude to obtain d'long in minutes, which is then used to automatically update the present longitude. Now, what does that mean in plain terms? "Departure" is the east-west distance the aircraft has travelled, measured in nautical miles. But longitude lines converge as you move toward the poles — one minute of longitude is not the same physical distance at the equator as it is at 60° north. So to convert a distance in nautical miles into a change of longitude in minutes, you have to account for the latitude you're at. The secant of the latitude — which is 1 divided by the cosine of the latitude — does exactly that scaling. Multiply your departure by secant of latitude, and you get the change in longitude in minutes. The excerpt then moves to the second stage of integration. At this stage, the east-west speed — which the system has already integrated from acceleration — is converted into east-west distance gone. That distance is the departure we just talked about. To convert this departure into a change of longitude, the question is: what mathematical operation do we perform? The correct answer is that it has to be multiplied by the secant of the latitude — exactly the same relationship we just saw. Let me be clear on the wrong options so you understand why they don't work. Dividing by secant of latitude would actually be multiplying by cosine, which would shrink the value — that's the opposite of what we need. Dividing by tangent of latitude gives a different geometric relationship entirely, not the one that converts departure to longitude change. And multiplying by cosine of latitude would also reduce the value, which is wrong because at higher latitudes, a given departure corresponds to a larger change in longitude, not a smaller one. Now, let me place this in the broader context of the Flight Management System, or FMS. The FMS is designed to improve navigation, aid fuel efficiency, and reduce crew workload. Computers fly the aircraft along complex routes using Lateral Guidance, abbreviated LNAV. Vertical Guidance, or VNAV, enables the system to calculate optimum cruise altitudes and determine the best combination of autothrottle control and speed during climb and descent. At all times when the crew are not actually controlling the aircraft by hand, they use the FMS controls to "fly" the aircraft. The controls of an FMS are, in effect, a miniature flight deck with fingertip control. Let me walk you through the system architecture shown in the schematic layout. The key component the crew interacts with is the Control and Display Unit, or CDU. Its primary function is to act as the interface between the aircraft and the crew. The CDU can command completely automatic control of the aircraft, or semi-automatic with varying degrees of pilot involvement, including full manual control. Two CDUs are usually fitted, one on either side of the centre console. The left CDU is normally the master. In the B747-400, they are joined by a third CDU placed on the centre console for use primarily by engineering staff. Each CDU comprises a monochrome or coloured cathode ray tube — CRT — display on which different "pages" of selected data can be shown, along with a selector key panel. Looking at the schematic in Figure 21.1, you can see the full system. The CDU connects to the Flight Management Computer, or FMC. There are two FMCs, each with its own database — Database 1 and Database 2. The FMCs receive inputs from multiple sources: two Air Data Computers, labelled ADC1 and ADC2; two Inertial Reference Systems, IRS1 and IRS2, plus a third IRS3; two VOR receivers; two DME receivers; two GPS receivers; and engine and fuel systems. The FMCs also connect to the Flight Guidance System, or FGS, which provides the actual guidance commands to the autopilot and flight director. So when the system is automatically updating your present longitude, here's the chain: the Inertial Reference System measures acceleration, integrates it to get east-west speed, then integrates that speed to get east-west distance gone — that's departure. Then, as we covered, the FMC multiplies that departure by the secant of your present latitude to get the change in longitude in minutes, and updates your present position accordingly. That's the precise mathematical step that makes inertial navigation work over long distances.